If a specific example will help: Suppose I have a thin cylinder-shaped piece of iron of length L and radius r constrained to move in z in a field that is dominantly in z: \vec{B} \approx (0,0,B_0/z^3). This is kind of like a dipole field far from the dipole. The field in the x and y directions is not exactly zero here - that doesn't follow Maxwell's Equations, but this doesn't matter because of the symmetry and the constraint:
Call the point the center of the cylinder (0,0,z
0). The energy removed (in appropriate units) is
B_0^2 \pi r^2 \int_{z_0 - L/2}^{z_0 + L/2} dz/z^6 =<br />
-5B_0 ^2\pi ^2 \left[ \frac{1}{(z_0+L/2)^5} - \frac{1}{(z_0-L/2)^5}\right]
\approx 5B_0^2 \pi^2 \frac{L/2}{z_0^6} = \frac{5V B_0^2}{2z_0^6}
Turning to the force (and replacing z
0 with z for clarity):
F = \frac{\partial}{\partial z} \frac{5VB_0^2}{2z^6} = \frac{5VB_0^2}{12z^7}
I'm sure I did some screwing up here, but the point is that this is calculable, but even in an oversimplified case, it's far from trivial. In reality the dependence is not this strong - it's only the 6th power of z. The reason is that if I had written a field that satisfied Maxwell's equations the amount of field eliminated is smaller. The field lines that were along z turn in x and y - they don't just disappear.
But the two points remain:
- Think energetics and not forces; if you want forces, derive them from energies.
- An actual calculation is in no way B-level.