Can someone solve these four simultaneous equations?

Question

I am solving a boundary value problem and stuck at simple algebra part. I've four equation as:

1. $E_i+E_r=E_++E_-$
2. $E_i-E_r=n(E_+-E_-)$
3. $E_t=E_+e^{i\phi}+E_-e^{-i\phi}$
4. $n(E_+e^{i\phi}-E_-e^{-i\phi})=E_t$

Here n and $\phi$ are constants. And I am trying to express $E_t$ and $E_r$ in terms of $E_i$ separately. Any kind of idea will be highly appreciated.

Thank you in advance.

Answer 1

Step 1: Use equations 1 and 2 to express $E_+$ and $E_-$ in terms of $E_i, E_r$ and $n$: $$ E_+ = \frac1{2n} (E_i -E_r + n(E_r + E_i))\\ E_- = \frac1{2n} (E_r -E_i + n(E_r + E_i)) $$ Step 2: Substitute for $E_+$ and $E_-$ in equations 3 and 4: $$ 3: E_t = \frac1{2n} \left( e^{i\phi} (E_r-E_i+n(E_r+E_i)) + e^{-i\phi} (E_r-E_i-n(E_r+E_i)) \right) \\E_t= \frac1n\left[ (n\cos\phi +i\sin\phi)E_i + (n\cos\phi -i\sin\phi)E_r) \right]$$ and$$ 4:E_t = (\cos\phi + ni\sin\phi)E_i -(\cos\phi -ni\sin\phi)E_r $$ Step 3: Equate those two expressions for $E_t$ and thus have one equation to solve for $E_r$ in terms of $n,\phi,E_i$: $$ E_r = -E_i \frac{(n^2-1\sin\phi}{(n^2+1)\sin\phi+2in\cos\phi} $$ Step 4: substitute $E_r$ into either of the $E_t$ expressions and simplify to get $$ E_t = E_i\frac{2in}{2in\cos\phi+(n^2+1)\sin\phi} $$ Don't be scared by the fact that the expressions imply complex values; i fact, in a general medium, the index of reflection $n$ is also complex.

Do be scared by the fact that in the first edition of Jackson, problem 7.5 is different, asking for the transmission coefficient when a plane wave strikes a thin or thick "excellent conductor." If I remember correctly, the solution to that problem is more difficult.

Answer 2

If $E_i$ is considered known, you have a system of four equations in four unknowns

$$\begin{cases}+E_r&&-E_+&-E_-&=-E_i\\ -E_r&&-nE_+&+nE_-&=-E_i\\ &+E_t&-E_+e^{i\phi}&-E_-e^{-i\phi}&=0\\ &-E_t&+nE_+e^{i\phi}-&nE_-e^{-i\phi}&=0.\end{cases}$$

You can solve it by Cramer or by Gaussian elimination.