Torque Acting on a Magnetic Dipole in a Uniform Magnetic Field

The product of pole strength and magnetic length of a magnetic dipole, which represents the strength of a magnet, is called its magnetic dipole moment.

The magnetic moment of an electron revolving around the nucleus of an atom, arising due to its orbital motion similar to a tiny current loop, is called the orbital magnetic moment of the electron.

The work done in rotating a magnetic dipole against the action of the torque in a uniform magnetic field, which is stored in the dipole, is called the magnetic potential energy of the dipole.

• Scalar form: τ = mB sin ⁡θ
• Vector form (rectangular current-carrying coil in uniform field): \[\vec τ\] = \[\vec M\] × \[\vec B\]

T = \[\frac {2π}{ω}\] = 2π\[\sqrt {\frac {1}{mB}}\]

A magnetic dipole placed in a uniform magnetic field experiences a torque given by τ = mB sin ⁡θ, or in vector form \[\vec τ\] = \[\vec M\] × \[\vec B\], where m is the magnetic dipole moment, B is the magnetic field, and θ is the angle between them. This torque tends to align the magnetic dipole moment vector with the magnetic field vector. It is responsible for various phenomena such as the behavior of compass needles aligning with Earth's magnetic field and the operation of electric motors based on the interaction between a magnetic field and current-carrying wires.

If a magnetic dipole is rotated against the action of the torque in a uniform magnetic field, work has to be done. This work is stored as the potential energy of the dipole and is given by U = −\[\vec M\] . \[\vec B\] = −mB cos ⁡θ.
Special cases:

• At θ = 0° (M parallel to B): U_min = −mB → bar magnet is in stable equilibrium with minimum potential energy.
• At θ = 90° (M perpendicular to B): U = 0 → neutral, zero potential energy.
• At θ = 180° (M antiparallel to B): U_max = +mB → bar magnet is in the most unstable state with maximum potential energy.

When a freely suspended magnetic dipole in a uniform magnetic field is slightly displaced from its equilibrium position, it performs angular oscillations with time period T = \[\frac {2π}{ω}\] = 2π\[\sqrt {\frac {I}{mB}}\], where I is the moment of inertia of the dipole, m is its magnetic dipole moment and B is the magnetic field strength.