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
With this definition can be anything without changing anything.
In the case of static currents where there is no time dependence we set and deduce the equation
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
In the time dependent case we define the vector potential A by
These definitions do not determine A and V completely.
Given any scalar field f, we can add 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 and j).