Show that a longitudinal wave is electrostatic

[Logarythmic]

Homework Statement

Show that all longitudinal waves must be electrostatic by using Faraday's law.

Homework Equations

Faraday's law:

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

The Attempt at a Solution

Where should I start??

 

[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?

 

[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.

 

[gabbagabbahey]

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.

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?

 

[Logarythmic]

Sometimes I feel so smart that I don't know what to do with myself. ;) Thanks!