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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chapter 1 Electromagnetic Model A field is a spatial distribution of a quantity（a scalar or a vector）, which may or may not be a function of time. A time-varying electric field is accompanied by a magnetic field and vice versa. （That is , time-varying electric and magnetic fields are coupled resulting in an electromagnetic field.） Under certain conditions, time-dependent electromagnetic fields produce waves that radiate from the source. 1 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （Postulating the existence of electric and magnetic fields and electromagnetic waves ） Field → wave ↑ （Time-varying field） In the transmitting unit ,when the length of the antenna is an appreciable part of the carrier wavelength a non-uniform current will flow along the open-ended antenna. 2 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU This current radiates a time-varying electromagnetic field in space, which propagates as an electromagnetic wave and induces currents in other antennas at a distance. The message is then detected in the receiving unit. screen RS A B B VS ( t ) AC RL e E Fig 1-1 Fig 1-2 3 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Force → Field → Wave （Gravity , Electricity Magnetism） （due to time-varying field） E mc2 Field：presence of energy Wave：signaling or action-at-a-distance 4 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chapter 2 Vector Analysis Vector Algebra Vector representation （1）A Vector A a AA Where A A and aA is a dimensionless unit vector A specifying the direction of A , i.e. , a A A （2）Equal vector A B a B B where AB and aA aB Even though they may be displaced in space. Fig 2-1 A B 5 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Vector Addition and Subtraction （1） A B C A head-to-tail rule C B A Fig 2-2 （Ex） A B D C D C B A Fig 2-3 6 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2） A B C key： C arrowhead points to that of A B C A Fig 2-4 7 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Vector Multiplication （1）Dot product A B AB cos AB , especially A A A A A2 A A A A B AB Key：the correlation of A and A B cos AB B Fig 2-5 8 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （Ex） Find AB C C Sol： A B C B AB B A B A B A 2 2 Fig 2-6 A B 2 A B cos AB A 2 B 2 2AB cos From definition ax ax 1 9 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note：Dot product （i.e., inner product）has two definitions N A B i i A B cos AB i 1 B is projected onto A or sum of product of their components on the same base. A B a x A1 a y A2 a z A3 a x B1 a y B2 a z B3 i.e. 3 Ai Bi a x a y 0..., etc. i 1 10 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2）Cross product A B a n A B sin AB （read “ A cross B ”） where an is a normal vector perpendicular to the plane containing A and B From a right hand, “ A cross B ”means the gingers rotate from A to B through AB B AB an A Fig 2-7 11 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note： (1)Length evaluation 12 C C C From definition az ax ay az az ax ay ay ay az ax ax Fig 2-8 where ax ay az (2)Area evaluation AB A B a A A a A B cos AB a A, B sin AB a n A B sin AB where a A a A, a n 12 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （3）Triple product (i) Scalar triple product A B C A a n BC sin BC Volume evaluation ： A B C Important identity A B C B C A C A B A C a BC n a CA n Fig 2-9 B 13 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note：The normal vectors for each cross product in this identity points to the interior of the volume. (ii) Vector triple product A B C A a n BC sin BC Note：The above vector manipulations do not involve the concepts of coordinate system. 14 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Orthogonal Coordinate System Introduction Z （1）Cartesian Coordinate ： P ( X , Y , Z) ‧A right- handed system 0 Y Base vector a x a y a z X e.g., az ax ay Fig 2-10 (i) Point P=（x , y , z） 15 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (ii) Position vector OP a x x a y y a z z x, y, z (iii) Vector A a x AX a YAY a ZAZ （x (iv) Vector field A , y , z） Scalar field r（x , y , z） 16 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (v) Vector differentials or products Vector differential line d a d ax dx ay dy az dz Z d a d d 0 Y ( x , y, z ) X Fig 2-11 17 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Vector differential surface d an dz d s a n ds dy For example ： dx Z d s a x dx a y dy d ds Y a n a x a y , and ds dxdy X Fig 2-12 i.e., the unit normal vector perpendicular to the plane containing d s 18 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Differential volume dv For example ： dv a z d z a x dx a y dy a z a n dxdydz whrer an ax ay dxdydz a z a n 1 19 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2） Cylindrical coordinates Z ‧A right-handed system P (X, Y, Z) Base vectors a r , a a z Z 0 r Y e.g., a z a r a X (i) Point P r, , z Fig 2-13 (ii) Vector A a r A r a A a z A z 20 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (iii) Vector field A r, , z Scalar field V r, , z *If A a r A r a z A z , i.e., A 0 then A is a position vector (iv) Vector differentials or products Vector differential line (or length) d a d a r dr a rd a z dz ,where r, , z ↑ Metric coefficient for expressing vectors (∵A vector consists of its length and direction.) 21 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Vector differential surface d ax dx ay dy az dz ds an ds rd For example： d s (a r dr) (a rd ) dz d Z d (a r a )(rdrd ) dr a z rdrd r Y Differential volume dv X Fig 2-14 For example： dv (a z dz)((a r dr) (a rd )) (a z a n )rdrddz( where a n a r a a z ) rdrddz( a z a n a z a z 1) 22 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note ： Vector representation in cylindrical coordinates A a r Ar a A a z A z Involves the concept of metric coefficient since a is a has vector for angle , not for length Therefore , A should contain a metric coefficient , so that A is the value of length 23 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （3）Spherical coordinates Z ‧A right-hand system P ( R , , ) θ R Base vector a R , a , a r Y X e.g. , a a R a (a) Point P ( R , , ) Fig 2-15 (b) Vector A a R A R a A a A 24 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (c) Vector field A ( R , , ) dR Scalar field V( R , , ) Z R sin d d d θ R Rdθ (d) Vector differentials or products r Y Vector differential line (or length) X d a Fig 2-16 a R dR a Rd a R sin d where ( R , , ) 25 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Vector differential surface d s a n ds For example ： d s (a R dR ) (a Rd ) a RdRd , ( a a R a ) Differential volume For example ： a dA d s dv (a R sin d ) ((a R dR ) (a Rd )) (a a )R 2 sin dRd d R 2 sin dRd d 26 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note ： (1) R：the radius of 3-dimensional sphere Y：the radius of 2-dimensional circle (2)Volume is not directional (3)The angle Φ is cylindrical coordinate require a metric coefficient r to convent “a differential angle change ” d to a differential length change” i.e., rd (4)Similar to (3), the metric coefficient corresponding to the angles θ and Φ in spherical coordinates are R and Rsinθ, respectively. 27 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Let the metric coefficients h 1, h 2, h 3 correspond to the space variables (u1, u2, u3) in a general coordinate system. Especially, u3 ˙In Cartesian coordinates (u1, u2, u3)=(x, y, z) h 3du 3 h1=1, h2=1, h3=1 h 2du 2 h1du1 u2 ˙ In Cylindrical coordinates (u1, u2, u3)=(r, Ø, z) h1=1, h2=r, h3=1 u1 Fig 2-17 ˙In Spherical coordinates (u1, u2, u3)=(R, θ, Ø) h1=1, h2=R, h3=Rsinθ 28 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (6) Dot product of A and B in Cartesian coordinates A B (a x A x a y A y a z Az ) (a x Bx a y By a z Bz ) A x Bx A y B y A z Bz A B a A A (a A B cos AB a A B sin AB ) A B cos AB ( a A a A 1, a A a A ' 0( a a a A ' )) 29 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (7) Cross product of A and B is Cartesian coordinates A B (a x A x a y A y a z Az ) (a x Bx a y By a z Bz ) a x ( Ay Bz Az B y ) a y ( Az Bx Ax By ) a z ( Ax B y Ay Bx ) aX aY aZ Ax Ay Az Bx By Bz 30 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (8) A vector in Spherical coordinates A a R AR a A a A ; a direction A length where A and A should contain metric coefficients for representing vector, since a vector consists of its length and direction. (9) Unlike the Cartesian coordinates, in cylindrical coordinates and Spherical coordinates, expressing a position vector is trivial , since a r A r a z A z and a R A R are respectively position vector. 31 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Coordinate Transforms (1) Point coordinates (a) (r,Φ, z) → (x, y, z) x= rsinΦ Z y= rsinΦ P z=z θ R (b) (R, θ, Φ) → (x, y, z) r Y ∵ r = Rsinθ X Fig 2-18 ∴ x= (Rsinθ)cosΦ y= (Rsinθ)sinΦ z= Rcosθ 32 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (2) Vector (differential) transforms Z d a x dx a ydy a z dz where l ( x, y , z ) d (a) (r,Φ, z) → (x, y, z) d 0 Y dx= cosΦdr – rsinΦdΦ X Fig 2-19 dy= sinΦdr + rcosΦdΦ dz= dz (Base on the corresponding point coordinates, the vector differentials are taken.) dx cos sin 0 dr dy sin cos 0 rd dz 0 0 1 dz 33 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Similarly, Let A a x Ax a y A y a z Az a r A r a A a z A z We have Ax cos sin 0 Ar Ay sin cos 0 A Az 0 0 1 Az Note : Since a vector consists of its length and direction we have to consider the metric coefficient for a vector differential, e.g., rdΦ. Then we can extend the transform of a vector differential to that of a vector. 34 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note : Another solution to the vector transform (r, Φ, θ) → (x, y, z) at D.K.Cheng, PP31-32 By dot product techniques, since A a r Ar a A a z Az a xAx a yA y a zAz We have A x A a x (a r A r a A a z A z ) a x (a r a x ) A r (a a x ) A az a x To see the details, refer to D.K.Cheng 35 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (b) (R, θ, Ø) → (x, y, z) From the corresponding point coordinates , we take the vector differential . dx = sinθcosΦ dR + RcosθcosΦ dθ- RsinθsinΦ dΦ dy = sinθsinΦ dR + RcosθsinΦ dθ- RsinθcosΦ dΦ dz = cosθdR – Rsinθdθ dx sin cos cos cos sin dR dy sin sin cos sin cos Rd dz cos 0 1 R sin d 36 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Similarly, let A a R A R a A a A a xAx a yAy a zAz We have Ax sin cos cos cos sin AR Ay sin sin cos sin cos A Az cos 0 1 A 37 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note : (1) In Cartesian coordinates (x, y, z), the base vector a x , a y , a z are position-invariant; i.e., the directions of these unit vectors are unchanged to represent a vector A a x Ax a y A y a z Az (2)In cylindrical coordinates (r, Φ, z) ,the base vectors a r and a are varied with position; i.e., the directions of the two unit vectors are dependent on the position on which the represented vector is located. 38 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Z A a r A r a A a z A z az A Therefore, the cylindrical coordinates can be r Y easily applied to describe a position vector X ar i.e., A a r A r a z A z Fig 2-20 if A 0 Z aZ a When Ais not a position vector, the component A a A plays an important role in representing A ar ar Z r Y ,since ar expresses only the direction of a position X ar vector projected onto the x-y plane for describing Fig 2-21 the location of the vector A 39 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU See the top view of the cylindrical coordinate to show the above point of view. ie., a r is change from a location to another. 