VIEWS: 41 PAGES: 19 CATEGORY: Graduate POSTED ON: 2/24/2010
basic concept of electromagnetic theory
Electromagnetic Theory Mohammad Imran Aziz Assistant Professor Physics Department Shibli National Post Graduate College Azamgarh,India.(aziz_muhd33@yahoo.co.in) pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! 3 Electromagnetic Theory and Light Light possesses both wave-particle manifestations. Classical electrodynamics based on Maxwell’s electromagnetic theory unalterably leads to the picture of a continuous transfer of energy by way of electromagnetic waves. Quantum electrodynamics describes electromagnetic interaction and the transport of energy in terms of massless elementary “particles” known as photons, which are localized quanta of energy. One of the basic tenets of quantum mechanics is that both light and material objects each display both wave-particle properties. In physical optics light is treated as an electromagnetic wave. 3.1 Maxwell’s equations The simplest statement of Maxwell’s equations governs the behavior of the electric and magnetic fields in free space. Maxwell’s equations are generalization of experimental results. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! B c E dl A t dS , (3.1) E c B dl 0 0 A t dS , (3.2) A E dS 0, (3.3) B dS 0 (3.4) A where 0 and 0 are the permeability and the electric permittivity of free space, respectively. It should be noted that except for a multiplicative scalar, the electric and magnetic fields appears in above equations with a remarkable symmetry. The mathematical symmetry implies a good deal of physical symmetry. Maxwell’s equations tell us that a time-varying magnetic field generates an electric field and a time-varying electric field generates a magnetic field. Maxwell’s equations above can be written in differential form by using following two theorems from vector calculus. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! Gauss divergence theorem F dS F dV (3.5) A V and Stocke theorem F dl F dS (3.6) C A By applying theorem (3.5) to Eqs. (3.1) and (3.2) and applying theorem (3.6) to Eqs (3.3) and (3.4), we obtain the following differential equations B E ( 3 .7 ) t E B 0 0 ( 3 .8 ) t E 0 ( 3 .9 ) B 0 ( 3 . 10 ) i j k E (3.11) x y z Ex Ey Ez E E E E x y z (3.12) x y Z pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! The consequent equations for free space are in detail as follows: E z E y B x, (i ) y z t E x E z B y, (ii ) (3.13) z x t E y E x B z, (iii ) x y t Bz B y E 0 0 x , (i ) y z t Bx Bz E 0 0 y , (ii ) (3.14) z x t B y Bx E 0 0 z , (iii ) x y t E x E y E z 0, (3.15) x y z Bx B y Bz 0, (3.16) x y z pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! 3.2 Electromagnetic waves Maxwell’s equations for free space can be manipulated into the form of two vector expressions: Taking the curl of Eq. (3.7) ( E ) ( B ) (3.17) t L. H .S ( E ) 2 E 2 E Eq.( 3.9 ) 2E Eq.( 3.8 ) R.H .S 0 0 2 t We have 2 E 2 E 0 0 2 (3.18) t Similarly we have 2B 2 B 0 0 2 (3.19) t pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! The Laplacian, , operates on each component of E and B , so that the two vector 2 equations (3.18) and (3.19) actually represent a total of 6 scalar equations. One of these expressions, in Cartesian coordinates, is 2 Ex 2 Ex 2 Ex 2 Ex 0 0 2 (3.20) x 2 y 2 z 2 t Each component of the electromagnetic field ( E , E , E , B , B , B ) therefore x y z x y z obeys the scalar differential wave equation 2 2 2 2 1 , (3.21) x 2 y 2 z 2 v 2 t 2 provided that 1 v . (3.22) 0 0 If we substitute the values of 0 and 0 into Eq. 3.22, the predicted speed of all electromagnetic waves travelling in free space would then be c= 3 x 108 m/s. This theoretical value was in remarkable agreement with the previously measured speed of light. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! The experimentally verified transverse character of light should be explained within the context of the electromagnetic theory. To that end, consider the fairly simple case of a plane wave propagating in the positive x-direction and write as E E ( x, t ) . Eq. (3.15) is reduced to E x 0 (3.23) x For a progressive wave, the solution of (3.23) is Ex=0. So the electric component must be perpendicular to the propagation direction, x. Let’s orient the coordinate axes so that the electric field is parallel to the y-axis: E E ( x, t ) j . From Eq. (3.13 y III), it follows that E y Bz . (3.24) x t This implies that the time-dependent B-field can only have a component in the z-direction. Clearly then, in free space, the plane electromagnetic wave is indeed transverse, as shown in Fig. 3.1. Now let’s write Fig. 3.1 Field configuration in a plane E y ( x, t ) E0 y cos[ (t x / c ) ], (3.25) harmonic electromagnetic wave. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! The associated magnetic field can be found by directly integrating Eq. (3.25), that is, E y Bz dt x E 0 y c sin[ ( t x c ) ]dt 1 E 0 y cos[ ( t x c ) ] c E y. (3.26) c Fig. 3.2 Orthogonal harmonic E-field and B_field. Clearly, E and Bz have the same time y dependence, E j ˆE y ( x, t ) and B kBz ( x, t ) are in phase at all points inspace. ˆ Moreover, E and B are mutually perpendicular, and their cross-product, E B, points in the propagation direction, as shown in Fig.3.2. It should be noted that plane waves are not only solutions to Maxwell’s equations. As we saw in the previous chapter, the differential wave equation allows many solutions including spherical waves. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! 