The electric field in this case is perforce parallel to the surface and hence and (for incident, reflected and refracted waves). Only two of the four equations above are thus useful. The equation is trivial. The equation requires us to determine the magnitude of the cross product of of each wave with . Let's do one component as an example.

Examining the triangle formed between
and
for the
incident waves (where
is the angle of incidence), we note
that
and thus:
1&mu#mu;|
_0 ×
| & = & 1&mu#mu; B_0
(&thetas#theta;_i)

& = & &mu#mu;&epsi#epsilon;&mu#mu; E_0 (&thetas#theta;_i)

& = & &epsi#epsilon;&mu#mu; E_0 (&thetas#theta;_i).

Repeating this for the other two waves and collecting the results, we
obtain:
E_0 + E_0'' & = & E_0'

&epsi#epsilon;&mu#mu; (E_0 - E_0'')(&thetas#theta;_i) & = &
&epsi#epsilon;'&mu#mu;' E_0'(&thetas#theta;_r)
This is two equations with two unknowns. Solving it is a bit tedious.
We need:
(&thetas#theta;_r) & = & 1 - ^2(&thetas#theta;_r)

& = & 1 - n^2n'^2^2(&thetas#theta;_i)

& = & n'^2 - n^2^2(&thetas#theta;_i)n'
Then we (say) eliminate
using the first equation:
&epsi#epsilon;&mu#mu; (E_0 - E_0'')(&thetas#theta;_i) =
&epsi#epsilon;'&mu#mu;' (E_0 + E_0'')
n'^2 - n^2^2(&thetas#theta;_i)n'
Collect all the terms:
E_0 ( &epsi#epsilon;&mu#mu; (&thetas#theta;_i) -
&epsi#epsilon;'&mu#mu;' n'^2 -
n^2^2(&thetas#theta;_i)n' ) =

E_0''(&epsi#epsilon;'&mu#mu;'n'^2 - n^2^2(&thetas#theta;_i)n'
+ &epsi#epsilon;&mu#mu; (&thetas#theta;_i) )
Solve for
:
E_0'' = E_0
( &epsi#epsilon;&mu#mu; (&thetas#theta;_i) -
&epsi#epsilon;'&mu#mu;' n'^2 -
n^2^2(&thetas#theta;_i) n' )
( &epsi#epsilon;&mu#mu; (&thetas#theta;_i) +
&epsi#epsilon;'&mu#mu;' n'^2 -
n^2^2(&thetas#theta;_i) n' )

This expression can be simplified after some tedious cancellations
involving
nn' = &mu#mu;&epsi#epsilon;&mu#mu;'&epsi#epsilon;'
and either repeating the process or back-substituting to obtain :
E_0'' & = &E_0
( n(&thetas#theta;_i) -
&mu#mu;&mu#mu;' n'^2 - n^2^2(&thetas#theta;_i) )
( n(&thetas#theta;_i) +
&mu#mu;&mu#mu;'n'^2 - n^2^2(&thetas#theta;_i) )

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