偏心 A Z a a r a ar Rsinθ 對準圓心 r θ R a Z - Z Y r Y X A a r A r a A ^^^^ 軸向大小與方向 Fig 2-22 X Fig 2-23 40 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The expression of vector transform (r,ψ,z) → (x,y,z) is a general form for the point transform (r,ψ,z) → (x,y,z) since a point in a coordinate system can be regarded as a position vector. A x cos sin 0 A r A y sin cos 0 A A 0 1 A Z z 0 A point vector a x x a y y a zz A a xAx a yA y a zAz a rAr a zAz ( A 0) A x x cos sin 0 A r A y y sin cos 0 0 A z 0 1 Az z 0 41 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU x r cos y r sin ie., the point coordinates (r,ψ,z)→(x,y,z) zz Therefore the vector representation in (r,ψ,z) is relevant to it’s location. If a vector A is treated as a position vector, its representation is changed since a r A r is changed to express a distinct radial component. 42 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (3) All the argument in this Note(2) hold for vector representation in spherical coordinates. A point p in (r,ψ,z) → (x,y,z) ←─→ A vector A in (r,ψ,z) → (x,y,z) or (R,θ,ψ) → (x,y,z) or (R,θ,ψ) → (x,y,z) This is because A a R A R a A a A , where a R means the unit vector to indicate the position vector of the location of A 43 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU If A is a position vector, we have A a R A R ( A A 0) Z a a r A a R AR a A a A R θ a r Y X Fig 2-24 44 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU When the vector A is translated and treated as a position vector, it’s vector representations in cylindrical coordinates and in spherical coordinates are changed. However the vector representation for A in Cartesians coordinates are invariant , and the vector A always has no change in its direction and magnitude Z in any coordinate system. A a R A R a A a A A a R AR Y X Fig 2-25 45 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Physical meaning of curl of a vector field A Fig 2-26 Quiz #2. 1. Give an interpretation of the curl of a vector field B and illustrate its meaning in detail. (60%) 2. Compute the divergence of the curl of a vector field B (i.e., ( ( B) ) and show your result. (40%) 46 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Vector Calculus •Gradient of a scalar field let ( 1 , 2 , 3 ) be a scalar function of space coordinates ( 1 , 2 , 3 ) and it may be constant along certain lines or surface. Consider the space rate of change of ( 1 , 2 , 3 ) in a specified direction, e.g., the direction of d , is a directional derivative. 47 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note： Gradient: a scalar field P3 d P2 α 1 d dn P1 ( 1 , 2 , 3 ) 1 Fig 2-27 for some 1 , 2 , and 3 48 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Illustrate the meaning of gradient of Φ and grad where ax ay az in Cartesian coordinates. x y z <Ans> The directional directive d d d d dn d al al al cos al grad al d al d l d dn d d d an al d d cos a an al l dn d d d Where grad an or grad an dn dn dn 49 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU From the above, we see that d grad a l d d grad a l d grad d 1 Total derivatives in Cartesian coordinates d dx x dy dz y z a x dx a y dy a z dz 2 a x x ay az y z d 50 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Compare (1) with (2), we have grad a x ay az x y z where a x ˆ ay az x y z Note： A B C A B C 不滿足結合律 scalar scalar ∴ grad d d grad d a d a 51 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU A vector field • Divergence of A div A Fig 2-28 A(X, Y, Z) A(X, Y, Z) q A(X 0 , Y0 , Z 0 ) (X 0 , Y0 , Z0 ) q (X 0 , Y0 , Z0 ) Fig 2-29 Fig 2-30 52 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Gauss’s divergence theorem Adv A ds v Fig 2-31 Helmholtz theorem： A vector field is determined if both its divergence and its curl are specified everywhere. 53 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Illustrate the meaning of divergence of A and A a x x a y y a z z Y a x A x a y A y a z A z dz dy A x A y A z SL SF ST SB Ax x y z Ax Ax dx x SR X 0 , Y0 , Z 0 X <Ans> S TB dx Consider the special case： Z Fig 2-32 S F flux at x 0 : A x dydz A x S B flux at x 0 dx : A x dx dydz A x x 0 dx dydz x a A x dx a ds x x 0 x A x A x x 0 dx A x x 0 x x 0 dx x 54 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Outward flux over SF and SB A x A x x 0 dx dydz A x x 0 dydz x x0 dxdydz x Similarly, we have the outward flux over S T and S B A y y y0 dxdydz and outward flux over S L and S R y A z z z 0 dxdydz z Therefore, the net outward flux at point (x 0 , y0 , z0 ) A x A y A z dxdydz A dv ax a z a x A x a y A y a z A z dv x ay x y z dv y z Ad s divAdv Ad s Ads Adv v 55 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU It follows that we can define the divergence of A divA A lim ˆ Ads v0 v where A x A y Az A and A ( x 0 , y 0 , z 0 ) is continuous and differentiable. x y z d A d s 56 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Fig 2-33 point (or an object) Fig 2-34 57 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Gauss’s Divergence theorem A dv A ds v The divergence theorem is an important identity in vector analysis. It converts a volume integral of the divergence of a vector to a enclosed surface integral of the vector, and vice versa. flux : A Ads Fig 2-35 58 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • Curl of a vector field Curl： A( x, y, z) Ax Ax dy y Ax Ax Ax dy regarded as y or Ax 2 dy 2dy y Ax dy 1dy y Fig 2-36 Ax 59 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Divergence: A( x, y, z) Ax Fig 2-37 Ax dx x dz dy dx Ax 0) y Fig 2-38 60 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Az Az dz z Ay Ay dy y Ax Ax Ax dx Fig 2-39 x Ay Az Ay (Ay )dS XZ AydS XZ y ax ay az Illustrate the meaning of curl of A and A x y z Ax Ay Az 61 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ay Fig 2-40 Ay dx x Ay Y Ax <pf> Consider the special case on Ax dy (X 0 , Y0 ) y Ax the x-y plane, we see that Ax Ay Ax dy Ay dx Xo y Ay Ax curlA z a z x X az ˆ x y (X 0 , Y0 , Z0 ) a X aY aZ Z ( A) Z Similarly, on the y-z plane Ay dx Y aZ x Ax Az Ay dy curlA x a x y ax ˆ y z (X 0 , Y0 ) Ax dy Ay (X 0 , Y0 ) y and on the z-x plane dx X x - aZ Ax Az ay curlA y a y ˆ Ay Ay z x aZ ( ) ( ) Z a z x x Z Fig 2-41 62 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ax ay az curl A a x curl A x a y curl A y a z curl A z ∴ x y z Ax Ay Az curl A A ˆ Note： Ay Az 1. A a x Y z y Az Ax ay x z X Ax Ay y x az Z Fig 2-42 63 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Illustrate Stokes’s theorem Consider the line integral along path a b c d on the x-y plane. A a x A x a y A y and d a x dx a y dy Path ab：Axdx Y Ax Ax dy Ay y Path bc： Ay dx dy x d c Ax A Ay Path cd： Ax Ay dy dy dx y x y Y a A b x X Ax a x Ax dy a x dx X X dx y Ay Ay dy Path da： Aydy a y Ay a y dy x Fig 2-45 Ay Ax Ad A dS xy Sxy x y Sxy abcd 64 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Similarly, in the 3 dimensional case, we have circulation A d Ads C S ds a n ds right-hand rule az ∵intuitively for example, az ax ay ay (a x Ax) (a y Ay) Fig 2-47 ax (Ax Ay)a z C Fig 2-46 where the surface S is bounded by the contour C 65 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU We may define the curl of A curlA A lim ˆ S0 S 1 a n A d C max Where S is the area enclose by the contour C Since A is a vector point function, the value of line integral A d depends on the determination of the contour C C curl A is a measure of strength of a vortex source. 66 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Contour C Orienting the contour C in the such a way that the circulation is a maximum. S (Curl A ) z Fig 2-48 Curl( A ) (Curl A ) y (Curl A) x Fig 2-49 67 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (5) TWO NULL IDENTITIES (1) V 0 It means that gradient (of a scalar field) is curl-free or invitational. (zero net circulation) <pf> By Stokes’s Theorem V ds Vdl dV 0 S C C V 0 68 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (2) A 0 It means that gradient (of a scalar field) is curl-free or invitational. (zero net circulation) <pf> By Divergence Theorem V Adv S Ads S1 Aan1 ds S2 Aan2 ds A d A d 0 C1 C2 (∵ C1 and C 2 traverse the same path in opposite directions ) V a n1 C1 S1 C2 S 2 a n2 Fig 2-50 outward normal vector (for evaluating outward flux) 69 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note： (1) Divergence Theorem： Outward flux of a vector field i.e. V Adv A ds S (2) Stokes’s Theorem： Net circulation a vector field i.e. Ads A d S C 70 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Divergence theorem For example, v E 0 1 Q E d s Edv v dv S V 0 V 0 71 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Stokes’s theorem For example, E 0 E d E ds 0 C S 72 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chap3. Static Electric Fields Virtual displacement ( Gauss s law ) Force Field Energy ( work ) ( Coulomb s law ) ( Field intensity ) Electric displacementD Potential V ( Electric dipoles ) ( Electric flux density ) ( Scalar field ) ( P polarization vector in material media ) ( J in a good conductor ) 73 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Cheng’s approad for static electric fields： Two postulates ：（in different form） （1） E v 0 （Gauss’s law） （2） E 0 （ E is irrotational i.e. it is conservation） The postulate 1 can be derived later. 74 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Two postulate of E filed in free space 1. E v 0 F where E lim q 0 q v ：volume charge intensity of free charges. q is small enough not to disturb the distribution of source charges. 0 ：permittivity of free space. 75 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ‧ Faraday Experiment in 1837 + + - -Q + - - r=a + +Q + + + + r=b + - - + - + + ( Electric displacement ) 76 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ＊ Definition of Electric field intensity F E lim q 0 ( V m or N C ) q where the force is measured in newtons（N）and charge q in Coulombs （C） 77 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU NOTE： 1. The test charge q , of course , cannot be zero in practice； as a matter of fact , it cannot be less than the charge on e 1 an electron ’e’ , where.602 10 19 （C） 2. Also , the test charge is small enough not to disturb the charge distribution of the source 78 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Coulomb’s law q 1q 2 In 1785 , Coulomb found that F12 q 2 E12 a r 12 40 r122 1 Where the permittivity of free space 0 109 ( Fm ) 36 r12a r 12 q1 q2 r12 F12 Field point charge Test point charge Fig.3-2 NOTE： q1 F12 q1q2 1 F12 2 and E12 ar , r12 40 r12 2 12 79 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3-2 Gauss’s Law Consider electric field intensity E going thru a differential surface dS a n dS and the corresponding flux. ( Assume that there is no charge outside the enclose hypothetical surface S ) E a rE E dS a n dS ar d Fig.3-4 +q P A point charge located at P S Fig.3-3 80 Dr. Gao-Wei Chang OptoelectronicSystemsLab., Dept. of Mechatronic Tech., NTNU d E dS Ea r dSa n q d cos 40 r 2 E q d 40 A partial area of sphere dS cos where d 2 is the solid angle involvingd cos r ( steradian ) and is the angle between a n and a r , （ i.e. dScos is called the effective area ） 81 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The total flux going thru the entire enclose surface S q q q d S d Sd 4 40 40 40 d E dS E dS an Fig.3-5 82 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Consider the point charge is located at P’ outside the closed surface S. To evaluate flux resulting from a point source , we introduce the concept of solid angle. ( Assume that there is no charge inside the closed hypothetical surface S ) dS2 E2 r2 2 dS'2 dS1' E1 d 1 +q S dS1 P' Fig.3-6 83 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU d E1 dS1 E 2 dS2 ' dS1 dS'2 q a r dS1 a r dS2 2 2 1 d1 d 2 1 2 0 40 r1 r2 40 a dS a dS ( ∵ d1 r 2 1 d 2 r 2 2 d ) 1 2 r1 r2 ( That d1 d , d 2 d ) ∴ 0 0 , q is located outside S ｛ ∴ EdS S q , q is located inside S 0 ……( 1 ) 84 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU NOTE： Any charged object may be regarded as a collection of an infinite number of point charges. ar ar dS2 dS1 d Fig.3-7 85 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Steradian or solid angle R sin dS R d Fig.3-8 86 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU dS Rd 2R sin 2R 2 sin d ∴ The area of a sphere： S 0 2R sin d 2R 2 2 0 sin d 2R 2 cos 0 4R 2 ∴ Steradian of a sphere is define as S 2 4 R In general , we define steradian S R2 87 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Suppose a collection of point charges q 1 , q 2 ,…. q n are distributed inside enclosed surface S. The Eq(1) can be rewritten as 1 n SEdS q i 0 i 1 Or for a charged object with volume density . Eq(1) can be rewritten as 1 SEdS v v dv ……（2） 0 Where v is volume charge density and 0 is permittivity of free space . 