3.3 Energy Energy density, u , which is the radiant energy per unit volume, is given by u 0E 2 (3.27) 1 2 or u B . (3.28) From Eqs. (3.27) and (3.28), we have 0 1 u EB (3.29) 0 c The amount of energy transported during a unit time and through a unit area perpendicular to the transport direction is (suppose the wave travels through an area of A and with a speed of c and with a time duration of t ) u ( c t A) 1 S uc EB (3.30) t A 0 The corresponding vector is called Poynting vector: 1 S E B. (3.31) 0 2 It’s along the wave propagation direction. Its SI unit is watt per square meter (W m ). pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! E and B are so closely coupled to each other that we need to deal with only one of them. Using B E c from Eq. 3.26, we can rewrite Eq. 3.30 as 1 2 S E 0cE 2 (3.32) c0 The time-averaged value of the magnitude of the Poynting vector, symbolized by S , is a measure of the significant quantity known as the irradiance, I . 1 2 I S 0c E 2 0cE0 (3.33) 2 Where E0 is the peak magnitude of E. Within a linear, homogeneous, isotropic medium, the expression for the irradiance becomes 1 I v E 2 vE0 2 (3.34) 2 2 For a point light source, its irradiance is proportional to 1 r . This is well-known inverse-square law. Fig. 3.3 shows that a point source emits electromagnetic waves uniformly in all directions. Let us assume that the energy of the waves is pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! conserved as they spread from the source. Let us also center an imaginary sphere of radius r on the source, as shown in Fig. 3.3. If the power of the source is Ps , the irradiance I at the sphere must then be Ps I 2 . (3.35) 4r Fig. 3.3 A point source emits light isotropically. In Quantum theory, light possesses quantum energy h hc / 12400 / ( 3 . 36 ) is is in unit of ev, in unit of angstrom, h is Plank constant. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! 3.4 Electromagnetic-photon spectrum Although all forms of electromagnetic radiation propagate with the same speed in vacuum, they differ in frequency and wavelength. Fig. 3.4 plots the vast electromagnetic spectrum. The frequency range for whole electromagnetic spectrum is from a few Hz to 1022 Hz. The corresponding wavelength range is from many kilometers to 10-14 m. Radiofrequency waves: a few Hz to 109 Hz Microwaves: 109 Hz to about 3 x 1011 Hz. Infrared: 3 x 1011 Hz to 4 x 1014 Hz. The infrared (IR) is often subdivided into four regions: the near IR (780- 3000 nm), the intermediate IR (3000-6000 nm), the far IR (6000-15,000 nm), and the extreme IR (15,000 nm-1.0 mm). Light: 3.84 x 1014 to 7.69 x 1014 Hz. An narrow band of electromagnetic waves could be seen by human eye. Color is not a property of light itself but a manifestation of the electrochemical sensing system-eye, nerves, and brain. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! Fig. 3.4 Electromagnetic-photon spectrum. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! Ultraviolet: 8 x 1014 Hz to 3.4 x 1016 Hz. X-rays: 2.4 x 1016 Hz to 5 x 1019 Hz. Gamma rays: 5 x 1019 Hz to 2.5 x 1033 Hz. Table 3.1. Frequency and vacuum wavelength ranges for various colors pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! 120 GaInP2 100 Emission intensity (a.u.) 80 60 40 20 0 590 600 610 620 630 640 650 660 670 Wavelength (nm) Fig. 3.5 Spectra of sunlight and the light from Fig. 3.6 Emission spectrum of GaInP2 a tungsten lamp. semiconductor under excitation of a He- Cd laser.. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! 3.5 Light in matter 3.5.1 Dispersion In a homogeneous, isotropic dielectric, the phase velocity of light propagation becomes v 1 . (3.37) The ratio of the speed of electromagnetic wave in vacuum to that in matter is known as the absolute index of refraction n and is given by c n . (3.38) v 0 0 For most materials, / 0 is generally equal to 1. So, the expression of n becomes n / 0 . (3.39) Actually, n is frequency-dependent, so called dispersion. When a dielectric is subject to an applied electric field E, the internal charge distribution is distorted, which generates electric dipole moment p=Lq, with L the position vector from the negative charge -q to the positive charge q. The dipole moment per unit volume is called the electric polarization P. with pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! P ( 0 ) E (3.40) Fig. 3.8 shows the dipole formation and a oscillator model for the vibration of electrons under an E-field. The negative electrons are fastened to a stationary positive nucleus. The natural frequency of the spring is 0 k / m with k and m being the spring constant and the electron mass. The force (FE) exerted on an electron of charge q by a harmonic wave E(t) with frequency is FE qE (t ) qE0 cos(t ) (3.41) Newton’s second law provides the equation of motion with the second term the restoring force. 2d 2x qE0 cos(t ) m x m 2 0 (3.42) dt Fig. 3.8 (a) Distortion of the Let x (t ) x0 cos(t ) we have electron cloud in response to an q/m applied E-field. (b) The x(t ) E (t ) (3.43) ( 02 2 ) mechanical oscillator model for an For medium with electron density N isotropic medium the electric polarization P is pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! q 2 N / mE ( t ) P (t ) from Eq .( 3 . 40 ) ( 02 2 ) q2N / m 0 p (t ) / E (t ) 0 ( 02 2 ) From Eq .( 3 . 39 ) 2 Nq 2 1 n ( ) 1 ( 2 ) ( 3 . 44 ) 0m 0 2 Eq. (3.44) indicates dispersion. When 0 n>1; When 0 n<1. Fig 3.9 shows dispersion of materials. Fig. 3.9 Index of refraction versus wavelength and frequency for several important optical crystals. pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!