Electric and magnetic field problems (curl/divergence)

[ParoxysmX]

Homework Statement

Consider the electric field E(t,x,y,z) = Acos(ky-wt)k

1. Find a magnetic field such that \partial_tB + \nabla X E = 0
2. Show that \nabla . E = 0 and \nabla. B = 0
3. Find a relationship between k and w that enables these fields to satisfy

\nabla X B = \mu_{0}\epsilon_{0}\frac{\partial E}{\partial t}

The Attempt at a Solution

Really the problem here is the first one. I understand (sort of) the curl operator, but how do you find \nabla X E? Would you start with a matrix of

[i j k
0 kAcos(ky-wt) 0
0 0 Acos(ky-wt)]

Then find the determinant, which is Acos(ky-wt)(1+k)i - 0j + 0k?

 

[bossman27]

No, that matrix is not correct. The cross product of the del operator \nabla and a vector function is just an alternate convention for denoting curl. You can why here: http://en.wikipedia.org/wiki/Del#Curl. This should also explain why divergence can be denoted: \nabla \cdot

So your matrix should be: \left[ \begin{array}{ccc}<br /> \hat{i} &amp; \hat{j} &amp; \hat{k} \\<br /> \frac{\partial}{\partial x} &amp; \frac{\partial}{\partial y} &amp; \frac{\partial}{\partial z}\\<br /> 0 &amp; 0 &amp; {\scriptsize A\cos(ky-wt)} \end{array} \right]

The rest of the problem should be relatively trivial once you know what the operators do.

 

[ParoxysmX]

Ah I see. Thanks for your help.