88 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU In view of the Gauss’s divergence theorem , Eq(2) becomes v q SEdS v dv v Edv ( 0 ) 0 v ∴ E （point form of Gauss’s law） 0 （Usually we don’t use the differential form since the derivative does not exist at boundary points or discontinuous points.） NOTE： A Gauss’s surface is a hypothetical surface over which Gauss’s law is applied and it is needed for the integral form of the law . 89 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：Electric field intensity due to a point charged . E ERa R q R Ra R Fig.3-9 Since a point charge has no preferred direction , its electric field must be everywhere radial and has the same intensity at all points on the spherical surface .（i.e. the Gauss’s surface） Due to the fact that ( electric force lines do not intersect with each other ) 90 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （The choice of Gauss’s surface is very important to simplifying the integration in Gauss’s law） q SEdS Sa R E R a R dS 0 or E R SdS E R 4R 2 q q => ER 0 40 R 2 Therefore q E a RER a R 40 R 2 91 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU NOTE： Electric field intensity of an isolated point charge at an arbitrary location P . R R0 R0 R Fig.3-10 92 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU q E P a qP 2 40 R R 0 Where the unit vector a qP drawn from q to P RR a qP 0 R R0 R 0 is the position vector of q . and R is the position vector of field point P . Thus , we have qR R 0 EP 3 40 R R o 93 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU NOTE： qa r 1. ∵ E f q is linear 40 r 2 ∴ f a1q1 a 2 q 2 a1f q1 a 2 f q 2 a1E1 a 2 E 2 2. A single（point）charge → Continuous charge distribution （given charge distribution） （linear , planar , spherical , disk ,….） 94 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Gauss’s law v v dv SEdS 0 Gauss’s law is particularly useful in determing the E-field of charge distributions with some symmetry conditions , such that the normal component of E is constant over an enclosed hypothetical surface（called a Gauss’s surface）. 95 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：Determine E of an infinitely long straight line charge . E ERa R dS a R dS E ERa R R l Fig.3-12 L Fig.3-11 96 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： Applying Gauss’s law , q SEdS 0 Where S is a Gaussian surface L l E R 2RL l => E a R ER a R 0 20 R 97 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：An infinite planar charge az A E a zEz S S + + + + + + L E a z E z Fig.3-13 98 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： Applying Gauss’s law , we have q SEdS 0 where S is Gaussian surface S A => E Z a Z a Z A E Z a Z a Z A 0 => EZ S 2 0 ﹛ aZ S z0 20 ∴ E= S z 0 aZ 20 99 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：A parallel charged plate EZ a zEz A S S + + + + + + E i 0 ( 相互抵消 ) + + + + + + Fig.3-14 100 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： Applying Gauss’s law , we have q SEdS 0 Where S is a Gaussian surface S A => a Z E Z a Z A 0 d ∴ EZ a Z S , z > 0 2 S z< d E Z a Z , 0 2 d z d EZ 0 , 2 2 101 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：A spherical cloud of electrons with a volume charge density v 0 for 0 R b （both 0 and b are positive） E ERa R Sout - - - - - Sin - - Radius b - - - Fig.3-15 102 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： To determine E , we consider the following two cases： ( 1 ) 0R b r dr Fig.3-16 103 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Constructing a Gaussian surface Sin inside the spherical cloud , we have , from Gauss’s law , 4 R 0 3 3 Sin EdS o 4 3 0 R 2 3 => E R a R 4R a R 0 => E o Ra R E R a R 3 0 104 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU (2) R>b Constructing a Gaussian surface Sout outside the spherical cloud , it follows that from Gauss’s law , 4 3 0 b 3 E a 4R 2 a Sout EdS 0 R R R 0 b3 => E Ea R aR , R>b 3 0 R 2 105 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：A uniform charged disk of radius b that carries a surface charge intensity S z P Ep L b y r d rddr dS x Fig.3-17 106 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Although the disk has circular symmetry , we cannot visualize a hypothetical surface around it over which the normal component E has a constant magnitude ; hence Gauss’s law is not useful for the solution of this problem . To solve this problem efficiently , we introduce the concept if electrical potential. Q （pending until V is introduced） 40 R 107 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： The electrical potential at the point P(0,0,z) referring to the point at infinity is s 2 b r V 0 0 2 2 12 drd 40 z r s 2 0 z r 2 dz 2 r 2 1 b 2 1 2 40 2 s 2 0 z r 2 2 1 b 2 0 s 2 0 z r 2 z 2 2 1 , z0 ( i.e. If z >0 and z < 0 ) a z s 1 zz 2 b 2 2 1 ﹛ ,z>0 V 20 ∴ E V a z = z 20 a z s 1 zz 2 b 2 2 1 ,z<0 108 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU NOTE： For determining E , （1）it is simplest to apply Gauss’s law if a symmetrical Gaussian surface enclosing the chargse can be found over which the normal component of the field is constant . （2）it is simpler to find V (a scalar) first , and then obtain E from V , if a proper Gaussian surface is not found . 109 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ‧Principle of conservation of electric charge （1）Electric charge is conserved ; that is , it can neither be created nor be destroyed . （This is a law of nature and cannot be derived from other principles or relations） 110 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2）Electric charges can move from one place to another and can be redistributed under the influence of an electromagnetic field ; but the algebraic sum of the positive and negative charges in a closed（isolated） system remains unchanged . （This principle must be satisfied at all times and under any circum stances） （Energy stored or Work doned does not depend on the different paths with the same starting and end points） 111 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ‧Principle of conservation of energy in a static electric field . （1）Analogous to the concept of potential energy in mechanics , the electric field E is conservative or irrotational . ∴ E 0（∵ cEd l 0（By Stokes’s thm）） Fd l cqEd l 0 （2）There exists a scalar field V s.t. E V （∵ a null identity V 0 ） NOTE： Postulating the conservation of energy in a static electric field is similar to postulating that in a gravitational field . 112 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3-3 Electric potential Consider the charge in energy（of a static electric field）due to the movement of a unit positive test point charge q along the direction of a differential displacement vector d l dl + Fe dWe Fe d l + qt qf Fig.3-18 113 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Where the sign “ - ” means the sign of dW is opposite to that of（ Fe d l ） As Fe d l is positive（“+”）, the energy stored in the field is released （or decreased）and thus dWe is negative . On the other hand , as Fd l is negative（“ - ”）, the energy stored is increased（i.e. external work is needed）and thus dW is positive . 114 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU NOTE： 1. If d l is along the direction of Fe , the mechanical work Wm is positive . 2. Principle of energy conservation： Wm We const => dWm dWe 0 => dWe dWm Fe d l 3. Principle of charge conservation： +q +q -q + + - Isolated point charge Induced charges from dielectric 115 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Electric potential（cont’d） In moving a unit charge from point P1 to point P2 , in an electric field E , the external work Wm must be done against the field and the energy stored in the electric field . We P Fe d l P qEd l P2 2P 1 1 116 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Many paths may be followed in going from P1 to P2 , due to the principle of conservation of energy in electric field . Let’s define the difference of potentials at P1 and P2 We P V12 V2 V1 P Ed l 2 q 1 Usually the zero potential point is taken at infinity . The potential at P2 （or any point P）is denoted by V V2 P2 Ed l （or dV Ed l ） or E V （This implies E 0 ∴ E is conservative） 117 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：A test point charge q 0 is moved from infinity to the position vector r qa r E 4 0 r 2 q0 + r +q dl Fig.3-19 118 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： q ar ( V Ed l r r 2 dl a l ) r 40 Where l lr, , r q dr q q r 2 4 (1)r 4 r r V 1 40 0 0 ( a r a l 1 and dl=-dr ) 119 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU V “+” E Fig.3-20 ∴ The electric potential V of a point at a distance r from a point charge q referred to that at infinity. q V 40 r 120 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Electric potential due to charge distribution The electrical potential at r due to a system of n discrete charges q1,q2,、、、,qn located at r1' , r2' ,、、、, rn' is by superposition , the sum of the potentials due to the individual charges： y q1 r r1 1 n qk V ' 40 k 1 r rk r1 r r r2 distance r2 q2 x Fig.3-21 121 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：An electric dipole： It consists of equal and opposite point charges +q and –q separated by a small distance d az P R+ R R aRR R- +q a dd d d << R d cos -q 2 Fig.3-22 122 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The potential V at P： q 1 1 V ….（1） 40 R R For d R , we write d d R R cos and R R cos 2 2 123 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Form Eq（1）, we have q d cos qd cos 1 P aR V 40 40 R 2 40 R 2 2 2 d d R cos 2 1 cos 2 2R a R a d a R a d cos cos 1 a R cos 1 V 2 R where P qd is called the electric dipole moment V V qd E V a R a a R 2 cos a sin R R 40 R 3 124 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex：A linear electric quadru-pole Pr, , r1 +q + r d r2 - -2q +q + Fig.3-23 125 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The electrical potential at P（by superposition） 1 q 2q q V r 40 1 r r2 Assume d << r , using the approximation method similar to that for an electric dipole , we have 1 2qd 2 3 cos 2 1 V 3 40 r 2 1 1 V 3 E V 4 r r For electric multiples , 1 1 V and E n2 r n 1 r where n represents the number of independent displacements between any two opposite charges 126 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ＊ Electric potential due to continuous charge distribution （1）For a line charge distribution, 1 ldl' (i.e. dq) V 40 L r ' length （2）For a surface charge distribution, 1 s ds ' dq V 40 S r ' surface （3）For a volume charge distribution, 1 v dv' dq 40 v r V ' volume 127 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Material media in static electric field consists of atoms In general，we classify materials，according to their electrical properties （or energy bands of atoms），into three types： （1）Conductors conductor band valence band Fig.3-24 128 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2）semiconductors E hf Energy gap ( typically 1ev ) Fig.3-25 （3）insulators ( or dielectric ) Energy gap ( >>1ev ) Bounded charges ( no currents ) 129 Fig.3-26 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU It is found that In good conductors the conduction current J takes place as an external electric field E is applied J ee E ( Conduction current ) conductivity In terms of the band theory of solids , we find that there are allowed energy bands for electrons , each band consisting of many closely spaced discrete energy states ( between these energy bands there may be forbidden regions or gaps where non-electrons of the solid atom can reside ) 130 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （1）Conductors have an upper energy band partially filled with electrons or an upper pair of overlapping bands that are partially filled so that the electrons in those bands can move from one to another with only a small charge in energy. （2）In semiconductors , the energy gap of the forbidden regions is relatively small and small amounts of external energy may be sufficient to excite the electrons in the filled upper band to jump into the next band , causing conductor. 