In Chapter 21 we showed that the nonuniform direction of the magnetic field of a magnet is responsible for the force F = I(2π R)B sin θ exerted by the magnet on a current loop. However, we didn't know the angle θ, which is a measure of the nonuniformity of the direction of the magnetic field (Figure 22.109). Gauss's law for magnetism can be used to determine this angle θ.
(a) Apply Gauss's law for magnetism to a thin disk of radius R and thickness Δ x located where the current loop will
be placed. Use Gauss's law to determine the component B3 sin θ of the magnetic field that is perpendicular to the
axis of the magnet at a radius R from the axis. Assume that R is small enough that the x component of magnetic
Field in Space field is approximately uniform everywhere on the left face of the disk and also on the right face of
the disk (but with a smaller value). The magnet has a magnetic dipole moment μ.
(b) Now imagine placing the current loop at this location, and show that F = I(2π R)B3 sin θ can be rewritten as F =
μloop|dB/dx|, which is the result we obtained from potential-energy arguments in Chapter 21.