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Course No. TEE-353 Course Title Electromagnetic Theory Credit break-up 2(2-1-0) Pre-requisite Engineering Mathematics-II (BPM-132) Catalogue Description: Vector approach to static electric and magnetic fields, Gauss Theorem, divergence, scalar and vector potential functions, Laplace and Poisson's equations, Curl and Stroke's theorem, energy storage, mapping of fields, forces on conductors in magnetic fields, capacitances and inductances, theory of images, inversion. Maxwell’s equation, Transmission Lines. Syllabus: Review: Scalar and vector fields; Vector representation of surfaces. physical representation of gradient, divergence and curl, Gauss's law, Stoke's theorem; Helmholtz theorem; different coordinate systems. The static electric field: The force between point charges and Coulomb's law; Electric Field Intensity; The electric field of several point charges and the principle of superposition of fields, The electric scalar potential; Use of Gauss's flux theorem, Solution of Laplace's and Poission's equation in one dimension; Method of images applied to plane boundaries; Electric dipoles; Polarizability; Electric flux density; Boundary conditions; Capacitance; Electrostatic shielding, Electrostatic energy. The static magnetic field: Ampere's force law; Magnetic flux density; Vector potential; ampere's circuital law; Magnetic dipoles; Magnetic field intensity; Polarization currents; Boundary conditions, Scalar potential; Faraday's law; Motional emf; Inductance. Time Varying fields: Continuity equation: Displacement current; Maxwell's equations; Boundary conditions; Wave equation and its solution in different media; Phaser notation, Polarization; Reflection and refraction of plane waves at plane boundaries. Transmission Lines: Coaxial, Two-wire and Infinite-plane transmission lines. The infinite uniform transmission lines, Comparison of circuit and field quantities, Characteristics-impedance determinations, The terminated uniform transmission line, Transmission line charts. Books: William H. Hayt, “Engineering Electromagnetics”, Tata McGraw-Hill Joseph A. Edminister, “Schaum’s Outline of Theory and Problems of Electromagnetics”, McGraw-Hill Evaluation: First Prefinal Exam: 25 Second Prefinal Exam: 25 Final Exam: 50 Vector Analysis Cartesian Coordinate System: Point P is represented as (x,y,z). Circular Cylindrical Coordinate System: Point P is represented as (,,z). Spherical Coordinate System: Point P is represented as (r,,). x: distance from the origin to the intersection of a perpendicular dropped from the point P to the x axis y: distance from the origin to the intersection of a perpendicular dropped from the point P to the y axis z: distance from the origin to the intersection of a perpendicular dropped from the point P to the z axis distance from the origin to the intersection of a perpendicular dropped from the point P to the xy plane (or z = 0 plane) or distance of the point P from the z axis in a plane normal to the z axis angle between x-axis and a line between the origin to the intersection of a perpendicular dropped from the point P to