How to use Solve properly

Question

I'm attempting to solve the following system of equations for k and $\kappa$ $$\mu\epsilon\omega^2=k^2-\kappa^2$$ $$\mu\sigma\omega=2k\kappa$$ but when I type into Mathematica

Simplify[Solve[{μ ϵ ω^2 == k^2 - κ^2, μ σ ω == 2 k κ}, {k, κ}]

I get the following output, which is pretty gnarly

The solution I was hoping to get (from a textbook) is $$k\equiv \omega\sqrt{\frac{\epsilon\mu}{2}}\left[\sqrt{1+\left(\frac{\sigma}{\epsilon\omega}\right)^2}+1\right]^{1/2}$$ $$\kappa\equiv \omega\sqrt{\frac{\epsilon\mu}{2}}\left[\sqrt{1+\left(\frac{\sigma}{\epsilon\omega}\right)^2}-1\right]^{1/2}$$ I'm not sure how to simplify the output from Mathematica to get it into that specific from I'm looking for. Are there extra assumptions I need to pass? I've tried the same command as Simplify only changing it to Reduce but it's still not quite clear the direction to take. For some context, this is looking at the complex wave number from Griffiths' Introduction to Electrodynamics.

Answer 1

Try this:

Simplify[
 Solve[Eliminate[{μ ϵ ω^2 == k^2 - κ^2, μ σ ω == 2 k κ}, κ], k], 
 {ω > 0, σ > 0, ϵ > 0, μ > 0}
 ]

yielding the results for k

The last two results look almost like you want. To bring them exactly to the needed form try this:

expr = Sqrt[μ ω (ϵ ω + 
     Sqrt[ σ^2 + ϵ^2 ω^2])]/Sqrt[2];

HoldForm[ω*Sqrt[(ϵ*μ)/2]]* Simplify[
   Expand[expr/(ω*Sqrt[(ϵ*μ)/2])], 
   {ω > 0, σ > 0, ϵ > 0, μ > 0}
   ] // ReleaseHold

returning the following:

The same you could do for kappa.

Have fun!