131 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （3）Insulators or dielectrics are materials with a completely filled upper band , so conduction could not normally occur because of the exisrence of a large energy gap to the next higer band. 132 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Polarization of a dielectric material Polarization E ( due to exertion of Columbs s force ) No E-field applied - - - - - - -q +q Electron - + - - + - cloud - + - - - - - - d Unpolarized atom Polarized atom Electron cloud Positive mucleus Fig.3-27 133 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU To analyze the macroscopic effect of induced dipoles , we define a polarization vector P electric dipole moment nv Pk P lim k 1 ( Induced bound charges appear in pair v o v i.e. electric dipoles ) Where n is the number of the induced dipoles per unit volume and the numerator represents the vector sum of the induced dipole moments contained in a very small volume v 134 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The electric potential corresponding to dP Pdv aR +q + dP a R P a R dv R dV 40 R 2 40 R 2 d 1 P aR V 40 v R 2 dv -q - Fig.3-28 Where the R is the distance from the elemental volume dv to a fixed field point 135 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The effects of the induced electric dipoles （1）Equivalent polarization surface charge density ps The bounded charge distributed over a specified surface S d cos S a n S + + + + d d' E - - - - a n d' Fig.3-29 136 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU v q b nqSd ' nqd cos S nqdS n Pk P S P a n S P lim k 1 v 0 v q b P a n c / m 2 Polarization vector => ps S dq b q ( or ps lim b ) dS S0 S （ That is , q b P a n S or dq b P a n dS PdS ） 137 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2）Equivalent polarization valume charge density pv The net charge remaining within the volume V is bounded by S （ surface charge q b , remaining charge q b ） q b Sdq b SP a n dS q r q b P dS v P dv ( by Divergence thm ) S vpv dv ( i.e. dq b P a n dS P dS ) S S ( q b P a n S ) pv P ……（1） （ 即 P pv ） （∵ The total charge of the dielectric after polarization must remain zero） 138 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Total charge S ps dS v pv dv SP a n dS vPdv 0 ( i.e SP a n dS vPdv ) 139 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Electric flux density and dielectric constant The electric field intensity due to a given source distribution volume density of free charges v in a dielectric . Dielectric + + - + - + - + - + - - + + - + - + - - + + - + - + - - + + - + - + - - Fig.3-30 140 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 1 E v pv 0 Using Eq(1) , we have 0 E P v Now we define a new fundamental field quantity , the electric flux density or electric displacement P such that D 0 E P ….（2） D v ….（3） 141 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU or vDdv vv dv q f sD dS ….（4） Eq(4) another form of Gauss’s law , states that the total outward flux of the electric displacement over any closed surface is equal to the total free charge enclosed in the surface. pv P By Gauss’s law , E v pv => 0 E P v 0 142 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU P E EP + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - E decreased to become E i E E P + - + - + - + - + - + - + - + - Ei induced electric field Dielectric Fig.3-33 143 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU v E0 E medium Fig.3-32 D unchanged（∵ including P q b d ） 144 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU When the dielectric properties of the medium are linear and isotropic we have P 0 Xe E （conductor： J E ）current density Xe 0 E insulator： D E electric displacement Where X e is a constant called electric susceptibility . From Eq(2) D 0 1 Xe E 0 r E E （for conductors , r 1 ∴ X e 0 ）（∵ P 0 ） Where r 1 X e is called relative permittivity or dielectric constant o and is the absolute permittivity .（often called simply permittivity） 145 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Boundary conditions for electrostatic fields To investigate the relations of the field quantities at an interface between two media . （1）A conductor-free space interface E1t free space ( or dielectric1 ) h W d a + + + + c + conductor ( or dielectric2 ) b Fig.3-34 146 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Inside a conductor , the field exerting a force on the charges and making them move away from one another , until all the charges reach the surface in such a way that . v 0 and E 0 Under static conditions , the E field on a conductor surface is everywhere normal to the surface , In other words , the surface of a conductor is an equipotential surface under static condition . To see this , let’s construct a small path abcd , as shown in Fig.3-34 , where the width ad cb W , and the height ab dc h sides ad and bc are parallel to the interface . 147 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Since E is conservation , we have Ed l E t W as h 0 （∵ E 0） abcd ∴ Et 0 That is the tangential component of E on a conductor surface is zero under static conditions . To evaluate E n . We construct a Gaussian surface in the form of a thin pillbox as shown in Fig.3-35 . 148 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU free space ( or dielectric1 ) S h conductor ( or dielectric2 ) Fig.3-35 Using Gauss’s law , we have S S SEdS E n S 0 S or E n 0 149 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU （2）A dielectric1-dielectric2 interface In Fig.3-34 since E is conservative , we have abcd Ed l E1 W E 2 W E1t W E 2 t W 0 ∴ E1t E 2 t , a n E1 E 2 0 150 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU This states that the tangential component of an E field is continuous across an interface . E1 t E1 n W a b 1 h 2 d c E2t W E2n Fig.3-36 151 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU To evaluate E n , we apply another form of Gauss’s law a n 2 D1n a t 2 D 2 n D1 an2 S2 a n 2 S S dielectric1 dielectric2 S1 a n1S a n1 D2 Fig.3-37 152 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU SDdS D1a n 2 D2 a n1 S a n 2 D1 D2 S S S ∴ a n 2 D1 D2 S or D 2 n D 1n S a n 2 a n 2 D1n a n 2 a n 2 D 2 n S => D 2 n D 1n S 153 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Capacitances and capacitors （Capacitances：due to the property of equipotential of a conductor） A conductor in a static electric field is an equipotential body（due to overlapping of the conduction band and valence band of its atom）and that charges deposited on a conductor will distribute themselves on its surface in such a way the electric field inside vanishes . 154 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The potential of the surface of an isolated conductor is directly proportional to the total charge on it , sine （1) E V （increasing the potential V by a factor of K increases E by K） （2）The boundary condition at a conductor-free space interface S E an 0 （as a result , S（or the total charge Q）increases by k when E increases by a factor of k） 155 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Therefore , the radio Q remains uncharged . For the isolated V conducting body , we define capacitance Q C V Of considerable importance in practice is the capacitor（or condenser）, which consists of two conductors separated by free space or dielectric media . The capacitance of a capacitor is a physical property of the two conductor system . It depends on the capacitor and on the permittivity of the medium . 156 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex1：Determine the capacitance of a parallel-plate capacitor y + + + …………… + S y1 d area A E dl y0 0 - - - …………… - Fig.3-38 157 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： ∵ V10 y Ed l y1 0 d 0 E dya y S 0 a y dya y d 0 d S y S 0 0 0 Q Q Q A ∴ C V V10 Q d d A 158 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ex2：Determine the capacitance of a cylindrical capacitor l a E b Gaussian surface S Fig.3-39 159 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Sol： By Gauss’s law , we have （neglecting the fringing effect of the field near the edges of the conductors） E dS S 2b l a r E r a r dS a r E r a r 2rl S 2b l S b ∴ E ar r b S b Vba a Ed l a a r a r dr b r r b 1 b S ln r S b ln a a 160 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chap. 4 Steady Electric Currents Electrostatic problems. Field problems associated with electric charges at rest. Charges in motion that constitute current flow . (Problems of current flow in a conductive medium are governed by Ohm’s law.) Question: How about the problems of current flow in a good insulator? 161 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Two types of electric current: (Caused by the motion of electric charges) (1) Convection current (2) Conduction current Convection current： The result of hydrodynamic motion involving a mass transport, are not governed by Ohm’s law）: (1) Electron beams in a cathode-ray tube (2) Violent motions of charged particles in a thunderstorm.. 162 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Conduction current： As an external electric field is applied on a conductor, an organized motion of conduction (or valence) electrons, which may wander from one atom to another in a random manner is produced. The conduction electrons collide with the atoms in the course of their motion, dissipating part of their kinetic energy as heat：thermal radiation. This phenomenon manifests itself as a damping force or resistance, to current flow. 163 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The redation mbetween conduction current density J（and electric field intensity E mgive us a point form of Ohm’s law), i.e., J E Where is a macroscopic constitutive parameter of the medium called conductivity . ( In a dielectric (or an insulator) the electric displacementis D given by D E Where is called permittivity.) Recall that A good condutor A good insulator Conduction band J Energy Valence gap >> band 1ev E Fig.4 3 164 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The same I J x, y but different lead todifferent magnetic field B effects, i.e. different Charge carrier Fig.4 4 165 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Convection current: Point charge(s) moving with a velocity v in free space. Free space qv Fig.4 5 Conduction current Metallic conductor is filled with free electrons. Under the influence of E conduction electrons collide with atoms and consequently conduction current produces. Fig.4 6 166 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Current density and Ohm’s law 1.Convection current: Consider the steady motion of one kind of charge carriers, each of charge q, across an elem of surface ΔS with a velocity through the surface s The amount of charge passing S an s Where N is the number of charge carriers per unit volume and the vector quantity u s = a n s Q From Eq(1), we have I v u s t Fig.4 7 167 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU In field theory, we are usually interested in events occurring at a point rather than within some large region, and we shall introduce the concept of current density J v 2 (where v Nq , is free charge per unit volume. ) I J s so that 168 