# Is exercise 4, part b of chapter one of Ashcroft and Mermin wrong?

This is from Solid State Physics. It is about helicon waves in a metal.

When I take the curl of the given electric field it does not equal zero even though the magnetic field is static. Is this an error in the book?

$$E_x=E_0 e^{i(kz-\omega t)}$$

$$E_y=E_x$$

$$E_z=0$$

$$\nabla\times E=\hat{x}(ikE_x-0)+\hat{y}(0-ikE_x)\neq 0$$

• Its the electric field of an EM wave, so there should be an associated magnetic field. It is possible that some quirk of the material properties or the boundary conditions is somehow suppressing the magnetic field (though I can't immediately think of a method of doing this). What I think is most likely is that in many situations the effect of the magnetic field is simply negligible compared to the electric field. Nov 21, 2018 at 10:21
• @GeneNaden - What do you mean by "the magnetic field is static"? Do you mean an externally imposed field or the field of the wave? If the latter, then it is an electrostatic wave in which case $E_{x} \rightarrow 0$ must be true such that $\nabla \times \mathbf{E} = 0$. If the former, then this says nothing about the field of the helicon wave, which certainly can be non-zero. Also be careful in case they are asking about plane wave boundary conditions. Nov 23, 2018 at 19:48
• @honest_vivere, I meant the externally imposed field. Obviously the helicon wave has it's own magnetic field, a fact that I overlooked. Nov 24, 2018 at 20:37