Electromagnetic Forces

Electromagnetic Forces

For the electrostatic case we calculated the force by subtracting the electrostatic potential from the total energy (h_bar omega ).  Let’s try generalizing this and subtract the electromagnetic 4-potential from the test charge energy-momentum 4-vector in the Klein-Gordon equation.  For the time being we will have to restrict ourselves to forces on stationary test charges, but we allow the source charge to move.  This means that in the Klein-Gordon equation the k term is initially zero, but there is a vector potential from the moving source charge:

We are trying to calculate the forces on a free particle.  If the potentials were constant in time this would imply that the total energy (h_bar omega ) is constant as well.  We will not make that assumption this time however.  We will assume that each term above is constant separately.  Let’s solve for omega and k, with the constant value of each term captured as omega_0 and k_0:

Where

Now expand each expression in a Taylor Series about the position r1, with respect to time and space. Keeping just the first linear terms gives:

Where x is the displacement from r₁.  We now put these expressions into the wave function:

We move the gradient of the potential to the k term, as before, to get:

And now take the time derivative of the exponent to get dp/dt, the 4-force:

Or more compactly:

Or, in terms of the 4-vector potential

The first (top) term is the rate of change of energy of the test charge.  We see that this depends only on the rate of change of the electric potential.  The remaining terms are all the conventional electro-magnetic force on a stationary particle.  It depends on the gradient of the electric potential and the rate of change of the magnetic vector potential.

Note that this result is incomplete in that it does not include magnetic forces, which are only apparent when the test charge is moving.  Before addressing magnetic forces, however, we need to derive the relativistically correct way to calculate the 4-potential for a particle moving very fast, at an appreciable fraction of the speed of light.  These are the so-called Lienard-Wiechert potentials.