How to evaluate the Lorentz force at a surface where the field is discontinuous?

Question

I'll take a simple case as an example. You have a constant and uniform magnetic field inside an ideal infinitely long solenoid, with currents circulating all around the thickless coils (so there's a surface current with a field discontinuity there). \begin{equation}\tag{1} \vec{B}_{\text{inside}} = \mu_0 \, n \, I \: \vec{z}, \end{equation} where $\vec{z}$ is the unit vector oriented along the solenoid's main axis. The field is 0 outside the solenoid (ideal case) : $\vec{B}_{\text{outside}} = 0$.

The magnetic field exerts a Lorentz force density $\vec{f} = \vec{J}_{\text{sol}} \times \vec{B}$ on the currents that are creating that same field, so there is magnetic pressure acting on the solenoid. That pressure should be proportional to the field's energy density.

Now, how do you evaluate the magnetic field, at the solenoid's surface, that should act on the current density $\vec{J}_{\text{sol}}$ ? $\vec{B}_{\text{inside}}$ given above ? $\vec{B}_{\text{outside}} = 0$ ? The average defined as this : \begin{equation}\tag{2} \vec{B}_{\text{average}} = \frac{\vec{B}_{\text{inside}} + \vec{B}_{\text{outside}}}{2} = \frac{1}{2} \, \mu_0 \, n \, I \: \vec{z} \quad ? \end{equation} or what else ? If it's the average (2), how can you justify it ?

Answer 1

It is indeed the average, as you suspect. To understand why, witness that the current is spread over a nonzero thickness. Begin with an azimuthal directed current density $J(r)$ (sketch this) and you get two equations:

$$\frac{\mathrm{d} \,B(r)}{\mathrm{d}\,r} = -\mu_0\,J(r)\quad\text{(}\vec{B}\text{ axially directed)}$$ $$\frac{\mathrm{d} \,F(r)}{\mathrm{d}\,r} = r\,\theta\,L\,J(r)\,B(r)\quad\text{(Lorentz force on sector of azimuthal subtense}\,\theta,\,\text{length}L)$$

Eliminate $J$ to find that $B$ varies with $F$ according to $\frac{\mathrm{d} \,F}{\mathrm{d}\,B} = -r/\mu_0$; this simple equation shows you very simply that the force on an infinitely thin current sheet is calculated with the average of the magnetic fields inside and outside the coil.

The same principle works in calculating the force exerted by a plane wave on a perfect conductor when the former is reflected or absorbed by the latter. You get the same answer whether you use half the magnetic field at the on the outer surface acting on the current sheet, or whether you do the full calculation with exponentially dwindling fields with depth owing to the skin effect.