the xy plane or z = 0 plane r: distance of the point P from the origin : angle between z-axis and a line between the origin to point P P: common intersection of three surfaces: Cartesian Coordinate System: x = constant, y = constant, and z = constant Circular Cylindrical Coordinate System: = constant, = constant, and z = constant 1 Spherical Coordinate System: r = constant, = constant, and = constant x = constant: infinite plane - x y = constant: infinite plane - y z = constant: infinite plane - z = constant: circular cylinder 0 = constant: half plane 0 2 r=constant: sphere (center at the origin) 0r = constant: cone (vertex at the origin) 0 x2 y2 r x2 y2 z2 z cos1 r y tan 1 x x cos y sin z r cos r sin x = distance from the yz plane y = distance from the xz plane z = distance from the xy plane = distance from the z axis r = distance from the origin = angle between z-axis & the line from origin to the point = angle between xz plane and a plane containing z axis and the point Position Vector: Position vector of a point P: Vector from the origin to the point P P(x, y, z) rP xa x ya y za z Vector from P to Q, RPQ rQ rP (Use this formula with caution in cylindrical and spherical coordinates.) Unit Vectors have fixed directions (independent of the location of P) only in the Cartesian system. ˆ ˆ ˆ ˆ ˆ ˆ ˆ All Unit vectors a , a , a , a , a , a , and a are normal to the corresponding surfaces and point x y z r 2 towards increasing coordinate value. All these systems are right handed. ax a y az ; ˆ ˆ ˆ a a a z ; ˆ ˆ ˆ ar a a ˆ ˆ ˆ ˆ A Ax a x Ay a y Az a z A a A a Az a z Ar ar A a A a ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A Unit Vector along the direction of A, aA ˆ A R PQ Unit vector in the direction from Point P towards Point Q, ˆ a PQ R PQ where RPQ = RPQ = distance between points P and Q (Use this formula with caution in cylindrical and spherical coordinates.) Dot Product: Scalar, Magnitude of one vector × Projection of another vector in the direction of the first A B AB cos AB Cross Product: Vector A B AB sin AB a N ˆ ˆ a N : a unit vector normal to the plane determined by A and B when they are drawn from a common point. The normal selected is in the direction of advance of a right-handed screw when A is turned towards B . a x a cos ˆ ˆ a x a r sin cos ˆ ˆ a a r sin ˆ ˆ a x a sin ˆ ˆ a x a cos cos ˆ ˆ a a cos ˆ ˆ ax az 0 ˆ ˆ a x a sin ˆ ˆ a a 0 ˆ ˆ a y a sin ˆ ˆ a y a r sin sin ˆ ˆ a a r 0 ˆ ˆ a y a cos ˆ ˆ a y a cos sin ˆ ˆ a a 0 ˆ ˆ ay az 0 ˆ ˆ a y a cos ˆ ˆ a a 1 ˆ ˆ az a 0 ˆ ˆ a z a r cos ˆ ˆ a z a r cos ˆ ˆ a z a 0 ˆ ˆ a z a sin ˆ ˆ a z a sin ˆ ˆ az az 1 ˆ ˆ a z a 0 ˆ ˆ a z a 0 ˆ ˆ Scalar Field: Scalar function of a position vector Vector Field: Vector function of a position vector Differential Volume, Surface, and Line Elements: 3 Line Elements: ˆ dxa x ˆ dya y ˆ dza z Surface Elements: ˆ dydza x ˆ dzdxa y ˆ dxdya z Volume Element: dv dxdydz Line Elements: da ˆ da ˆ ˆ dza z Surface Elements: ddza ˆ dzda ˆ dda z ˆ Volume Element: dv dddz Line Elements: ˆ dra r rda ˆ r sin da ˆ Surface Elements: r 2 sin dda r ˆ r sin drda ˆ rdrddaˆ Volume Element: dv r 2 sin drdd Coulomb’s Law and Electric