Find an expression for a magnetic field from a given electric field

[TechieDork]

Homework Statement

Find an expression for a magnetic field B(X,t) from a given electric field in a monochromatic plane wave solution E(X,t) using Maxwell's equations.

Relevant Equations

E(X,t) = εe^i(kX-ωt) , X = (x1,x2,x3) , k = (kx1,kx2,kx3) , ω = c|k| , k·ε = 0
ε is a complex vector

Here this is my attempt :

Reference Textbook : Zangwill's Modern Electrodynamics

I stuck at the last step , I really have no idea what to do next.

 

[pasmith]

You have not split $\mathbf{E}$ into its components correctly. The $\mathbf{x}_1$ component of $\mathbf{E}$ should be $\epsilon_1 e^{i(\mathbf{k}\cdot \mathbf{x} - \omega t)} = \epsilon_1 e^{i(k_1x_1 + k_2x_2+k_3x_3 - \omega t)}$, etc.

But you don't need to split this into components.

The electric field depends on $\mathbf{x}$ only through a constant vector times a scalar function of the dot product, so its curl is computed most easily by the chain rule: $$\nabla \times (\mathbf{c}f(\zeta(\mathbf{x}))) = f'(\zeta) ((\nabla \zeta) \times \mathbf{c}).$$