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

In summary, The conversation is about correctly splitting the electric field into its components and computing its curl. One person is stuck at the last step and doesn't know what to do next. The other person explains that the electric field can be represented as a constant vector times a scalar function of the dot product, making it easier to compute the curl using the chain rule. The other person also mentions that there is no need to split the electric field into components.
  • #1
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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 :

SSZ.jpg


Reference Textbook : Zangwill's Modern Electrodynamics

I stuck at the last step , I really have no idea what to do next.
 
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  • #2
You have not split [itex]\mathbf{E}[/itex] into its components correctly. The [itex]\mathbf{x}_1[/itex] component of [itex]\mathbf{E}[/itex] should be [itex]\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)}[/itex], etc.

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

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