Field Intensity Coulomb’s Law: 1 Q1Q2 F2 ˆ a12 4 R12 2 F2 = Force on Q2 due to Q1 R12 =Directed line segment from Q1 to Q2 R12= | R12 | ˆ a12 = R12 / R12 4 R12 = R21; F1 = F2; a12 a 21 ; F1 F2 ˆ ˆ For free space, =0=8.854×10-12 F/m Electric Field Intensity: 1 Q E ˆ aR 4 R 2 R =Directed line segment from the location of Q to the point at which E is desired. R=| R | ˆ a R = R /R Charge Distributions: Volume charge, Surface (Sheet) charge, Line charge, and Point charge Volume charge density: lim Q v v 0 v Q v dv V v aR ˆ E dv V 4 R 2 Surface charge density: lim Q S S 0 S Q S dS S S aR ˆ E dS S 4 R 2 Line charge density: lim Q l l 0 l Q l dl L l aR ˆ E dl L 4 R 2 Streamlines of Fields: The equation of a streamline is obtained by solving the differential equation E y dy E x dx 5 Electric Flux Density, Gauss’s Law, and Divergence Electric flux density, D = E 1 Q aˆ a ˆ aˆ D a R v R dv S R dS l R dl ˆ 4 R 2 V 4R 2 S 4R 2 L 4R 2 Electric flux, = D dS S Gauss’s Law: The electric flux passing through any closed surface is equal to the total charge enclosed by that surface. Q = D dS . S The solution is easy if we are able to choose a closed surface which satisfies two conditions: 1. D is everywhere either normal or tangential. 2. Wherever it is normal, D = constant. First step is to investigate the symmetry of the field by asking 1. With which coordinate does the field vary? 2. Which components of the field are present? D due to a Uniform Line Charge: Dz = DΦ = 0 D : tangential at surface 1 & 3 : normal at surface 2 | D |=Dρ = constant at surface 2 Applying Gauss’ Law, Q D dS D dS D dS D dS 1 2 3 1 2 3 0 D 2 L Q Q/L D l 2 L 2 2 D l aˆ 2 l E ˆ a 2 D due to a Uniform Surface Charge: Dρ = DΦ = 0 D : tangential at surface 2 : normal at surfaces 1 & 3 | D |= Dz = constantat surfaces 1 & 3 Applying Gauss’ Law, 6 Q D dS D dS D dS D dS 1 2 3 1 2 3 ˆ ˆ D z a z a z 0 D z a z 2 a z 2 ˆ ˆ D z 2 0 D z 2 Q S Dz 2 2 2 D S az ˆ 2 E S az ˆ 2 Divergence: div A lim A dS v 0 v Divergence in Cartesian Coordinates: A dS A dS A dS A dS A dS S front back left right A dS A dS top bottom x Ax x Ax Axo yz Axo yz 2 x 2 x y A y y A y A yo xz A yo xz 2 y 2 y z Az z Az Azo xy Azo xy 2 z 2 z A A y A y x x xyz y y A A y A y x x v y y A A y A y div A x x y y Divergence in Cylindrical Coordinates: A A 1 Az 1 div A z Divergence in Spherical Coordinates: r Ar r sin sin A r sin A 1 2 1 1 div A 2 r r Divergence Theorem: S A dS div Adv V Gauss’s Law in Point Form: dQ Q D dS div Ddv div D v S V dv Energy and Potential 7 Work Done in Moving a Point Charge: dW= Fa dl = F dl = QE dl final final W Q E dl Q E L dl init init W Q( E L1 L1 E L 2 L2 E L 3 L3 .......) Potential of a Point and Potential Difference between Two Points: Potential of point A with respect to point B: Work done in moving a unit positive charge from B to A A V AB E dl B VAB = VAC - VBC = VA - VB (If reference point C is at zero potential e.g. infinity) VA =Potential of point A VB =Potential of point B Potential Field of a Point Charge: 1 Q 1 Q E aR ˆ ˆ ar 4 R 2 4 r 2 1 Q a r dra r rda r sin da 1 Q dV E dl ˆ ˆ ˆ ˆ dr 4 r 4 r 2 2 A rA B 1 Q Q 1 1 V AB dV dr r 4 r 4 A rB 2 B rB Q 1 1 