A bar magnet is often modeled as a magnetic dipole with one end labeled the north pole N...

A bar magnet is often modeled as a magnetic dipole with one end labeled the north pole N and the opposite end labeled the south pole S. The magnetic field for the magnetic dipole is symmetric with respect to rotation about the axis passing lengthwise through the center of the bar. Hence we can study the magnetic field by restricting ourselves to a plane with the bar magnet centered on the x-axis. For a point P that is located a distance r from the origin, where r is much greater than the length of the magnet, the magnetic field lines satisfy the differential

Equation

and the equipotential lines satisfy the equation

(a) Show that the two families of curves are perpendicular where they intersect. [Hint: Consider the slopes of the tangent lines of the two curves at a point of intersection.]

(b) Sketch the direction field for equation (4) for You can use a software package to generate the direction field or use the method of isoclines. The direction field should remind you of the experiment where iron filings are sprinkled on a sheet of paper that is held above a bar magnet. The iron filings correspond to the hash marks.

(c) Use the direction field found in part (b) to help sketch the magnetic field lines that are solutions to (4).

(d) Apply the statement of part (a) to the curves in part (c) to sketch the equipotential lines that are solutions to (5). The magnetic field lines and the equipotential lines are examples of orthogonal trajectories. (See Problem 32 in Exercises 2.4, pages 62–63.)†