Torque on Current Loop, Magnetic Dipole

Torque on a Current Loop: The Core Idea

A rectangular loop carrying current, placed in a uniform magnetic field, experiences a net torque that tends to rotate it. This torque arises because the magnetic forces on opposite sides of the loop are equal in magnitude but opposite in direction, forming a couple (a pair of forces that produce rotation but no net translation).

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Step-by-Step Derivation

Consider a rectangular loop $PQRS$ of length $l$ and breadth $b$, carrying a steady current $I$. It is placed in a uniform magnetic field $\mathbf{B}$. Let the plane of the loop make an angle $\theta$ with the direction of $\mathbf{B}$.

1. Forces on the arms:

   • On arms $PQ$ and $RS$ (length $l$), the current is perpendicular to $\mathbf{B}$. The force on each is $F = I l B$, directed perpendicular to both the current and $\mathbf{B}$. These forces are equal, opposite, and parallel — they form a couple.
   • On arms $SP$ and $QR$ (breadth $b$), the current is parallel (or anti-parallel) to $\mathbf{B}$. The force on each is zero ($F = I b B \sin 0^\circ = 0$).

2. Lever arm of the couple:

   • The perpendicular distance between the two forces $F$ (on $PQ$ and $RS$) is $b \sin \theta$.
   • The torque $\tau$ due to the couple is:

3. Final expression for torque:
   • Since area $A = l \times b$, we get:

*   This is the magnitude of the torque. The direction is such that it rotates the loop to align its plane perpendicular to $\mathbf{B}$ (i.e., to make $\theta = 90^\circ$).

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Magnetic Dipole Moment

The torque expression is analogous to the torque on an electric dipole in an electric field ($\tau = pE \sin \theta$). This motivates defining a magnetic dipole moment $\mathbf{m}$ for the current loop.

• Definition: The magnetic dipole moment of a current loop is: $\mathbf{m} = I A \, \hat{\mathbf{n}}$ …