]> 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 $j ⟶$ ).