Q Q VA r 4 r V 4 r 4 A A Potential of a Charge Distribution: 8 v dv dS l dl V S V 4 R S 4 R L 4 R Conservative Property of the Electrostatic Field: Closed line integral: zero E dl 0 L E dl E dl E dl VBA V AB (VB V A ) (V A VB ) 0 B A B I II A I II Potential Gradient: dV = -E dl cos dV = -E cos dl dV Maximum of is E, and it occurs when = 180o, i.e., field is opposite dl to the direction in which V is increasing most rapidly. dV dV E aN ˆ a N grad V ˆ dl m ax dN dV dV grad V ˆ aN dl m ax dN E grad V If dl is directed along an equipotential surface, dV = -E dl cos90o = 0 Gradient in Cartesian Coordinates: dV E dl grad V dl grad V dl dl dxa x dya y dza z ˆ ˆ ˆ V V V dV dx dy dz x y z V V V dx dy dz grad V dxa x dya y dza z grad V x dx grad V y dy grad V z dz ˆ ˆ ˆ x y z grad V x V x V grad V y y grad V z V z V V V grad V grad V x a x grad V y a y grad V z a z ˆ ˆ ˆ ax ˆ ay ˆ ˆ az x y z Gradient in Cylindrical Coordinates: V 1 V V grad V a ˆ a ˆ ˆ az z Gradient in Spherical Coordinates: 9 V 1 V 1 V gradV ar ˆ a ˆ ˆ a r r r sin The Dipole: Q 1 1 Q R2 R1 Q d cos V R 4 R R 4 r 2 4 1 R2 1 2 Qd cos Qd sin E grad V ar ˆ ˆ a 2 r 3 2 r 3 p Qd d a r d cos ˆ p ar ˆ V 2 r 2 Energy Density in the Electrostatic Field: Consider a system of n point charges Q1, Q2,… Qn. Vij = potential at the location of Qi due to the presence of Qj. n Vi = potential at the location of Qi= Vij j 1 j i Wi = work to position Qi, charges are being positioned in a sequence Q1, Q2,… Qn. Wi’= work to position Qi, charges are being positioned in a sequence Qn, Qn-1,… Q1. i 1 Wi Qi Vij j 1 in Wi ' Q i V j i 1 ij n i 1 in 1 n 1 n n 1 n WE Wi Wi ' 1 Qi Vij Qi Vij Qi Vij QiVi 2 i 1 j 1 2 i 1 2 i 1 j 1 j i 1 2 i 1 j i 1 2 V WE vVdv 10 Tutorial Sheet-1 1. Transform (i) point P(1,2,3) from Cartesian to cylindrical coordinates, (ii) point P(1,30o,3) from cylindrical to Cartesian coordinates, (iii) point P(1,2,3) from Cartesian to spherical coordinates, (iv) point P(1,30o,3) from cylindrical to spherical coordinates, (v) point P(1,2,3) from Cartesian to spherical coordinates, (vi) point P(1,30o,45o) from spherical to Cartesian coordinates, and (vii) point P(1,30o,45o) from spherical to cylindrical coordinates. ˆ ˆ ˆ 2. Given points A(-1,-3,-4) and B(2,2,2), find: rA , rB , R AB , rA , rB , a A , a B , R AB & a AB . 3. Given points A(2,5,-1), B(3,-2,4) and C(-2,3,1), find: RAB RAC ; the angle between R AB and R AC ; the length of the projection of R AB on R AC ; the vector projection of R AB on R AC , R AB R AC ; the area of the triangle defined by the points A, B, and C; and a unit vector perpendicular to the plane in which the triangle is located. 4. Given points A(x = 2, y = 3, z = -1) and B(ρ = 4,Φ = -50o, z = 2), find the distances from A to the origin, B to the origin, and A to B. Also find a unit vector in cylindrical coordinates at point A directed toward point B. 5. Given points A(x = 2, y = 3, z = -1) and B(r = 4, θ = 25o, Φ = 120o), find the spherical coordinates of A, the Cartesian coordinates of B, and the distance from A to B. ˆ 6. Transform each of the following vectors to cylindrical coordinates at the point specified: 5a x at P(ρ = 4,Φ = 120 , z = 2), 5a x at Q(x = 3, y = 4, z = -1), and 4a x 2a y 4a z at A(x = 2, y = 3, z = 5). o ˆ ˆ ˆ ˆ 7. Express the field F 2xyza x 5x y z a z in cylindrical coordinates. Find F at P(ρ = 2,Φ = 60o, z = ˆ ˆ 3). ˆ 8. Transform each of the following vectors to spherical coordinates at the point specified: 5a x at P(r = 4, θ = 25 , Φ = 120 ), 5a x at Q(x = 2, y = 3, z = -1), and 4a x 2a y 4a z at A(x = -2, y = -3, z = 4). o o ˆ ˆ ˆ ˆ 9. A closed surface is defined in spherical coordinates by 3<r<5, 0.1π< θ<0.3π, 1.2π< Φ <1.6π. Find the volume enclosed, the total surface area and the distance from A(r = 3, θ = 0.1π , Φ = 1.2π) to B(r = 53, θ = 0.3π , Φ = 1.6π). [Ans: 14.91, 36.8, 3.86]. 10. Find the total volume defined by 4< ρ<6, 30o< Φ<60o, 2<z<5. What is the length of the longest straight line that lies entirely within the volume? Find the total area of the surface. Tutorial Sheet-2 1. A 2 mC positive charge Q1 is located in vacuum at (3,-2,-4), and a 5 C negative charge Q2 is at (1,-4,2). Find the vector force on the negative charge. What is the magnitude of the force on the positive charge? ˆ ˆ ˆ [Ans: 0.616 a x +0.616 a y -1.848 a z N, 2.04 N]. 2. Calculate E at A (3,-4,2) in free space caused by(a) a charge Q1 = 2 C at B(0,0,0); (b) a charge Q2 = 3 C at B(-1,2,3); (c) a charge Q1 = 2 C at B(0,0,0) and a charge Q2 = 3 C at B(-1,2,3). [Ans: 345 a x - ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 460 a y +230 a z V/m; 280 a x -419 a y -69.9 a z ; 625 a x -880 a y +160.3 a z ]. 3. Find the total charge inside each of the volumes indicated: (a) v = 10z2e-0.1xsiny, -1 x 2, 0 y 1, 3 z 3.6; (b) v = 4xyz2, 0 2, 0 /2, 0 z 3; (c) v = 3 cos2 cos2/[2r2(r2+1)], universe. [Ans: 119.5 C, 72 C, 15.5 C]. 4. Charge is distributed uniformly along an infinite straight line with constant density l. Develop the expression for E at the general point P. 11 5. Develop an expression for E due to charge uniformly distributed over an infinite plane with density S. 6. An infinitely long, uniform line charge is located at y = 3, z = 5. If l = 30 nC/m, find E at: (a) the ˆ ˆ ˆ ˆ ˆ origin; (b) A (0.6.1); (c) B (5.6.1). [Ans: -47.6 a y -79.3 a z V/m; 64.7 a y -86.3 a z V/m; 64.7 a y - ˆ 86.3 a z V/m]. 7. Four infinite uniform sheets of charge are located as follows: 20 pC/m2 at y = 7, -8 pC/m2 at y = 3, 20 pC/m2 at y = 7, 6 pC/m2 at y = -1, and -18 pC/m2 at y = -4. Find E at the point: (a) A(2,6,-4); (b) ˆ ˆ ˆ B(0,0,0); (c) C(-1,-1.1,5); (d) D(106, 106, 106). [Ans: -2.26 a y V/m; -1.355 a y V/m; -2.03 a y V/m; 0]. 8. Obtain the equation of the streamline that passes through the point P(-2,7,10) in the field =: (a) 2(y-1) ˆ ˆ ˆ ˆ a x +2x a y ; (b) ey a x +(x+1)ey a y . [Ans: (y-1)2-x2=32; 2y-(x+1)2=13]. 9. A uniform line charge of l = (2×10-8/6) C/m lies along the x axis and a uniform sheet of charge is located at y = 5 m. Along the line y = 3 m, z = 3 m the electric field E has only a z-component. What is S for the field? [Ans: 125 pC/m2]. 