Potential energy of moving charge in field

[It's me]

Homework Statement

1. Homework Statement [/B]

Prove the potential energy of interaction between an electric charge $q$ moving with velocity $\vec{v}$ and an electromagnetic field with potentials $V$ and $\vec{A}$ is given by $U = qV-q \vec{v} \cdot \vec{A}$

Homework Equations

Lorentz force $\vec{F}=q(\vec{E}+\vec{v} \times \vec{B})$

$\vec{B} = \nabla \times \vec{A}$

$\vec{E} = -\nabla V - \frac{\partial \vec{A}}{\partial t}$

The Attempt at a Solution

I don't really know how to begin this problem. I was thinking of using vector properties on the Lorentz force until I reach the desired property, since the force is the gradient of potential energy. Would that work?

 

[Simon Bridge]

I don't really know how to begin this problem. I was thinking of using vector properties on the Lorentz force until I reach the desired property, since the force is the gradient of potential energy. Would that work?

Try it and see. (Note: how do you normally work out the potential energy function?)