$\mbox{\boldmath$E$}$ Parallel to Plane of Incidence

Now the magnetic field is parallel to the surface so $\mbox{\boldmath$B$}\cdot\hat{\mbox{\boldmath$n$}} = 0$ and $\vert\mbox{\boldmath$B$} \times \hat{\mbox{\boldmath$n$}}\vert = 1$. This time three equations survive, but they cannot all be independent as we have only two unknowns (given Snell's law above for the reflected/refracted waves). We might as well use the simplest possible forms, which are clearly the ones where we've already worked out the geometry, e.g. $\mbox{\boldmath$E$}_0 \times \hat{\mbox{\boldmath$n$}} = E_0\cos(\theta_i)$ (as before for $\mbox{\boldmath$B$}_0$). The two simplest ones are clearly:

$$(E_0 - E_0'')\cos(\theta_i) = E_0'\cos(\theta_r)$$ (9.73)

$$\sqrt{\frac{\epsilon}{\mu}}(E_0 + E_0'') = \sqrt{\frac{\epsilon'}{\mu'}} E_0'$$ (9.74)

(from the second matching equations for both $\mbox{\boldmath$E$}$ and $\mbox{\boldmath$B$}$ above).

It is left as a moderately tedious exercise to repeat the reasoning process for these two equations - eliminate either $E_0'$ or $E_0''$ and solve/simplify for the other, repeat or backsubstitute to obtain the originally eliminated one (or use your own favorite way of algebraically solving simultaneous equations) to obtain:

$$E_0' = E_0 \frac{ 2nn'\cos(\theta_i) }{ \frac{\mu}{\mu'} n'^2\cos(\theta_i) + n\sqrt{n'^2 - n^2\sin^2(\theta_i)} }$$ (9.75)

$$E_0'' = E_0 \frac{ \frac{\mu}{\mu'} n'^2\cos(\theta_i) - n\sqrt{n'^2 - n^... ...} }{ \frac{\mu}{\mu'} n'^2\cos(\theta_i) + n\sqrt{n'^2 - n^2\sin^2(\theta_i)} }$$ (9.76)

The last result that one should note before moving on is the important case of normal incidence (where $\cos{\theta_i} = 1$ and $\sin(\theta_i) = 0$). Now there should only be perpendicular solutions. Interestingly, either the parallel or perpendicular solutions above simplify with obvious cancellations and tedious eliminations to:

$$E_0' = E_0 \frac{ 2n }{n' + n}$$ (9.77)

$$E_0'' = E_0\frac{n' - n}{n' + n}$$ (9.78)

Note well that the reflected wave changes phase (is negative relative to the incident wave in the plane of scattering) if $n>n'$. This of course makes sense - there are many intuitive reasons to expect a wave to invert its phase when reflecting from a ``heavier'' medium starting with things one learns studying wave pulses on a string. If this doesn't make sense to you please ask for help.