In Jackson section 5.16 he discusses different ways of calculating the energy in a magnetic field. One way (eq. 5.149) expresses it in terms of the magnetic vector potential and the current density: $$ W = \frac12 \int \mathbf{J}\cdot \mathbf{A} \: d^3 x $$ However, he does not discuss the situations in which this expression is correct, only stating "assuming a linear relation between $\mathbf{J}$ and $\mathbf{A}$". Is he assuming we pin down the vector potential with eq. (5.32)? $$ \mathbf{A}(\mathbf{x}) = \frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|} \: d^3x' $$ If we do a gauge transformation on the vector potential, does that render the energy expression incorrect? And how do we handle situations where the magnetic vector potential cannot be calculated by (5.32), as in the case of an infinite wire with uniform current density $\mathbf{J}$?
1 Answer
In the magnetostatic case, we know that $\nabla\cdot \vec{J} = 0$. Further, we assume that the currents are localized, so we may say that $\vec{J}\to 0$ at infinity. With these in mind, we can examine the effect of a gauge transformation.
$A\to A + \nabla f$.
\begin{align}
\int d^3 x\ \left(\vec{A} + \nabla f\right)\cdot \vec{J} &= \int d^3x \ \left[ \vec{A} \cdot \vec{J} + \nabla f \cdot \vec{J} \right] \\
&= \int d^3 x\ \left[ \vec{A} \cdot \vec{J} + \nabla \cdot \left( f \vec{J} \right) - f \nabla \cdot \vec{J} \right]\\
&= \int d^3 x \ \vec{A} \cdot \vec{J} + \oint_{S_\infty} dA\ f \vec{J} - \int d^3 x \ f \nabla \cdot \vec{J}
\end{align}
Now the second term (a surface integral at infinity) vanishes because of the locality of the currents and the third term vanishes because we are in a magnetostatic situation.
We are therefore left with exactly the original integral.
