Prozorov_24 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 24 Electromagnetic waves PART I y Faraday’s law: Line integral over a rectangular circuit (sides Δx, a) in the xy plane.: Ey x x ,t a E y x ,t a E dl Ey x,t Ey x x,t a Bz x,t x Bz x x,t z x If Δx is small B a x x x d B E dl dt Bz x ,t t Bz x ,t d B a x dt t Bz x ,t dx ~ aBz x ,t x E y x x ,t E y x ,t a a x E y x x ,t E y x ,t x x 0 PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University Bz x ,t t Bz x ,t t E y x ,t x 19 October 2011 2 y Ampere’s law: Line integral over a rectangular circuit (sides Δx, a) in the xz plane: Ey x,t Bz x x ,t Bz x ,t a 0 0a x 0 0 E y x ,t t a Bz x,t z d E B dl 0 0 dt Ey x x,t x Bz x x,t x E y x ,t t Bz x x ,t Bz x ,t x 0 0 x 0 E y x ,t t PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University Bz x ,t x 19 October 2011 3 0 0 E y t Bz 0 0 x t Bz t 2Bz 2Bz t x x t x Bz t 2E y t 2 2Bz t x E y x x 0 0 E y x 2y 1 2y v 2 t 2 x 2 This value is essentially identical to the speed of light measured by Foucault in 1860! (3108 m/s) Maxwell identified light as an electromagnetic wave. PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University t 2 2E y x 2 Similarly, we can obtain Remember the wave equation from 221? 2E y 0 0 2Bz t 2 2Bz x 2 This is a wave with speed v 1 0 0 19 October 2011 4 using Maxwell Equations PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 5 in vacuum A A 2 A B E E , E 0, B 0 0 t t 2 E 2 E E E B 0 0 2 t t 2E 2E 2 2 0 0 2 2 E 0 2 c E 0, t t 1 1 8 m c 2.9979×10 0 0 s 4 ×107 ×8.85×1012 PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 6 plane electromagnetic wave Let us assume that we have: • E-field in the y direction, uniform along yz plane • B-field in the z direction, uniform along yz plane • propagation in the x direction 2 Ey d 2 Ey 2E 2 2 c E 0 c2 0 2 2 2 t dx t 2 B 2 2 2 Bz d 2 Bz 2 c B 0 c 0 2 2 2 t t dx PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 7 harmonic solution Let’s try a wave-like solution: E y Emax cos kx t 2 Ey c2 d 2 Ey 0 t dx 2 Emax cos kx t c 2 k 2 Emax cos kx t 0 2 2 2 c 2 k 2 0 with Maxwell equation: k 2 c 2 f f c c dE y B E , E kEmax sin kx t z t dx dE y dBz E kEmax sin kx t z dt dx k Bz Emax cos kx t Bz Emax E cos kx t , Bmax max c c PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 8 alternatively with another Maxwell equation: y dB 1 E , B z y c 2 t dx dBz 1 E y 2 2 Emax sin kx t dx c t c B Bz Emax cos kx t z kc 2 E Bz max cos kx t c PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University x 19 October 2011 9 how is the EM wave moving? Let’s focus on one point (e.g. when E=Emax) and see how it behaves in time it is moving with velocity c in a positive x direction! E y Emax cos kx t 1 kx t 0 x t k x ct PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 10 harmonic EM waves summary for harmonic EM wave propagating in the x direction: E Emax cos kx t B Bmax cos kx t with 2 k 2f c f 1 k 0 0 E is perpendicular to B propagation direction is E B wave speed is c 1 0 0 E and B are always in phase E cB (magnitudes) PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 11 In-class example: radio stations Radio stations broadcast at frequencies that range from 540 kHz (low end of AM band) up to 108 MHz (high end of FM band). What is the range of wavelengths associated with these frequencies? c f max min 3 108 m/s 540 10 Hz 3 3 108 m/s 108 10 Hz 6 555 m 2.78 m significant range! Very different physics (we’ll come back to this: diffraction and reflection) PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University 19 October 2011 12 EM waves not in vacuum E field inside a material is characterized by dielectric constant or the dielectric permittivity 0 Similarly: B field inside a material is characterized by relative permeability m or the permeability 0 EM wave speed in general: v 1 c m c n PHYS222 - Lecture 24 - Prof. Ruslan Prozorov - Iowa State University n m 1 always Refraction index 19 October 2011 13 ...
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