Torque on Current Loop — Core Principles

Core Principles

A current-carrying loop placed in an external magnetic field experiences a torque. This torque arises because the forces acting on different segments of the loop, due to the magnetic field, are generally not collinear, even though the net force on the loop in a uniform magnetic field is zero.

The magnitude of the torque ($\tau$) on a loop with $N$ turns, carrying current $I$, enclosing area $A$, and placed in a magnetic field $B$, is given by $\tau = NIAB \sin\theta$. Here, $\theta$ is the angle between the normal to the plane of the loop (which defines the magnetic dipole moment $\vec{M}$) and the magnetic field $\vec{B}$.

The magnetic dipole moment is defined as $\vec{M} = NIA \hat{n}$, where $\hat{n}$ is the unit vector normal to the loop's plane. In vector form, the torque is $\vec{\tau} = \vec{M} \times \vec{B}$. The torque tends to align the magnetic dipole moment $\vec{M}$ with the magnetic field $\vec{B}$.

Maximum torque occurs when the plane of the loop is parallel to the field ($\theta = 90^\circ$), and zero torque occurs when the plane is perpendicular to the field ($\theta = 0^\circ$ or $180^\circ$).

This phenomenon is fundamental to the operation of electric motors and galvanometers.

Important Differences

vs Force on a Current Loop

Aspect	This Topic	Force on a Current Loop
Definition	Torque on a Current Loop: The turning effect experienced by a current-carrying loop in a magnetic field, tending to rotate it.	Force on a Current Loop: The net translational push or pull experienced by a current-carrying loop in a magnetic field.
Cause	Caused by non-collinear forces acting on different segments of the loop.	Caused by the Lorentz force acting on individual charge carriers within the conductor segments.
Uniform Magnetic Field	Can be non-zero. $\tau = NIAB \sin\theta$.	Always zero. $\vec{F}_{net} = 0$ for a closed loop in a uniform field.
Non-uniform Magnetic Field	Can be non-zero.	Can be non-zero. A net force can exist if the field varies across the loop.
Effect	Causes rotation (angular acceleration).	Causes translation (linear acceleration).
Vector Representation	$\vec{\tau} = \vec{M} \times \vec{B}$	$\vec{F} = I \oint (d\vec{l} \times \vec{B})$ (which is zero for uniform B)
Alignment	Tends to align the magnetic dipole moment $\vec{M}$ with $\vec{B}$.	If non-zero, tends to move the loop towards regions of stronger magnetic field (for paramagnetic materials) or weaker field (for diamagnetic materials).

While both force and torque originate from the Lorentz force, their manifestations on a current loop differ significantly, especially in a uniform magnetic field. A current loop in a uniform magnetic field experiences a net force of zero, meaning it won't undergo translational motion. However, the forces on its various segments can form a couple, leading to a net torque that causes rotational motion. This torque aims to align the loop's magnetic dipole moment with the external magnetic field. In contrast, in a non-uniform magnetic field, a current loop can experience both a net force and a net torque.