Show that a longitudinal wave is electrostatic

1. Jan 13, 2009

Logarythmic

1. The problem statement, all variables and given/known data
Show that all longitudinal waves must be electrostatic by using Faraday's law.

2. Relevant equations

$$\frac{\partial \vec{B}}{\partial t} = - \nabla \times \vec{E}$$

3. The attempt at a solution
Where should I start??

2. Jan 13, 2009

gabbagabbahey

A good place to start would be to write down any equations for the electric and magnetic fields (or auxiliary field H) for longitudinal EM waves. Then calculate dB/dt (or dH/dt)....what do you get?

3. Jan 13, 2009

Logarythmic

I tried with

$$\vec{B} = B_0 \sin{[i(kx-\omega t)]}$$

so

$$\nabla \times \vec{E} = i \omega B_0 \cos{[i(kx-\omega t)]}$$

But that doesn't really help me.

4. Jan 13, 2009

gabbagabbahey

Don't you mean:

$$\vec{B} = \vec{B_0} \sin{[i(kx-\omega t)]}$$

and

$$\nabla \times \vec{E} = \Re[i \omega \vec{B_0} \cos{[i(kx-\omega t)]}]$$

....what is the real part of a purely imaginary number?

5. Jan 13, 2009

Logarythmic

Sometimes I feel so smart that I dunno what to do with myself. ;) Thanks!