# Derive the wave equation for fields E, B from the potentials

Tags:
1. Feb 14, 2017

### phenomenologic

• Thread moved from the technical forums, so no Homework Template is shown
I'm studying for my electrodynamics exam and one of the past exam questions is:

From the scalar and vector potentials, derive the homogenous wave equations for E and B fields in vacuum.

I did derive the wave equation for the B field by simply taking the curl of the homogenous wave equation for the vector potential A. But I got stuck deriving the wave equation for the E field. I think it has to come from a tricky combination of the following four equations:
1) Definition of the E field E = -Φ - ∂A/dt
2&3) Wave equations for Φ and A
4) Lorenz gauge condition: ∇.A - (1/c^2)(∂^2Φ/∂t^2) = 0

Any help is appreciated!

2. Feb 14, 2017

### TSny

If Φ satisfies the wave equation, can you show that -Φ also satisfies the wave equation?

3. Feb 14, 2017

### phenomenologic

Hmm, so I write -(ΔΦ-(1/c^2)(∂^2Φ/∂t^2))=Δ(-Φ)-(1/c^2)(∂^2(-Φ)/∂t^2)=0. And this operation (i.e, taking nabla 'inside' the Laplacian and the partial time derivative) is allowed, right? There are no restrictions on it as far as I know but I'm not 100% sure.

Then I perform the same trick for -∂A/∂t. Similarly, I take the partial derivative inside (again, assuming it's allowed), showing the equation is satisfied for -∂A/∂t. Then add up the new equations, use formula 1) above and done?

4. Feb 14, 2017

### TSny

5. Feb 14, 2017

### phenomenologic

That was what I wasn't 100% sure. Thanks!