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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 ⟶ ).

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