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ue e E (m / s ) 5 E ue ue E Where e is the electron mobility measured in ( m 2 / v s ) ( This is because conduction currents are the result of the drift motion of charge carriers under the influence of applied electric field intensity.) Table the electron motilities for some conductors Conductor Copper Aluminum Silver e 3.2 1.4 5.2 10 3 10 4 10 3 169 Dr. Gao-Wei Chang (Unit in m / v s ) 2 Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 6 5.2 4 3.2 2 electron mobility 0.4 0 copper aluminum silver Tab.4 1 electron mobility 3.2 0.4 5.2 Each in 10 3 m 2 / v s From Eqs (4) and (5) , we have J e e E E (Point form of Ohm’s law) Where the negative quantity e N e is the volume charge density of the drifting electrons and the conductivity e e (A/vm or Siemens per meter(s/m)) 170 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU For semiconductors, conductivity depends on the concentration and mobility of both electrons and holes： i i e e n n i mobility concentration where the subscript h denotes hole. e Ne Nq 171 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Equation of continuityand kirchhoff’s current law Conservation of charge： Electric charges may not be created or destroyed (just transferred from one place to another) 172 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Continuity of current： If a net current I flows across an enclosing surface（封閉面） out of the bounded volume V, the net charge Q in V must decrease at a rate that equals current dQ d I J ds i v dv S dt dt V Qo By Divergence theorem, we have Qi v V J dv V t dv For arbitrary choice of V, it follows that J v 1 t Fig.4 8 173 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Qi Qo 0 Qo Qi dQo dQ That is , I o dt dt Outward current Qi Qo dQ I Qi Qo Q Q 0 dt Fig.4 9 This point relationship derived from the principle of conservation of charge is called the equation of continuity (of current). For steady electric currents charge density does not vary v with time, 0 v is fixed.）Equation (1) becomes J 0 （ t Thus, steady currents are divergenceless or solenoidal. 174 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Over any enclosed surface, we have, by Divergence theorem, J ds 0 S Which can be written as Ij j 0 2 Equation (2) is an expression of kerchief’s current law. <Ex>consider that charges introduced（引進） in the interior（內部） of a conductor will move to the conductor surface and redistribute（重新分佈） themselves in such a way as to make v 0 and E 0 inside under equilibrium（平衡，均勢） conditions. Please calculate the time it tables to reach（取得，抓到） an equilibrium. 175 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU <Pf> : From Ohm’s law, the equation of continuity becomes v J E t Where is the conductivity of the conductor In a simple medium, Gauss law From the above eqs, we have E v 176 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU v 0 t J E v v t It can be readily obtained that v 0 e ( / )t (c/m3). Where 0 is the initial charge density at t=0. For a good conductor such as copper, 5.8 10 ( s / m) 7 ,ε≒ 0（like vacuum no electric dipole） 8.85 10 12 ( F / m) constant / 1.53 10 sec ,a very short time indeed. 19 177 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Power dissipation and joule’s law Under the influence of an electric field E , conduction electrons conductor under go a drift motion macroscopically, and they collide with atoms on lattice sites. (Energy is thus transmitted from the electric field to the atoms in thermal vibration.) The work ΔW done by E in moving a charge q a distance is w Fe qE W dl Which corresponds to a power P lim qE q E u t 0 t dt F l F l dl W lim qE t t 0 t dt 178 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU ＳＷ 1 0 LC Ｃ Ｌ Energy storage in E field Energy storage inB field Ｒ Power dissipation Fig.4 10 The total power delivered to all the charge carriers in a volume is dv 179 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU dP dP E( Ni qiui )dv E J dv i i i P and E J (w) v Is thus the point form of a power density under steady-current conditions. For a given volume V, the total electric power converted into heat is P E J dv Which is known as Joule’s law. V 180 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU <Ex> Show that in a conductor of a constant cross section , we have P=I2R(w) <Pf>： Where d is measured in the direction of dv ds d ds d P E J dv Edl Jds V I V L S Fig.4 11 Since V=RI , we have P=I2R(w) 181 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Governing equation for steady current density. Basic quantity： Current density vector 182 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Governing eqs Jds 0 (1) J 0 or J 2n a2n J 2n S means J is divergence less. J 1n a1n J 1n At an interface between two different conductors, Fig.4 12 J 1n J 2 n ( A / m 2 ) 2 (2) ( J / ) 0 (∵ E 0 and J E ) J1t J 2t 1 1 or C Jdl 0 Fig.4 13 1 2 at on interface between two different conductors J 1t J 2t J 1t 1 1 2 or J 2t 2 183 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Resistance calculations V LEdl LEdl R= I Jds Eds S S <Ex> Derive the voltage-current relationship (i.e. resistance) of a piece of homogenous material of conductivity σ, length l, and uniform cross section S, as shown below. 184 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU <Sol>： The potential difference or voltage between terminals 1 and 2 is － 0 V12 E V12 V 12 Ed ＋ J a Where E a E and a E Ｓ ０ 2 The total current is １ I Jds J S E S S V12 ∴ R () I S Fig.4 14 185 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chap 5 Static Magnetic Fields in Free Space 186 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU About Magnetic Field Field and Wave Electromagnetics due to Force exertion Field existance ( ) Maxwell's Equations (action - at - a - distance) due to Energy t ransfer Wave motion ( ) Poy ting's thm (or Energy flow) 187 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space ＊ A magnetic field can be cause by （1）a permanent magnet（like the magnetized lodestone） （2）moving charges （3）a current flow 188 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space I I (a) (b) 189 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space A magnetic field can be characterized by a so-called magnetic flux density B ,which is defined in terms of Fm experienced by a moving charge q ,i.e., Fm qu B (N) 190 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 5-2 Static Magnetic Field in the Free Space where u (m/s) is the velocity of the moving charge and B 2 is measured in webers per square meter (Wb/ m ) or teslas (T). teslas = 104 Gauss （Here , B has been not yet defined） 191 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space ＊ Lorentz’s force equation When a test charge q is placed in an electric field E and it is also in motion in a magnetic B the total electromagnetic force on it is F Fe Fm q (E u B) ……(1) which is called Lorentz’s force equation 192 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space Magnetic force is a kind of transverse force analogous to the electric force （or Coulomb’s force） 193 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space Note： （1） B and u are perpendicular to Fm This phenomenon is found by Oersted. Magnetic force Fm is a transverse force found by Oersted. Specifically Fm is perpendicular to both B and u 194 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Static Magnetic Field in the Free Space （2） 195 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Charges ＊ Law of magnetic force between two moving point charges （Popovic ,Introductory Engineering Electromagnetics,1971） The magnetic force Fm12 exerted by a charge q 1 on the other charge q 2 is found by indirect experiments, involving steady current system, to be (q2u2 ) (q1u1 ar12 ) Fm12 km 2 ……(2) r12 km (q1u1 ar12 ) B r2 12 196 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Charges (the permeability of a vacuum) in MKSA system, and r12 a r12 r12 is the vector length from q 1 to q 2 （Equation(1) may be compared with Coulomb’s law in electrostatic fields.） r12 a r12 r12 B q1 q2 u1 u2 u1 B F Fm12 u 2 m12 Test moving Field moving charge q1 charge q 2 197 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Charges ＊ The concept of magnetic field magnetic flux density B from Eqs(1) and (2) we can see that 0 q1u1 ar12 Fm12 q2u2 4 2 q2 u 2 B r12 0 q1u1 ar12 where B 4 2 r12 is the definition of the flux density vector produced by a point charge q 1 moving with a velocity u1 q 1 u 1：field moving charge q 2 u 2：test moving charge 198 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Charges 0I 0 Id sin s 0 I sin d s B B 2 dB 2 0 r 2 dB 2a 4 r2 0 sin sin a r s2 a 2 , s2 a 2 0I ad s B 2 dB 0 2 0 s 2 a 2 3 2 0I s I 0 2a (s 2 a 2 ) 12 2a 0 199 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Charges Note： If a test charge q t resides at a fixed location outside the current-carrying conductor, it will be acted on neither by the electric force nor by the magnetic force. （1）The moving charges inside the conductor are compensated so that there is no appreciable electric field outside (nor in side) the conductor. （2）Since charge q t is stationary according to the law of magnetic force between two moving point charges the magnetic force on q t is also zero. （In metallic conductors, charge carriers are called conduction electrons. In a good insulator (or dielectric), induced charges are called bound charges.） 200 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Loops ＊Ampere’s law of force Consider two idealized complete circuits C1 C 1 and C 2 , consisting of two very thin I1 dl1 C2 R r2 r1 dl2 conducting loops(wires) carrying filamentary I2 r1 r2 currents I1 and I 2 respectively. O In Ampere’s extensive experiments, he found that in free space, ( I 2 d 2 ) ( I1d 1 aR ) c2 c1 F12 0 4 R2 F12 ( I 2 ) is linear where R a R R 201 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Loops This equation is referred to Ampere’s law of force and it constitutes the foundation of magnetostatics. （Usually, the magnetic force due to two moving charge acting on them is very relatively small. For example, in a conductor, charges forming steady current are moved by both electric force and magnetic force. However, the magnetic force is much small than the electric force. As a result, the average drift velocity v is governed by the electric field intensity E , i.e.,） J E (i.e., J E ) v E , where is the mobility of the charges. (see popovic ) 202 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Loops Suppose n charges q 1、 2 , …….., q moving with velocities q n u 1 , u 2 …….u n 。 According to the superposition principle, the total magnetic force Fm on a test charge q t moving with a velocity u t 0 n qi ui arit Fm qt ut 4 ……(3) i 1 rit2 qt ut B 203 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Force between Two Loops Where a rit are with vectors directed ( at the time instant considered ) from charges q 1、 2 , …….., q n toward the test charge q qt 0 n q i u i a rit B 4 i 1 r 2 ……(4) it and rit a rit rit 204 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law ＊The magnetic field of steady electric current： The Biot-Savart law (for conduction current) Suppose that the number of free charges per unit volume of a conductor is N。 Then，inside a small volume v ，there are Nv charges moving with the same velocity v ，since v is supposed to by very small。 Equation (4) becomes 0 Nq u ar B 4 r 2 V V ……(5) J a 0 2 r V 4 V r 205 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law where J Nq u V u represents the current density vector at point inside the volume element v and V is volume charge density。 If V is assumed to be “ physically small ” the magnetic flux density due to the steady current in the conductor is given by 0 J ar B 4 V r 2 dV ……(6) 206 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law where is the unit vector directed from the volume element dv r towards the fixed point at which B is being determined. v' V J ar r ar r qt Position of a testing point a small charge q t in the field of charges moving inside a conductor. 