Finding the magnetic vector potential

[Je m'appelle]

I'm supposed to find the magnetic field, the scalar electric potential and magnetic vector potential for the following electromagnetic wave:

$$\vec{E} = E_0 cos (kz - \omega t) \left \{ \hat{x} + \hat{y} \right \}$$

Alright, the magnetic field goes as

$$\vec{B} = \frac{1}{c} \hat{k} \times \vec{E}$$

$$\vec{B} = \frac{E_0}{c} cos (kz - \omega t) \left \{ \hat{z} \times \hat{x} + \hat{z} \times \hat{y} \right \}$$

$$\vec{B} = \frac{E_0}{c} cos (kz - \omega t) \left \{ \hat{y} - \hat{x} \right \}$$

I used $\hat{k} = \hat{z}$ since the wave seems to be traveling through the z-axis.

Now when it comes to the potentials, I'm lost. I know, for once, that I can't directly apply $V = - \int \vec{E} . \vec{d \ell}$ since the electric field is no longer constant.

Any hints?

 

[blue_leaf77]

You need the additional information about the presence of free charge and current density to solve for the potentials.

 

[vela]

These are vacuum solutions of the wave equation, so you can assume $\rho=0$ and $\vec{J}=0$.

A good place to start (and the reason the template you deleted is supposed to be there) is to find the relevant equations which relate the fields to the potentials.

 

[Brian T]

Now when it comes to the potentials, I'm lost. I know, for once, that I can't directly apply $V = - \int \vec{E} . \vec{d \ell}$ since the electric field is no longer constant.

For the line integral $V = - \int \vec{E} . \vec{d \ell}$, you do not need a constant function to do the integral (maybe review some vector calculus?). You also need to consider what gauge you're using since you have time varying fields.