Home  18.013A  Chapter 29 


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 $\stackrel{\u27f6}{B}$ , has vanishing divergence, can be written as the curl of a vector potential in a similar way.
We define the vector potential $\stackrel{\u27f6}{A}$ so that $\stackrel{\u27f6}{\nabla}\times \stackrel{\u27f6}{A}=\stackrel{\u27f6}{B}$ .
With this definition $\stackrel{\u27f6}{\nabla}\xb7\stackrel{\u27f6}{A}$ can be anything without changing anything.
In the case of static currents where there is no time dependence we set $\stackrel{\u27f6}{\nabla}\xb7\stackrel{\u27f6}{A}=0$ and deduce the equation
We can solve this equation in all of space with the boundary condition that $\stackrel{\u27f6}{A}$ go to $\stackrel{\u27f6}{0}$ 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 $\stackrel{\u27f6}{A}$ by
These definitions do not determine $\stackrel{\u27f6}{A}$ and $V$ completely.
Given any scalar field $f$ , we can add $\stackrel{\u27f6}{\nabla}f$ to $\stackrel{\u27f6}{A}$ and $\frac{1}{c}\frac{\partial f}{\partial t}$ to $V$ and neither $\stackrel{\u27f6}{B}$ nor $\stackrel{\u27f6}{E}$ will change at all. Such a change is called a "change of gauge", and these expressions for $\stackrel{\u27f6}{B}$ and $\stackrel{\u27f6}{E}$ are said to be "gauge invariant" because they are unaffected by changes in gauge.
Exercise 29.4 Find the equations satisfied by $\stackrel{\u27f6}{A}$ and $V$ implied by Maxwell's Equations (including sources $\rho $ and $\stackrel{\u27f6}{j}$ ).
