]> 29.5 Potentials

29.5 Potentials

A vector field that has no curl can be written as the gradient of a potential function. As a consequence we can describe the electrostatic field as such a gradient.

When we do so, we find that we could find a solution for the potential produced by a distribution of charge in empty space as an integral of the charge density multiplied by the potential produced by a unit charge at the point of integration.

A vector field that, like B , has vanishing divergence, can be written as the curl of a vector potential in a similar way.

We define the vector potential A so that × A = B .

With this definition · A can be anything without changing anything.

In the case of static currents where there is no time dependence we set · A = 0 and deduce the equation

× ( × A ) = 4 π j c + 1 c E t = ( · ) A = 4 π j c

We can solve this equation in all of space with the boundary condition that A go to 0 at infinity just as we solved for V . The result, exactly like that for V in the last chapter is

A ( P ) = j ( P ' ) c | P P ' | d V '

In the time dependent case we define the vector potential A by

B = × A
and
E = 1 c A t V

These definitions do not determine A and V completely.

Given any scalar field f , we can add f to A and 1 c f t to V and neither B nor E will change at all. Such a change is called a "change of gauge", and these expressions for B and E are said to be "gauge invariant" because they are unaffected by changes in gauge.

Exercise 29.4 Find the equations satisfied by A and V implied by Maxwell's Equations (including sources ρ and j ).