Energy Conservation in Total Internal Reflection · RMPH 2018 #B2.4

Physics · electromagnetism

Problem

For any value of the incidence angle, the relationship between the amplitude of the field of the reflected wave and that of the incident wave was derived by the French physicist Augustin Fresnel (1788 – 1829):

E_{0r} = \frac{n_1 \cos\theta_1 - n_2 \cos\theta_2}{n_1 \cos\theta_1 + n_2 \cos\theta_2} E_{0i}

Physical note: If the perturbation produced by a wave in a point in space at a given moment is expressed using complex numbers, then the wave intensity has the mathematical expression

I = \frac{1}{2}\varepsilon_0 c |E_0|^* |E_0| = \frac{1}{2}\varepsilon_0 c |E_0|^2

, where

E_0^* = a - ib

is the complex conjugate of the complex number

E_0 = a + ib

. Here

\varepsilon_0

is the vacuum permittivity and

c

is the speed of light in vacuum. Prove that the totally reflected wave has the same intensity as the incident wave.

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