Physics · Ch 3 — Magnetism and Magnetic Effects of Electric Current

Potential Energy of a Bar Magnet in a Uniform Magnetic Field

3.3.1

Potential Energy of a Bar Magnet in a Uniform Magnetic Field

To rotate a dipole from angle $\theta'$ to $\theta$ against the restoring torque $\tau_B = p_mB\sin\theta$ , an external torque must do work $dW = \tau_B\,d\theta = p_mB\sin\theta\,d\theta$ for each small angular step. The total work done, which is stored as potential energy, is

W = \int_{\theta'}^{\theta} p_mB\sin\theta\,d\theta = p_mB(\cos\theta' - \cos\theta)

Choosing the reference orientation $\theta' = 90°$ (so $\cos\theta'=0$ ) gives the standard formula for the potential energy of a magnetic dipole in a uniform field:

$\boxed{U = -p_mB\cos\theta = -\vec p_m \cdot \vec B}$ …

Figure 3.17A bar magnet (magnetic dipole) in a uniform magnetic field

What this figure shows. A bar magnet of magnetic length 2l is drawn tilted at angle theta from the direction of a uniform field B, with the perpendicular separation between the two poles' lines of action marked as 2l sin(theta) -- the same moment-arm geometry used to compute both the torque and, by integrating that torque over angle, the potential energy stored as the magnet is rotated away from alignment wi …

Misc Example 3.7Energy of the parallel and anti-parallel orientations

Worked out. Before an external field is switched on, a dipole has no preferred orientation and its energy is taken as U=0. Once the field B is switched on, if the dipole settles parallel to B (theta=0), its energy is U_parallel = -p_m B cos(0) = -p_m B -- this is the minimum possible energy, confirming this is the stable orientation. If instead it were forced anti-parallel (theta=180 degrees), its energy would be U_antiparallel = -p_m B cos(180) = +p_m B, the maximum possible energy and hence the least stable orientation -- consistent with a compass needle always settling parallel …

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