tsl319 - t (F) and x (A): 2 E y tx =- 2 B z t 2 ,- 2 B z x...

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Electromagnetic Plane Wave (1) Maxwell’s equations for electric and magnetic fields in free space (no sources): Gauss’ laws: I ~ E · d ~ A = 0 , I ~ B · d ~ A = 0 . Faraday’s and Ampère’s laws: I ~ E · d ~ = - d Φ B dt , I ~ B · d ~ = μ 0 ² 0 d Φ E dt . Consider fields of particular directions and dependence on space: ~ E = E y ( x, t ) ˆ j, ~ B = B z ( x, t ) ˆ k. Gauss’ laws are then automatically satisfied. Use the cubic Gaussian surface to show that the net electric flux Φ E is zero, the net magnetic flux Φ B is zero. E y x B tsl319 – p.1/4
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Electromagnetic Plane Wave (2) Faraday’s law, I ~ E · d ~ = - d Φ B dt , applied to loop in ( x, y ) -plane becomes [ E y ( x + dx, t ) - E y ( x, t )] dy = - ∂t B z ( x, t ) dxdy ∂x E y ( x, t ) = - ∂t B z ( x, t ) (F) Ampère’s law, I ~ B · d ~ = μ 0 ² 0 d Φ E dt , applied to loop in ( x, z ) -plane becomes [ - B z ( x + dx, t ) + B z ( x, t )] dz = μ 0 ² 0 ∂t E y ( x, t ) dxdz ⇒ - ∂x B z ( x, t ) = μ 0 ² 0 ∂t E y ( x, t ) (A) z B E y x dx dz dy dx tsl320 – p.2/4
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Electromagnetic Plane Wave (3) Take partial derivatives ∂x (F) and ∂t (A): 2 E y ∂x 2 = - 2 B z ∂t∂x , - 2 B z ∂t∂x = μ 0 ² 0 2 E y ∂t 2 . 2 E y ∂t 2 = c 2 2 E y ∂x 2 (E) (wave equation for electric field). Take partial derivatives
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Unformatted text preview: t (F) and x (A): 2 E y tx =- 2 B z t 2 ,- 2 B z x 2 = 2 E y tx . 2 B z t 2 = c 2 2 B z x 2 (B) (wave equation for magnetic field). c = 1 (speed of light). Sinusoidal solution: E y ( x, t ) = E max sin( kx-t ) B z ( x, t ) = B max sin( kx-t ) tsl321 p.3/4 Electromagnetic Plane Wave (4) For given wave number k the angular frequency is determined, for example by substitution of E max sin( kx-t ) into (E). For given amplitude E max the amplitude B max is determined, for example, by substituting E max sin( kx-t ) and B max sin( kx-t ) into (A) or (F). k = E max B max = c. The direction of wave propagation is determind by the Poynting vector: ~ S = 1 ~ E ~ B. tsl322 p.4/4...
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tsl319 - t (F) and x (A): 2 E y tx =- 2 B z t 2 ,- 2 B z x...

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