Curl of B for plane wave propagating in z-direction

[adichy]

Homework Statement

The electric and magnetic fields in a plane wave propagating in free-space in the z-direction can we represented by (in complex-exponential notation)

E(x, y, z, t) = E_0 e^i(kz−wt+d ) and B(x, y, z, t) = B_0e^i(kz−wt+d )

Starting with the Ampère-Maxwell law, ∇ x B =μ_0 ε_0 dE/dt , show that E_0 = −c(z×B0).

Homework Equations

The Attempt at a Solution

I'm only having problems when I do curl B (differentiating E is no problem). I'm using the matrix method, but by just looking, it seems like it should be zero since B is in the z direction and there's no field in the x or y direction i.e

x_______y________z

d/dx___d/dy_____d/dz

0_______0____B_0e^i(kz−wt+d )

I should be getting something like ike^(...), but I'm not. Any insight would be helpful

Thanks

 

[jedishrfu]

While B is in the z direction only, the curl B will have components in the x and y directions and not the z direction. So you should review the curl definition again and see where you went astray.

 

[vela]

The wave is propagating in the z-direction. The magnetic field doesn't point in the z-direction.

 

[adichy]

Sorry, what I meant was from doing the calculation, I found the answer to be zero. That's what I found confusing. It shouldn't be zero.

Am I assuming wrong that d/dx and d/dy of B will come out as 0 since it's differentiating constants as there are no x or y variables in the field?

Basically, this is what I got from doing curl B:

x_hat[d/dy e^(...) - d/dz (0)] -y_hat[d/dx e^(...) - d/dz(0)] + z_hat[0-0]

 

[adichy]

The wave is propagating in the z-direction. The magnetic field doesn't point in the z-direction.

Ahh, of course, they're supposed to be orthogonal. I was under the impression that the magnetic field was in the z direction.