207 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law Filamentary current I ＊ Biot-Savart law (cont’d) r ar r J 0 J ar B Field point dV ……(6) 4 V r 2 In practice ，the current is very often flowing thru thin conducting wires. Suppose the cross-section area of the wire is S and then V S d where d is the vector differential element of the wire Equation (6) yields 0 I d ar B 4 C r 2 208 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law where the current element I d J a n S d J S d J dV S d r ar r and S a n S qtut Conductor current I( conduction current in a flamentary conductor ) (or thin wrie ) This important formula is known as the Biot-Savart law. 209 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law Note ： 0 J ar B 4 V ' r 2 dV 0 J ar S d 4 C r2 0 J S d ar 4 C r2 0 I d ar 4 C r 2 0 I d ar r 4 C r3 210 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law 【EX】 Find the magnetic flux density B at a point located at a distance r from the current-carrying straight wire ，as shown below 。 <sol> The distance vector from the source element dz' to the field point p is R ar r ( a z z ' ) d R az dz ar r az z a rdz 211 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law By applying Biot-Savart law ，we have 0 I z L rdz ' B a 4 z' r L 3 2 2 2 L 0 I L Id l az dz ' a source element 2r L2 r 2 z' p I d aR （∵ dB 0 a z dz ' r ar r 4 R2 Field point I d aR 0 R3 4 -L 1 R Z ' R 2 2 2 ） 212 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law 【EX】 Find the magnetic flux density at center of a planar square loop , with side W carrying a direct current I. z <sol> From the preceding example, we have O I I W/2 2 2 0 I W B 4a z 0 az 2 W W 2 W 2 W 2 2 2 B的方向和迴路中電流的方向遵循右手規則 (homework or exercise) 213 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law 【EX】 Find the magnetic flux density at a point on the axis of a circular loop of radius b that carries a direct current I. P(0,0,z) R <sol> a zz d b I y We apply Biot-Savart law to the circular loop x d abd R azz arb R z 2 b2 1/ 2 214 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Biot-Savart Law Recall that R is the vector from the source element d to the field point P. d R a bd a z z a r b a r bzd a z b 2 d (differential length vector distance vector in free space) We need only consider the a z component of this cross product since the a r component is canceled due to cylindrical symmetry. 0 I 2 b 2 d 0 Ib2 B az az Teslas 4 0 z 2 b 2 3/ 2 2z b 2 2 3/ 2 215 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The Curl and Divergence of B ＊ The curl and divergence of B 0 J a r Recall B 4 v r 2 dv whereB B(r) J J (r) and r(x,y,z) is the position vector of the field point p ( x, y, z) V J r ar r P(x,y,z) 216 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The Curl and Divergence of B ar 1 Since r2 r 0 1 We have B r 4 v J dv ……(1) From the vector identity J 1 1 J J ……(2) r r r it follows that 0 J B r 4 v dv 217 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The Curl and Divergence of B In space change rate J B ( 0 4 V' r dv') ……(3) Let’s define the vector magnetic potential 0 J A 4 V' r dv' ……(4) Where the source coordinates are primed. Therefore B ( A) 0 ∴ B 0 for determining B 218 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The Curl and Divergence of B Furthermore, suppose the stationary current I in a thin wire It appears that the vector magnetic potential 0 J A 4 V' r dv' I d 0 4 L ' r According to Helmholtz’s theorem, a vector field is determined if both its divergence and its curl are specified everywhere Therefore, we need to further evaluate B for determining B 219 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The Curl and Divergence of B Note： Since it is found that J B ( 0 4 V ' r dv') We thus define the rector magnetic potential 0 J A 4 V ' r dv' Where the source coordinates are primed. 220 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU The Curl and Divergence of B Therefore B ( A ) 0 (due to the null identity) or B 0 (any where) Source free Thad is, B is rotational. B 0 That is, B is rotational. B 0 dS ∴ B 0 anywhere 221 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Ampere’s circuital law A A A 2 2 2 2 ax ay az 2 2 2 2 x y z x y z 2V 2V 2V V 2 2 2 2 x y z 2 2 2 2V a x a y a z a x a y a z V 2 2 2 V x y z x y z x y z 222 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Consider B A A A … (1) 2 from vector identities , where 2 2 2 A a x Ax a y Ay a z Az 2 and A ax Ax a y Ay az Az 1. The first term on the right hand side of Eq . (1) 0 Jdv' A 0 J dv' v' r … (2) 4 4 v ' r 223 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law From vector identities , J 1 1 J J … (3a) r r r or J ' 1 J r … (3b) r ( r a x x x ' a y y y ' a z z z ' , r x x ' y y ' z z ' 2 2 2 1 2 and ' is the differentiation with respect to the source coordinate x' , y ' , z.' ) 224 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law From vector identities , J J r ' 1 1 ' ' J J r … (4) r r ( for any source element the field point is specified ) 225 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Eqs . (3b) and (4) yields J ' J 1 ' J r … (5) r r (Since J 0 is a necessary condition for static ' magnetic fields (i.e. , steady current) . ) Eqs . (2) and (5) yields 0 ' J A dv' r … (6) 4 v ' 226 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law From Divergence theorem , it follows that 0 J A ds ' … (7) 4 s ' r where S ' encloses the volume V ' . Since J is the volume current density ( i.e. ,all currents are enclosed inside S ' ) its normal component is always equal to zero . J ds ' 0 227 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Equation (7) becomes A 0 …(8) This is a general form for Coulomb condition . A 0 2. For the second term on the right side of Eq . (1) , 0 2 J 0 2 1 A v' r dv' 4 v' J r dv' 2 … (9) 4 228 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law If the field point P is not located inside the volume V ' 1 ( i.e. , r x x' y y ' z z ') , 0 2 2 2 2 we have 1 2 0 r 1 2 1 ( 2 1 r 0 for r 0) r 2 r r r r 1 For 2 0 , the field point must be located inside r the volume V ' and it is infinitely close to a source element . 229 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law r 0 , if r is located inside the volume V ' i.e. , the field point is infinitely close to a source point ( i.e. , , , ). x x' y y' z z' Therefore , Eq . (9) becomes 0 2 1 '2 1 A v ' J dv' 0 v '0 J r dv' 2 4 r 4 0 J 1 J ' 1 4 v '0 s '0 r ds ' 2 A ' ' dv' 0 r 4 ( ∵ Jis constant , as V ' 0) ( By Divergence theorem , and r' r , but r ' r ) 230 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law we have 0 J ar ' ds ' 4 s '0 2 A ( ar ' ar ) r '2 0 J 4 s '0 d ' ( d ' ds' ) r '2 0 J … (10) 231 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law From Eqs . (1) and (10) , We have B 0 J … (11) which is called Ampere’s circuital law . ( or simply called Ampere’s law . ) 232 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Also , by Stokes’s theorem , Bds Bdl 0 Jds ' 0 I s c s … (12) where the surface S is enclosed by the contour C . In addition , from Eq . (10) , A 0 J 2 is called a vector Poisson’s equation . 233 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law In Cartesian coordinates , 2 A ax Ax a y 2 Ay az 2 Az 2 where 2 Ax 0 J x … (11a) 2 Ay 0 J y … (11b) and 2 Az 0 J z … (11c) 234 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Hence , the solution for Eq . (11a) is 0 J Ax x dv' 4 v ' r ∴Consistently , 0 J A dv' 4 v' r 235 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law Note : In a static electric field v E 0 ( Point form only valids v for the point having ) 0 236 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Ampere’s Circuital Law In a static magnetic field B 0 J ( Point form only valids for the point having 0 J ) 237 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chap 6 Magnetic Dipole & Behavior of Magnetic Materials 238 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole ＊ Magnetic Dipole Let’s evaluate the magnetic flux density at a distance point of a small circular loop of radius b that carries a current I ( a magnetic dipole ) . 239 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole We choose a spherical coordinate system s.t. the field point p( R, , ) is located in the yz plane for convenience 2 z p ( R , , ) 2 R Φ=π/2 R1 y p' ' ' p' dl' x 240 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole It is intended to find the vector magnetic 0 I dl 1 l dl ' R 4 0 R A analogous to 4 L and then B A is determined analogous to E V 0 I dl aR1 (∵ B 4 L' R12 ) 241 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole From the top view of the small loop dl ' (-ax sin 'ay cos ' )dl' or dl ' (-ax sin 'ay cos ' )bd ' dl ' ax ' b p ay dl ' ' ' ax 242 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole I 2 b sin ' d ' A a x 0 4 0 R1 0 Ib 2 sin ' d ' 2 2 or A a R1 ( the a y component is canceled due to the source element I dl ' is symmetric to the y axis ) The law of cosine gives R12 R 2 b 2 2bR cos ' or R12 R 2 b 2 2bR sin sin ' ( left as an exercise ) 243 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole It follow that 1 1 1 b 2 2b 2 1 2 sin sin ' R1 R R R 1 1 1 2b 2 1 b or 1 sin sin ' 1 sin sin ' R1 R R R R b2 ( assuming R 2 b 2 i.e. 2 1 ) R 0 Ib 2 b 2R 2 ∴ A a (1 sin sin ' ) sin ' d ' R 244 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole 0 Ib 2 A a 2 sin 4R as a result 0 Ib 2 B A 3 (a R 2 cos a sin ) 4R 245 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole Note：The vector magnetic potential can be rearranged as 0 m aR A 4R 2 法線方向是 az 方向 where m a z Ib a z IS a z m 2 is defined as the magnetic dipole moment 246 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole also ,We can rewrite the magnetic flux density vector as 0m B a R 2 cos a sin 4R 3 Comparing with the similar expressions for the electric dipole in static fields , we have P aR V and 4 0 R 2 E V P a R 2 cos a sin 4 0 R 3 where p qd qda z p a z is the electric dipole and the magnetic dipole are also similar. 247 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Dipole + X - Magnetic dipole Electric dipole 248 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density ＊ Magnetization and Equivalent Current Density J mv , J ms Suppose the orbiting electrons in a material cause circulating currents and form microscopic magnetic dipoles . The application of an external magnetic field causes both an induced magnetic moment due to a change in the orbital motion of electrons and an alignment of the magnetic dipole Nv moments of the spinning electrons mk M lim k 1 Let’s define a magnetization vector v '0 v' 249 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density where m k is the magnetic dipole moment of an atom and N stands for the number of atoms per unit volume . 