Now the *magnetic* field is parallel to the surface so
and
. 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.
(as before
for
). The two simplest ones are clearly:
(E_0 - E_0'')(&thetas#theta;_i) & = & E_0'(&thetas#theta;_r)

&epsi#epsilon;&mu#mu;(E_0 + E_0'') & = &
&epsi#epsilon;'&mu#mu;' E_0'
(from the second matching equations for both
and
above).

It is left as a moderately tedious exercise to repeat the reasoning
process for these two equations - eliminate either
or
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
2nn'(&thetas#theta;_i)
&mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) +
nn'^2 - n^2^2(&thetas#theta;_i)

E_0'' & = & E_0
&mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) -
nn'^2 - n^2^2(&thetas#theta;_i)
&mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) +
nn'^2 - n^2^2(&thetas#theta;_i)

The last result that one should note before moving on is the important
case of *normal incidence* (where
and
). 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 2n n' + n

E_0'' & = & E_0n' - nn' + n

Note well that the reflected wave changes *phase* (is negative
relative to the incident wave in the plane of scattering) if
.
This of course makes sense - there are many intuitive reasons to expect
a wave to invert its phase when reflecting from a ``heavier''
medium^{11.7}.