The Potential Experienced by a Moving Charge

We now move on to the potential experienced by a moving charge in an electromagnetic field. Potentials are related to the force exerted on a particle. For a particle of charge q and velocity u in an electromagnetic field with electric field strength E and magnetic 

It is possible to derive the potential corresponding to F directly from (3.41) (see Goldstein 1950). Here we take a somewhat simpler approach and start with a suggested form of the potential and show that the application of (3.43) to this satisfies the Lorentz force equation. We choose the potential 

The expression for the generalized potential, (3.43), has two parts. We first evaluate the part involving the gradient  

The gradient of a scalar product of two vectors a and b has the general form 

We also note that x, and m, are independent canonical variables, and therefore 

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