Torque on Magnetic Dipole

Consider a rectangular loop of width $a$ and height $b$, carrying current $I$, placed in a uniform magnetic field $\vec{B}$ directed along the $x$-axis. The loop lies in the $xy$-plane initially, and its normal vector $\hat{n}$ points along the $z$-axis. The angle $\theta$ is measured between $\vec{\mu}$ and $\vec{B}$.

Using $\vec{F} = I\vec{L} \times \vec{B}$, the two sides of length $b$ (parallel to $\hat{y}$) carry currents in opposite directions. Side 1 experiences force $F_1 = IbB$ directed out of the page; Side 2 experiences $F_2 = IbB$ directed into the page. The two sides of length $a$ (parallel to $\hat{x}$, hence parallel to $\vec{B}$) experience zero force because $\hat{x} \times \hat{x} = 0$.

The two forces $F_1$ and $F_2$ form a couple. Each force has a moment arm of $\frac{a}{2}\sin\theta$ from the central axis of rotation. The net torque is therefore: $$\tau = F_1 \cdot \frac{a}{2}\sin\theta + F_2 \cdot \frac{a}{2}\sin\theta = IbB \cdot a\sin\theta = I(ab)B\sin\theta$$ Since $ab$ is the area $A$ of the loop, we get $\tau = IAB\sin\theta$.

We define the magnetic dipole moment as $\vec{\mu} = NIA\hat{n}$, where $N$ is the number of turns, $I$ is the current, $A$ is the area, and $\hat{n}$ is the unit normal given by the right-hand rule. For a single-turn loop, $|\vec{\mu}| = IA$. The torque equation becomes the elegant vector form: $$\vec{\tau} = \vec{\mu} \times \vec{B}$$ with magnitude $\tau = \mu B \sin\theta$. This result holds for any shape of loop, not just rectangles.

The torque is zero at $\theta = 0°$ (stable equilibrium, $\vec{\mu}$ parallel to $\vec{B}$) and at $\theta = 180°$ (unstable equilibrium). The associated potential energy is $U = -\vec{\mu} \cdot \vec{B} = -\mu B\cos\theta$. The system has minimum energy ($U = -\mu B$) when aligned and maximum energy ($U = +\mu B$) when anti-aligned — exactly analogous to an electric dipole in an electric field.

• \tau — Torque (N·m): The rotational force trying to align the loop with the magnetic field.
• \vec{\mu} — Magnetic dipole moment (A·m²): A vector perpendicular to the loop plane (right-hand rule), with magnitude $\mu = NIA$.
• N — Number of turns (dimensionless): How many times the wire is wound into a coil; multiplies the effective dipole moment.
• I — Current (A (amperes)): The electric current flowing through the loop.
• A — Area of loop (m²): The area enclosed by one turn of the current loop.
• B — Magnetic field strength (T (tesla)): The magnitude of the uniform external magnetic field.
• \theta — Angle (degrees or radians): The angle between the magnetic dipole moment vector $\vec{\mu}$ and the field $\vec{B}$.

Problem. A circular coil has 50 turns, a radius of 4.0 cm, and carries a current of 2.0 A. It is placed in a uniform magnetic field of magnitude 0.30 T. (a) Calculate the magnetic dipole moment of the coil. (b) Calculate the maximum torque the field can exert on the coil. (c) Calculate the torque when the plane of the coil makes a 30° angle with the field.

Answer. Magnetic dipole moment = 0.503 A·m²; maximum torque = 0.151 N·m; torque at 30° plane angle = 0.131 N·m.

• Magnetic Force on a Current-Carrying Conductor — The torque on a loop is built up from the individual forces on each current-carrying segment, making this the direct prerequisite concept.
• Magnetic Dipole Moment — The magnetic dipole moment μ = NIA is the key quantity that appears in the torque formula; understanding its definition and direction is essential.
• Potential Energy of a Magnetic Dipole — The torque formula integrates directly to the potential energy U = −μ·B, connecting force, torque, and energy for magnetic dipoles.
• Electric Motors and Galvanometers — Both devices operate by exploiting the torque on a current loop in a magnetic field, making this concept the theoretical heart of their design.
• Torque on an Electric Dipole in an Electric Field — The mathematics is identical: τ = p × E for electric dipoles, providing a powerful analogy that deepens understanding of both phenomena.