Reflection and Transmission of Electromagnetic Waves

We can show that the E- and H-vectors are perpendicular to one another and obtain a relation between their relative magnitudes from the V x E = —B equation. Assume the direction of propagation is in the -direction and that the E-vector oscillates in the x-y plane. Expanding the curl V x E and retaining only the x-variation of the y-component of E, 

Now take the indicated derivatives of the solutions to the wave equations, Ey = Eoe and = and put them into the Equations 24.13 and 24.14 above. 

The energy flux propagated by the electromagnetic wave is give by P, the Poynting vector defined as P = E x H, or 

Now let us consider what happens when an electromagnetic wave traveling through vacuum encounters a material with different /t and s. For simplicity, we consider only the case of normal incidence. The E and H vectors must be conserved so that, if part of the incident wave is reflected, the remainder must be transmitted. This can be expressed as 

Since the reflected intensity is ErP, the reflection coefficient can be written as 

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