250 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density since dm Mdv' , we have 0 M aR dA dv 4 R 2 0 1 M ' ( )dv' 4 R 1 1 (∵ ' ( ) 2 aR ) R R 0 1 v' ∴ A dA 4 v' M ' dv ' R ( where v' is the volume of the magnetized material ) M ' 0 ' M 0 ' 4 v ' dv ' ( )dv ' 4 v R R 251 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density ( From the vector identities , ' M 1 ' M ' 1 M R R R 1 ' 1 M M ' R R J mv J ms 0 ' M 0 M an' 4 v ' R dv' 4 S ' R ds' 252 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density Define the equivalent magnetization volume current density vector ' J mv M ( analogous to pv P ) and the magnetization surface current density J ms M a n ' ( analogous to ps P an ) ( For notational simplicity, we omit the primes ) Consider Bi J mv M 0 253 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density ( compared to Ampere’s law B 0 J in free space ) where the internal flux density B is produced by M In addition , we see that due to the free current density J Be J 0 where B e denotes the external magnetic flux density 254 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density Thus , the resultant magnetic flux density in the presence of a magnetized is changed by an amount B i ；i.e. B Be Bi 0 J J mv Note：The application of an external magnetic field causes both (1) an induced magnetic moment in a magnetic material (2) an alignment of the internal dipole moment and 255 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetization and Equivalent Current Density Fig . The induced magnetic dipole moment are partially aligned along dl by an Externally applied magnetic field magnetic dipole moment m Ids dl 256 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability ＊ Magnetic Field Intensity and Relative Permeability since the magnetic flux density B in the magnetic material 1 can be express by B J J mv J M 0 257 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability We define the magnetic field intensity H B H M 0 A Thus , H J ( m2 ) ( another form of Ampere’s law ) where J is the volume density of free current 258 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability According to Stokes’s theorem , we have S ( H )ds J ds S Hdl I C where C is the contour bounding the surface S and I is the total free current passing thru S . 259 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability H ds H C S The above formula holds in a nonmagnetic as well as a magnetic medium When the magnetic properties of the medium are linear and isotropic, the magnetization vector M xm H 260 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability where x m is a dimensionless quantity called magnetic susceptibility Therefore , B 0 ( H M ) 0 (1 x m ) H 0 r H H where the dimensionless quantity r 1 x m 0 is called the relative permeability of he medium and ( H/m ) is known as the ( absolute ) permeability. 261 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability Note： xm : magnetic susceptibility x e : electric susceptibility r : relative permittivity r : relative permeability r 1 xe r 1 xm 0 r 0 r 262 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability Note： 1. Electric dipole vs Magnetic dipole S + d X - I Electric moment P qd Magnetic moment m IS 263 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability Electric charge vs X (no magnetic charge) P 0 xm E + - + - + - + - + - + - + - + - Ed Ei E0 Ed 264 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability 2. The magnetic field intensity H is introduced as the basic quantity of the fields, the generalized Ampere’s law H J holds across any media 265 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability 3. Recall that the potential due to the polarized dielectric 1 P aR V 4 0 V ' R 2 dv' where R aR R is the distance vector from dv' to a fixed field point. 1 aR Since ' 2 R R 266 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability 1 1 4 0 V ' it follows that V P ' dv' R 1 P 1 V ' ' dv' V ' 'Pdv' R 4 0 R ( By the vector identity ' f A f ' A A ' f ) ps pv 1 P a 'n 1 'P dv' 4 0 S ' R V ' R ds' 4 0 267 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability where a' n is the outward normal from the surface element ds ' a ' n ds' of the dielectric + - S’ + - + - V’ Dielectric 268 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability Therefore , the electric potential due to a polarized dielectric 1 pv 1 ps V 4 0 V' R dv' 4 0 S' R ds' where pv 'P ( polarized volume charge density ) and ps P an ( polarized surface charge density ) 1 ∵ E v pv in the dielectric 0 0 E v P v 0 E P D 269 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability where D 0 E P is called electric displacement vector . since P 0 xe E , we have D 0 E 0 x e E 0 1 x e E 0 r E E 270 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Field Intensity and Relative Permeability 4. Polarization & Magnetization Polarization q p P Thru V pv , ps Magnetization I m M Thru A J ms , J mv 271 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials Magnetic Materials can be roughly classified into three main groups in accordance with their r values. (1)Diamagnetism if r 1 ( m 0 and m 0 ) (The word “dia” in Greek mean “across”) (2)Para magnetism, if r 1 ( m 0 and m 0 ) (The word “Para” in Greek mean “along”) (3)Ferro magnetism, if r 1 ( m 1 ) 272 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials nucleus + az B electron in orbital ds motion I B is produced due to current loop I electron in spinning motion (or magnetic dipole moment m Id s ) 273 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials m + - I Ue （a） （b） ms （c） 274 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials (1) Diamagnetic materials: As B appl 0, ( ie, no external magnetic field ) morb mspin 0 ( for an atom ) as is applied, B int B appl and B int B appl B appl This is because the induced magnetic moment always apposes the applied field according to Len’s law of electromagnetic induction. As a result, the magnetic flux density is reduced. The effect is equivalent to that of a negative magnetization ( ie, m 0 ) and it is usually very small. For diamagnetic materials,copper,lead,……etc. m is of the order of -10. 5 m 275 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials (2) Para magnetism materials: As B appl 0, morb mspin is small as B appl is applied, morb mspin is aligned in the direction of the applied field. s.t. B int B appl and B int B appl However, the alignment process is impeded by the forces of random thermal vibrations; as a result the paramagnetic effect is temperature dependent in contrast to that of diamagnetic materials.For paramagnetic materials, e.g, aluminum, tungsten,…..etc. m is usually of the order of -10-5. 5 276 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials (3) Ferro magnetic materials: As B appl 0, m spin m orb As B appl is applied B int B appl Due to the postulate of magnetized domains proposed by Weiss in 1907 (Called Weiss’ domains) 277 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials (a) (b) Fig. schematic of an unmagnified (a) paramagnetic and (b) ferromagnetic material. The arrows qualitatively show atom magnetic moments. 278 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 15 16 These domains, each containing about 10 or 10 atoms and usually having the linear dimension of about 105 m, are fully magnetized in the sense that they contain aligned magnetic dipoles resulting from spinning electrons even in the absence of an applied magnetic field. There are strong coupling forces between the magnetic dipole moments of the atoms in a domain, holding the dipole moments in parallel. Between adjacent domains there is a transition region about 100 atoms thick called a domain wall. 279 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials ( a) (b) (c) Applied magnetic field Fig. (a) unmagnetized (b) magnetic-domains translated (c) magnetic-domains rotated ferromagnetic materials Above a certain temperature, called the curie temperature, the thermal vibrations completely prevent the parallel alignment of molecule magnetic moments, and ferromagnetic materials become paramagnetic. This temperature is 770 C for iron 280 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials H H ( a) (b) (c) 281 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials B b Br c d a g 0 H f e Hysteresis loop in B-H plane for ferromagnetic materials 282 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials (4)Anti ferromagnetic materials As B appl 0, m spin m orb As B appl is applied B int B appl (5)Ferromagnetic materials As Bappl 0, m spin m orb As B appl is applied B int B appl 283 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials (a) (b) (c) Fig. Schematic atomic spin structures for (a) ferromagnetic, (b) antiferromagnetic, and (c) ferrimagnetic materials 284 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials Note（1） 1.For convection currents, ( which does not satisfy ohm’s law)the amount of moving charges q Nq (ut ) s vu s t l S q J s ( J v u ) t I J s l 285 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 2.For conduction currents, (which leads to ohm’s law and KCL) the volume current density vector J N i qi u i i u i i i where more then one kind of charge carriers qi drifting with different velocities vi . For metallic conductors, we write the drift velocity. u e e E Where e is the electron mobility measured in (m2/V.S) J e e E E where e Ne and conductivity e e 286 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 3.Equation of continuity dQ d I J ds v dv dt dt V S flowing outward current negative decreasing net charge rate ( where Q is locally existing charge) By Divergence thm, d d Jds V J dv dt V v dv S J v dt 287 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials Note（2） .static magnetic field Fm (qu) B ………….Lorentz Force equ. A point charge moving at a const. speed v go thru a magnetic field B Magnetic force acting on the moving charge due to the convection current is negligible 288 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 2. law of magnetic force between two moving point charges (q2u2 ) (q1u1 ar12 ) Fm12 km 2 ……..testing charge in motion r12 (convection current) 0 where km 4 According to Lorentz’s force equ. 0 q1u1 ar B 12 is called magnetic flux density 4 r12 2 Ampere’s law of force {for conduction current (specifically filamentary currents)} 0 ( I d ) ( I1d 1 aR) F 12 2 2 where R aR R 4 c 2 c1 R 2 289 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 3.Biot-Savart law: ( B due to large amount of charges in motion)specifically free charges per unit volume of a conductor is relative to a specified location (or a field point ) 0 N q u ar B 4 r 2 v v 0 J ar r 2 v 4 v Where J N q u v u 0 d J ar B 4 c r 2 dv 290 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 4.Point charge q compared to N q dv v dv( d) compared to qu ( N q dv) u v u dv J dv J S d Id 291 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials N q u ar 5.Since B 0 v 4 v r 2 ( r ar r is the distance vector from the source element N q u or J to a specified field point) 0 ( I dl ) ar or B 0 ( J dv) a r we have B 4 v r 2 4 L r 2 (Biot-Savart law) analogous to 1 ( v dv) ar 1 ( d) a r E 4 0 v r2 or E 4 0 L r2 For source element (a point charge in motion) 0 q u a r B 0 N q u ar B v as v 0 4 or r2 4 r 2 292 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials By superposition, we obtain the Biot-Savart law as formulated above Also, for the source element (a point charge) in a static electric field 1 q ar 1 N q ar E or E v as 4 0 r 2 4 0 r 2 v 0 Again, by superposition E due to a volume distribution of charge or it due to a line charge is obtained as show above, respectively. 293 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Behavior of Magnetic Materials 6.The vector magnetic potential J 0 I d A 0 4 V 1 r dv or A 4 V r analogous to the electric potential 1 v 1 v d V 4 0 V r dv or V 4 0 L r q for a point charge V 4 0 r By superposition, we obtain the above eq. for a volume charge distribution or for a line charge. 