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

\overset{⟶}{B}

, has vanishing divergence, can be written as the curl of a vector potential in a similar way.

We define the vector potential

\overset{⟶}{A}

so that

\overset{⟶}{\nabla} \times \overset{⟶}{A} = \overset{⟶}{B}

With this definition

\overset{⟶}{\nabla} \cdot \overset{⟶}{A}

can be anything without changing anything.

In the case of static currents where there is no time dependence we set

\overset{⟶}{\nabla} \cdot \overset{⟶}{A} = 0

and deduce the equation

\overset{⟶}{\nabla} \times (\overset{⟶}{\nabla} \times \overset{⟶}{A}) = \frac{4 π \overset{⟶}{j}}{c} + \frac{1}{c} \frac{\partial \overset{⟶}{E}}{\partial t} = (\overset{⟶}{\nabla} \cdot \overset{⟶}{\nabla}) \overset{⟶}{A} = 4 π \frac{\overset{⟶}{j}}{c}

We can solve this equation in all of space with the boundary condition that

\overset{⟶}{A}

go to

\overset{⟶}{0}

at infinity just as we solved for

V

. The result, exactly like that for

V

in the last chapter is

\overset{⟶}{A} (P) = ∭ \frac{\overset{⟶}{j} (P')}{c | P - P' |} d V'

\overset{⟶}{B} = \overset{⟶}{\nabla} \times \overset{⟶}{A}

\overset{⟶}{E} = \frac{1}{c} \frac{\partial \overset{⟶}{A}}{\partial t} - \overset{⟶}{\nabla} V

Given any scalar field

f

, we can add

\overset{⟶}{\nabla} f

\overset{⟶}{A}

and

\frac{1}{c} \frac{\partial f}{\partial t}

V

and neither

\overset{⟶}{B}

nor

\overset{⟶}{E}

will change at all. Such a change is called a "change of gauge", and these expressions for

\overset{⟶}{B}

and

\overset{⟶}{E}

are said to be "gauge invariant" because they are unaffected by changes in gauge.

Exercise 29.4 Find the equations satisfied by $\overset{⟶}{A}$ and $V$ implied by Maxwell's Equations (including sources $ρ$ and

\overset{⟶}{j}

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