Question

Derive an expression for the torque acting on a rectangular current loop suspended in a uniform magnetic field.

Hint:

Torque on a current loop in a magnetic field is maximum when the angle between the magnetic moment and the field is 90 degrees.

Answer 1

Torque on a Rectangular Current Loop in a Uniform Magnetic Field 

Considerations: 

• A rectangular loop of length \( l \) and breadth \( b \)
• Current \( I \) flows through the loop
• The loop is placed in a uniform magnetic field \( \vec{B} \)
• The angle between the magnetic field and the normal to the plane of the loop is \( \theta \)

Step-by-Step Derivation:

Let the rectangular loop be placed in such a way that its plane makes an angle \( \theta \) with the direction of the magnetic field \( \vec{B} \).

Magnetic force on a current-carrying conductor of length \( \vec{l} \) is given by:
\( \vec{F} = I (\vec{l} \times \vec{B}) \)

In a rectangular loop, the opposite sides experience equal and opposite forces, but these forces do not act along the same line. Hence, they form a couple which produces a torque.

Magnitude of Torque:

Let the area of the rectangular loop be:
\( A = l \times b \)

Torque \( \tau \) is given by:
\( \tau = IAB \sin \theta \)

Where:

• \( I \) is the current
• \( A \) is the area of the loop
• \( B \) is the magnetic field strength
• \( \theta \) is the angle between normal to the loop and the magnetic field

 

Vector Form:

Define the magnetic moment \( \vec{m} \) of the loop as:
\( \vec{m} = I \vec{A} \)
(Direction of \( \vec{A} \) is given by the right-hand rule perpendicular to the plane of the loop)

Then, torque in vector form is:
\( \vec{\tau} = \vec{m} \times \vec{B} \)

Conclusion:

The torque acting on a rectangular current loop placed in a uniform magnetic field is:
\( \tau = IAB \sin \theta \), and in vector form, \( \vec{\tau} = \vec{m} \times \vec{B} \)