10. The circular region, < a, z=0, carries a uniform surface charge density S. Find E at P(0,0,h), h>0. Tutorial Sheet 3 1. A 25 C point charge is located at the origin. Calculate the electric flux passing through: (a) that portion of the sphere r = 20 cm bounded by = 0 and , = 0 and /2; (b) the closed surface = 0.8 m, z = 0.5 m; (c) the plane z = 4 m. [Ans: 6.25 C, 25 C, 12.5 C] 2. Find D in Cartesian coordinates at P(6, 8, -10) caused by: (a) a point charge of 30 mC at the origin; (b) a uniform line charge l = 40 C/m on the z axis; (c) a uniform surface charge density s = 57.2 C/m2 on the plane x = 9. [Ans: 5.06 a x +6.75 a y -8.44 a z C/m2 ; 0.382 a x +0.509 a y C/m2 ; -28.6 a x C/m2 ] ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3. Let D =r a r /3 nC/m2 in free space. Find: (a) E at r = 0.2 m; (b) total charge within the sphere r = 0.2 m; (c) total electric flux leaving the sphere r = 0.3 m. [Ans: 7.53 V/m; 33.5 pC; 113.1 pC] 4. Find the total electric flux leaving the spherical surface r = 2.5 m given the charge configuration: (a) Q = 2 x nC on the x axis at x = 0, 1, 2, …. M; (b) a line charge l = 1/(z2+1) nC/m on the z axis; (c) a 2 surface charge s = 1/(x2+y2+4) nC/m2 on the z = 0 plane. [Ans: 2.125 nC, 2.38 nC, 2.96 nC] 5. Surface charge densities of 200, -50, and sx C/m2 are located at r = 3, 5, and 7 cm, respectively. Find D at r =: (a) 2 cm; (b) 4 cm; (c) 6 cm. (d) Find sx if = 0 at r = 7.32 cm. [Ans: 0, 112.5 a r C/m2, - ˆ 11.22 C/m ]2 ˆ ˆ ˆ 6. Let D =y2z3 a x +2xyz3 a y +3xy2z2 a z pC/m2 in free space. Find the total electric flux passing through the surface x = 3, 0 y 2, 0 z 1 in a direction away from the origin. Find E at P(3,2,1). Find the total charge contained in an incremental sphere having a radius of 2 m centered at P(3,2,1). [Ans: 0.667 pC, 4.31 V/m, 2.61 × 10-27 C] 7. Find the volume charge density that is associated with each of the following fields: (a) D = xy2 a x +yx2 a y +z a z C/m2; (b) D = z2sin2 a +z2sincos a +2zsin2 a z C/m2; (c) a r C/m2. [Ans: ˆ ˆ ˆ ˆ ˆ ˆ ˆ x2+y2+1 C/m3; z2+2zsin2 C/m3; 2/r C/m3] ˆ ˆ 8. Given the flux density D =(2cos/r3) a r +(sin/r3) a C/m2, evaluate both sides of the divergence theorem for the region defined by 1 < r < 2, 0 < < /2, 0 < < /2. [Ans: 0; 0] 9. Let D =x a x and find the value of DdS over the surface of the sphere r = 1. [Ans: 4.19] ˆ S 12 10. Given that D =0z a z in the region -1 z 1 and D =(0z/|z|) a z elsewhere, find the charge density. [Ans: ˆ ˆ 0 for -1 z 1 and 0 elsewhere] Tutorial Sheet-4 1. An electric field is given as E = 6y2zax + 12xyzay + 6xy2az V/m. An incremental path is represented by ∆L = -3ax + 5ay -2az m. Find the work done in moving a 2C charge along this path if the location of the path is at: (a) A(0,2,5); (b) B(1,1,1); (c) C(-0.7,-2,-0.3). Ans: 720pJ; -60pJ; -60pJ. 2. Find the work done in moving a 5C charge from the origin to P(2,-1,4) through the field E = 2xyzax + x2zay + x2yaz V/m via the pat: (a) straight line segments: (0,0,0) to (2,0,0) to (2,-1,0) to (2,-1,4); (b) straight line: x = -2y, z = 2x; (c) curve: x = -2y3, z = 4y2. Ans: 80J; 80J; 80J. 3. A time-varying E field need not be conservative. Let E = xay V/m at a certain instant of time, and calculate the work required to move a 3C point charge from (1,3,5) to (2,0,3) along the straight-line segments joining: (a) (1,3,5) to (2,3,5) to (2,0,5) to (2,0,3); (b) (1,3,5) to (1,3,3) to (1,0,3) to (2,0,3). Ans: 18J; 9J. 4. Let E = (-6y/x2)ax + (6/x) ay + 5 az V/m and calculate: (a) VPQ given P(-7,2,1) and Q(4,1,2); (b) VP if V = 0 at Q; (c) VP if V = 0 at (2,0,-1). Ans: 8.21V; 8.21V; -8.29V. 5. A point charge of 6nC is located at the origin in free space. Find VP if point P is located at P(0.2,- 0.4,0.4) and: (a) V = 0 at infinity; (b) V = 0 at (1,0,0); (c) V = 20V at (-0.5,1,-1). Ans: 89.9V; 36.0V; 73.9V. 6. Assume a zero reference at infinity, and find the potential at P(0,0,10) that is caused by this charge configuration in free space: (a) 20nC at the origin; (b) 10 nC/m along the line x = 0, z = 0, -1 < y < 1; (c) 10 nC/m along the line x = 0, y = 0, -1 < z < 1. Ans: 17.98V; 17.95V; 18.04V. 7. If V = (60 sin )/r2 V in free space and point P is located at r = 3 m, = 60o, = 25o, find: (a) VP; (b) EP; (c) dV/dN at P; (d) aN at P; (e) v at P. Ans: 5.77V; 3.85ar – 1.111a V/m; 4.01 V/m; - 0.961ar + 0.277a; - 7.57pC/m3. 8. A dipole of moment p = - 4ax + 5ay + 3az nC-m is located at D(1,2,-1) in free space. Find V at: (a) PA(0,0,0); (b) PB(1,2,0); (c) PC(1,2,-2); (d) PD(2,6,1). Ans: -1.835 V; 27.0 V; -27.0 V; 2.02 V. 9. Point charges of +3C and -3C are located at (0,0,1 mm) and (0,0,-1 mm), respectively, in free space. (a) Find p. (b) Find E in spherical components at P(r = 2, = 40o, = 50o). (c) Find E in spherical components at (1,2,1.5). Ans: 6az nC-m; 10.33ar + 4.33a V/m; 3.08ar + 2.29 a V/m. 10. Find the energy stored in free space for the region, 0 < r <a, 0 < < , 0 < z < 2, given the potential field V = : (a) V0r/a; (b) V0r/a cos 2. Ans: 1.5710V02; 1.3740V02. Tutorial Sheet-5 1. The vector current density is given as J = (4/r2) cos ar + 20e-2r sin a - r sin cos am(a) Find J at r = 3, = 0, (b) Find the total current passing through the spherical cap r = 3, 0 < < 20o, 0 < < 2, in the ar direction. Ans: 0.444 arm1.470 A. 2. Assume that an electron beam carries a total current of -500A in the az direction, and has a current density Jz that is not a function of and in the region 0 m. If the electron velocities are given by vz = 8 107 z m/s, calculate v at = 0 and z =: (a) 1 mm; (b) 2 cm; (c) 1 m. Ans: - 0.1989 C/m3; -9.95 mC/m3; -198.9 C/m3. 3. Find the magnitude of the electric field intensity in a sample of silver having = 6.17 107 mho/m and e = 0.0056 m2/V-s if: (a) the drift velocity is 1 mm/s; (b) the current density is 107 A/m2; (c) the sample is a cube, 3 mm on a side, carrying a total current of 80 A; (d) ) the sample is a cube, 3 mm on a 13 side, having a potential difference of 0.5 mV between opposite faces. Ans: 0.1786 V/m; 0.1621 V/m; 0.1441 V/m; 0.1667 V/m. 4. An aluminum conductor is 1000 ft long and has a circular cross-section with a diameter of 0.8 in. If there is a dc voltage of 1.2 V between the ends, find: (a) the current density; (b) the current; (c) the power dissipated. Ans: 1.504 105 A/m2; 48.8 A; 58.5 W. 5. A potential field is given as V = 100e-5x sin 3y cos 4z V. Let point P(0.1,/12,/24) be located at a conductor-free space boundary. At point P, find the magnitude of: (a) V; (b) E; (c) EN; (d) Et; (e) s. Ans: 37.1 V; 233 V/m; 0; 2.06 nC/m2. 