Torque on a Current Carrying Loop Placed in a Magnetic Field

When a current-carrying loop is placed in a magnetic field, it experiences a torque that tends to rotate the loop. This phenomenon is fundamental in the operation of electric motors and measuring instruments like galvanometers.

Understanding the Basics

Magnetic Moment

The magnetic moment ((\vec{\mu})) of a current-carrying loop is a vector quantity that represents the magnetic strength and orientation of the loop. It is given by the product of the current (I) flowing through the loop and the area vector ((\vec{A})) of the loop:

[ \vec{\mu} = I \cdot \vec{A} ]

where (\vec{A}) is perpendicular to the plane of the loop and its magnitude is equal to the area of the loop.

Magnetic Field

A magnetic field ((\vec{B})) is a vector field that exerts a force on moving electric charges and magnetic dipoles. It is measured in teslas (T) in the SI system.

Torque on the Loop

The torque ((\vec{\tau})) experienced by a current-carrying loop in a uniform magnetic field is given by the cross product of the magnetic moment and the magnetic field:

[ \vec{\tau} = \vec{\mu} \times \vec{B} ]

In terms of magnitude, the torque can be expressed as:

[ \tau = \mu B \sin(\theta) ]

where (\theta) is the angle between the magnetic moment vector and the magnetic field vector.

Important Points

• The torque is maximum when the plane of the loop is perpendicular to the magnetic field ((\theta = 90^\circ)).
• The torque is zero when the plane of the loop is parallel to the magnetic field ((\theta = 0^\circ) or (180^\circ)).
• The direction of the torque is given by the right-hand rule: if the fingers of the right hand point in the direction of the current, and the thumb points in the direction of the magnetic field, then the palm faces the direction of the torque.

Table of Differences and Important Points

Property	Description
Magnetic Moment	Vector quantity representing the magnetic strength and orientation of the loop.
Magnetic Field	Vector field that exerts a force on moving charges and magnetic dipoles.
Torque	Vector quantity representing the rotational effect on the loop due to the magnetic field.
Angle ((\theta))	Determines the magnitude of the torque; maximum at (90^\circ), zero at (0^\circ) and (180^\circ).

Formulas

• Magnetic Moment: (\vec{\mu} = I \cdot \vec{A})
• Torque: (\vec{\tau} = \vec{\mu} \times \vec{B})
• Magnitude of Torque: (\tau = \mu B \sin(\theta))

Examples

Example 1: Maximum Torque

A square loop of side (l) carries a current (I) and is placed in a uniform magnetic field (B) such that the plane of the loop is perpendicular to the field. Calculate the torque on the loop.

Solution:

The area of the loop is (A = l^2), and the magnetic moment is (\mu = I \cdot A). Since the loop is perpendicular to the field, (\theta = 90^\circ), and (\sin(90^\circ) = 1).

The torque is:

[ \tau = \mu B \sin(90^\circ) = I \cdot l^2 \cdot B ]

Example 2: Zero Torque

A circular loop of radius (r) carries a current (I) and is placed in a uniform magnetic field (B) such that the plane of the loop is parallel to the field. Calculate the torque on the loop.

Solution:

The area of the loop is (A = \pi r^2), and the magnetic moment is (\mu = I \cdot A). Since the loop is parallel to the field, (\theta = 0^\circ), and (\sin(0^\circ) = 0).

The torque is:

[ \tau = \mu B \sin(0^\circ) = 0 ]

The loop experiences no torque in this orientation.

Understanding the torque on a current-carrying loop in a magnetic field is crucial for the design and operation of various electromagnetic devices. The principles outlined here form the basis for much of electromechanical engineering and are essential knowledge for students and professionals in the field.