294 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chap 7 Time-Varying Fields and Maxwell’s Equations 295 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Faraday’s law of electromagnetic EM induction X X dw awdw X d B as ds From Lorentz’s force eq., the magnetic force l exerted on the conductor shown in the figure is X X Area Σ expressed as Fm qu B qEi X dl al dl X X enclosed by L A moving conductor in a where E is denoted as impressed electric field intensity. magnetic field i dw d dw d Ei u B ( aw ) ( ) (aw ) (as ) dt ds dt ds dw d d dw ( ) ( aw as ) ( )( )( al ) dt ds l dw dt d Ei dt or Ei dl t B ds L 296 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU By Stokes’s thm, for an arbitrary area B Ei dl ( Ei ) ds ( t ) ds L B where the subscript i is omitted is called Faraday’s law We see that E t of EM induction. In addition, the voltage across the terminals a and b of the conductor b d Vab ( Ei dl ) a dt can be applied to that of a coil with N turns. Nd di Vab L dt dt 297 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU where the polarities of the voltage are plus and minus on the terminals a and b, respectively and the inductance L is defined as Nd L di 298 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques Hall effect Consider a uniform magnetic field B a z B0 B a z B0 and a uniform direct current flows in the y-direction: J ay J0 z J ay J 0 Nqu y d o Vn where N is the number of charge carriers per x unit volume, moving with a velocity v , and b J ay J0 q is the charge on each charge carrier. B a z B0 It can be observed that (1)The magnetic force tends to move the charge carriers in the positive x-direction, creating a transverse electric field. Fm is transverse to B (i.e. Fm qu B , the same direction as that of J , ) 299 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques (2)This will continue until the transverse field is sufficient to stop the drift of the charge carriers. In the steady state, the net force on the charge carrier is zero: Eh u B 0 or E h u B ( Fm Fe 0 qu B qEh 0) This is known as the Hall effect, and E h is called the Hall field. Eh (a y u0 ) az B0 axu0 B0 A transverse potential (denoted as Vh and called Hall voltage) appears across the sides of the material. Thus, we have 0 0 Vh Eh dl a x u0 B0 a x dx u0 B0 d for electron carriers. d d 300 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques The Hall effect can be used for measuring the magnetic field and determining the sign of the predominant charge carriers (distinguishing an n-type from a p-type semiconductor). Forces on current-carrying conductors dl2 (dl1 aR12 ) Recall that F12 0 I 2 I1 4 c2 c1 2 R12 dl1 a R12 R12 R12 It is an inverse-square relationship and should I1 be compared with Coulomb’s law of force dl 2 between two stationary charges. I2 301 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques Torques on current-carrying Conductors Consider a small circular loop of radius b and carrying a current I in a uniform magnetic field of flux density B B// dl1 dl sin B dl o x I T dl2 (b) (a ) It is convenient to resolve B into B perpendicular and B// parallel to the plane of the loop, i.e. . B B B // 302 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques Obviously, B tends to expand the loop but exerts no net force to move the loop and B// produces a torque that tends to rotate the loop about the x-axis in such a way as to align the magnetic field (due to I) with the external B// field. The differential torque produced by dF1 and dF2 is dT a x (dF )2b sin a x ( Idl sin B// )2b sin dT a x 2 Ib2 B// sin 2 d ( dF dF1 dF2 and dl dl1 dl2 bd ) 303 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques The total torque acting on the loop is then T dT a x 2 Ib B// sin d a x I (b ) B// a y I (b ) a z B// m B 2 2 2 2 0 ( sin 2 d ) 0 2 where a y is nthe unit vector of the surface of the loop, a m B m ( B B// ) m B// and m a y I (b2 ) an I (b2 ) an I s Therefore, we have T m B m) (N The principle of operation of direct-current (d-c) motors is based on this equation. 304 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques Forces and torques in terms of stored magnetic energy The principle of virtual displacement is an alternative method of finding magnetic forces and torques. Let’s explore it in the following two cases. (1)System of circuits with constant flux linkages: The mechanical work F dl done bythe system is at the expense of a decrease in the stored magnetic energy, where F denotes the force under the constant- flux condition. Thus F dl dwmg wmg dl That is, F wmg If the ckt is constrained to rotate about an axis, say the z axis, the mechanical work wm done by the system will be (T ) z d and (T ) z m) (N 305 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques (2)System of circuits with constant currents Since dws I k d k is the energy supplied by the system source k ( dwk vk ik dt ik d k dws dwk I k d k for ik I k ) k k We have dws dw dwmg 1 1 In addition,since dwmg 2 I k dk dws 2 We have dw FI dl dwmg (wmg ) dl or FI wmg ( N ) If the ckt is constrained to rotate about the z-axis, the z-component of the torque acting on the ckt is wm (TI ) z ( N m) 306 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Forces and Torques (1) In electrostatic field dwm dwe Fe dl Fe dl dwe qEdl qdv dwm dwe 0 ( wm we cons tan t ) External work Electrostatic energy (2) p v i 307 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Boundary Conditions for Magnetic Fields (1)The normal components of magnetic field B1n B1 by divergence theorem, since ( B 0) by divergence thm an 2 an v Bdv B ds B1n (an s) B2n (an s) 0 s 1 B1 t B2t h 0 Assuming B1n an B1n and B2 n an B2 n 2 We have a n1 a n B2 n B2 (an B1n )( an s) (an B2 n )( an s) 0 B2 n B1n (Note that we may assume B1n an B1n and B2n an B2n As a result, B2 n B1n following the directions we assume. ) Therefore, the normal component of B is continuous across an the interface. For linear media, B1 1 H 1 and B2 2 H 2 , we have 1 H 1n 2 H 2 n 308 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Boundary Conditions for Magnetic Fields (2)The tangential components of magnetostatic field an 2 an Since ( H J ) by Stokes’s thm H1 H 1n ( H )ds Hdl Jds I s c s b a H 1t 1 h 0 H 2t 2 abcda dl H1t w H 2t w J sn w c d H w H 2 H 2n ( a x H 1t )( a x w) ( a x H 2t )( a x w) ax H1t H 2t J sn ( A / m) or an 2 ( H 1 H 2 ) J s where an 2 is the outward unit normal from medium 2 at the interface. Thus, the tangential component of H is discontinuous across an interface where a free surface current exists. (However, when the conductivities of both media are finite, currents are defined by volume current densities and free surface current do not exist on the interface.) 309 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Inductances and Inductors Consider two neighboring closed loops, S2 C1 and C2 bounding surfaces S1 and S2, respectively. S1 12 B1ds2 ( Wb) C2 s2 From Biot-Savart laws (determining B1 due to the filamentary current I1) I1 C1 Two magnetically I dl 'a R coupled loops B1 0 1 4 c ' R 2 Note: L↑ means energy increasing stored in a magnetic N N L12 2 12 12 2 I1 I1 I1 s2 B1 ds2 field. N L11 11 1 I1 I1 s1 B1 ds1 ( H ) 310 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Inductances and Inductors <Ex>Find the self-inductance of a closely wound toroidal coil. I Sol： 2 b c Bdl (a B )( a rd ) 2rB 0 NI 0 h r a NI dr B 0 2r 0 NI 0 NIh b dr 0 NIh b Bds (a 2 a r )(a hdr) ln s s 2r 2 a N 0 N 2 h b L ln (H ) I I 2 a 311 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Inductances and Inductors Cause : Capacitance: (involving the concept of 1Q 2Q charge storage) Effect : 1V 2V I1 Cause : Inductance: (involving the concept 1I1 2 I1 of flux linkage) Effect : 11 21 312 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Inductances and Inductors Loosely wound L small Tightly wound L large Tightly Loosely neighboring neighboring C large C small 313 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Energy Consider a single closed loop with a self-inductance L1 in which the current i1 increases from zero to I1. At the same time, an electromotive force (emf) is induced to oppose the current charge. The work that must be done to overcome this induced emf is I1 1 w1 v1i1dt L1 i1di1 L1 I12 0 2 di1 where v1 L1 is the voltage across the inductance. dt Obviously, this work required is stored as magnetic energy. 314 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Energy Now consider two closed loops, C1 and C2 carrying currents i1 and i2, which are initially zero and are to be increased to I1 and I2, respectively. Note : dw f Fe dl qEdl qdv dwe dwe qdv Consider q cv, We have dwe cvdv 1 2 we cv ( Assume we 0 initially) 2 For constsnt , ch arg e case, dwe qdv C1 1 or for constant voltage case i1 v1 dwe vdq In the constant voltage case, i dwe dq p v vi dt dt 315 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Energy To find the amount of work required, we consider the following three cases: Case1: Keeping i2 0 and increasing i from zero to I 1 1 The work required in loop C1 C1 12 w1 1 L1 I12 v1 i1 11 2 C2 The current i1 linking with magnetic flux 1 11 12 316 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Energy Case2: Keeping i1 at I 1 and increasing i2 from zero to I 2 Because of mutual coupling, some of the magnetic flux due to i2 will link with loop C1 giving rise to an induced emf that must be overcome di2 by a voltage v21 L21 in order to keep i1 constant at its value I1. dt This work involved is I2 w21 v21 I1dt L21 I1 di2 L21 I1 I 2 0 317 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Magnetic Energy Case3: At the same time, a work w22 must be done in loop C2 in order to counteract the induced emf is increased from o to I2. 1 i1 I w22 L2 I 2 2 2 v2 The total amount of work done in raising v21 the currents in loops C1 and C2 from 0 to I1 and I2, respectively. 1 1 1 2 2 wm w1 w21 w22 L1 I1 L21 I1 I 2 L2 I 2 L jk I j I k 2 2 2 2 2 j 1 k 1 1 2 2 wm L jk I j I k ) (J 2 j 1 k 1 For a current I following in a single inductor with inductance L, the stored magnetic energy is 1 2 wm LI ) (J 2 318 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Wave Equations In free space, we have E 0.......( M 1) H E 0 M .......( 2) t H 0.......( M 3) (B 0 H ) E H 0 M .......( 4) t 319 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Wave Equations From (M2) and (M4), we have E 2 ( E ) 0 ( H ) or ( E ) 0 0 t t 2 According to the vector identities, we see ( E ) ( E ) E 2 ( E 0) E 2 That is ( E ) E 0 0 2 2 t E 2 1 E 2 It follows E 0 0 2 2 2E 2 .......( W 1) t or t 1 where the real number ( 0 0 ) 2 320 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Wave Equations In a similar fashion, we can obtain the same Eq(W1). Let’s use the notation U to stand for E or H . It appears that 1 2 U 2 2 U .......( W 2) 2 which is called wave eq. t Assume U ( x, y, z, t ) axU x a yU y azU z For one-dimensional cases, one kind of the sols to Eq(W2) is U ( x, t ) U m sin(kx (t )) 321 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Wave Equations U ( x, y, z, t ) a u a u a u x x y y z z U a u a u a u 2 x 2 x y 2 y z 2 z U U ( x, t ) U sin( kx (t )) X m U(x,t) Um X X -Um 322 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Wave Equations 2 V (t ) Vm sin(t ) T U (t ) U m sin(kx (t )) 1 3 10 (m / s ) 8 0 0 Vm t T -Vm 323 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations Faraday`s law of electromagnetic induction Electrostatic model Electromagnetic (modified) model (due to time- varying field) : (modified) Magnetostatic Maxwell`s eqs. model Equation of continuity 324 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations Electrostatic model E 0 E 0 B E t This means that a changing magnetic field induces an electric field. 325 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations Magnetostatic model B 0 B J ( B ) 0 J 0 J 0 t V 326 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations Q Q 0 0 i Qi Q Q O dv i t t t V QO t I Jds Jdv Z V v J V Q t O 327 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations B 0 H 0 J J ( D) 0 t D (J )0 t D ( B ) 0 ( ( J )) t 0 328 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations D E 0 v H 0 J t D H J t D H J t E H J t 0 329 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations B E t D H J t D v B 0 330 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Maxwell’s Equations d C E d dt dD C H d I S dt ds D ds Q s B ds 0 s 331 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Reference 1.張國維老師上課手稿 2.楊國輝、廖淑慧。民89。應用電磁學。臺北市：五南。 3.徐在新、宓子宏。民83。從法拉第到麥克斯韋。新竹市：凡異。 4.David K. Cheng(1993), Fundamentals of Engineering Electromagnetics, Addison Wesley. 5.王奕淳、張友福、張毓華、郭志成、林漢璿、廖家成上課作業。 332 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Acknowledgement Thanks to 王奕淳、林漢璿等人 For typing the lecture notes Adjust：林裕軒 2005/10/20 333 Dr. Gao-Wei Chang

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