6. A point charge of 25 nC is located in free space at P(2,-3,5), and a perfectly conducting plane is at z = 2. Find: (a) V at (3,2,4); (b) E at (3,2,4); (c) s at(3,2,2). Ans: 11.78 V; 0.985ax + 4.92 ay – 4.69az V/m; - 57.6 pC/m2. 7. A certain homogeneous slab of lossless dielectric material is characterized by an electric susceptibility of 0.12 and carries a uniform electric flux density within it of 1.6 nC/m2. Find: (a) E; (b) P; (c) the average dipole moment if there are 2 1019 dipoles per cubic meter; (d) the voltage between two equipotentials 1 in apart. Ans: 161.3 V/m; 171.4 pC/m2; 8.57 10-30 C; 4.10 V. 8. The region y < 0 contains a dielectric material for which R1 = 2.5, while the region y > 0 is characterized by R2 = 4. Let E1 = -30ax + 50ay + 70az V/m, and find: (a) EN1; (b) Et1; (c) Et1; (d) E1; (e) 1; (f) DN2; (g) Dt2; (h) D2; (i) P2; (j) 2. Ans: 50 V/m; -30ax + 70az V/m; 76.2 V/m; 91.1 V/m; 56.7o; 1.107 nC/m2; 2.7 nC/m2; -1.062ax + 1.107ay + 2.48az nC/m2; -0.797ax + 0.830ay + 1.859az nC/m2; 67.7o. 9. Find the relative permittivity of the dielectric material used in a parallel plate capacitor if: (a) C = 40 nF, d = 0.1 mm, and S = 0.15 m2; (b) d = 0.2 mm, E = 500 kV/m, and S = 10 C/m2; (c) D = 50 C/m2 and the energy density is 20 J/m3. Ans: 3.01; 2.26; 7.06. 10. Find the capacitance of: (a) 20 cm of 58C/U coaxial cable having an inner conductor 0.0295 in in diameter, an outer conductor having an inside diameter of 0.116 in, and a polyethylene dielectric; (b) a conducting sphere 1 cm in diameter, covered with a layer of polyethylene 1 cm thick, in free space; (c) a conducting sphere 1 cm in diameter, covered with a layer of polyethylene 1 cm thick, and surrounded by a concentric conducting sphere 1.5 cm in radius. Ans: 18.37 pF; 0.885 pF; 1.886 pF. Tutorial Sheet-9 1. Given the magnetic flux density, B = 6 cos 106t sin 0.01x az mT, find: (a) the magnetic flux passing through the surface z = 0, 0 < x < 20 m, 0 < y < 3 m, at t = 1 s; (b) the value of the closed line integral of E around the perimeter of the surface specified above, at t = 1 s. Ans: 19.39 mWb; 30.2 kV. 2. Find the amplitude of the displacement current density: (a) in the air near a car antenna, where the field strength of an FM signal is E = 80 cos (6.277 108t – 2.092y) az V/m; (b) in an air space within a large power transformer where H = 106 cos (377t + 1.2566 10-6z) ay A/m; (c) inside a capacitor where R = 600 and D = 3 10-6 sin (6 106t – 0.3464x) az C/m2; (d) inside a typical metallic conductor where f = 1 kHz, = 5 107 mho/m, R = 1; and the conduction current density is J = 107 sin ( 6283t – 444z) ax A/m2. Ans: 0.445 A/m2; 1.257 A/m2; 18 A/m2; 11.13 nA/m2. 3. The unit vector 0.48ax – 0.6ay + 0.64az is directed from region 2 (eR2 = 2.5, mR2 = 2, s2 = 0) toward region 1 (eR1 = 4, mR1 = 10, s1 = 0). If H1 = (-100ax – 50ay +200az) sin 400t A/m at point P in region 1 adjacent to the boundary, find the amplitude at P of: (a) HN1; (b) Ht1; (c) HN2; (d) H2. Ans: 110 A/m; 201 A/m; 550 A/m; 586 A/m. 4. A coaxial transmission line with 14 15

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