Linearly polarised Electromagnetic waves

Question

Question: Define the linearly polarised electromagnetic wave with electric field given by,

$$E = A \left( \begin{array}{ccc} a \cos(\frac{\sqrt{3}}{2}x_1-\frac{{x_3}{{}}}2-ct) \\ 0 \\ \sqrt{3} \cos(\frac{\sqrt{3}}{2}x_1-\frac{{x_3}{{}}}2-ct) \end{array} \right) $$

Where A is a given constant. Find the dimnesionless factor a.

My attempt:

I'm not quite sure what this question is asking me to do.

I know we can write a linearly polarised EM wave as:

$$E(x,t)= \epsilon .sin(K.x - ct)$$

where $\epsilon$ is a constant vector satisfying $\epsilon.K=0$

Therefore I can say that in this case, $K= (\frac{\sqrt{3}}{2},0,-\frac{1}{2})$

But how would I find a?

Answer 1

If ${\bf E}$ is an electromagnetic wave that it should be a solution to Maxwell's equations; in particular, ${\bf E}$ should be divergenceless: $\nabla \cdot {\bf E}=0$ for all $x_{i}$ and $t$. This should be enough to find $a$.