Question

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

Hint:

The torque on a current loop in a magnetic field is maximum when the magnetic moment is perpendicular to the magnetic field. It tends to align the loop with the magnetic field.

Answer 1

The torque \( \tau \) acting on a current loop in a magnetic field is given by: \[ \tau = \vec{m} \times \vec{B} \] where \( \vec{m} \) is the magnetic moment of the loop, and \( \vec{B} \) is the magnetic field. The magnetic moment \( \vec{m} \) is given by: \[ \vec{m} = I A \hat{n} \] where \( I \) is the current, \( A \) is the area of the loop, and \( \hat{n} \) is the unit vector normal to the plane of the loop. The magnitude of the torque is: \[ \tau = m B \sin \theta \] where \( \theta \) is the angle between the magnetic moment and the magnetic field. Substituting for \( m \), we get: \[ \tau = I A B \sin \theta \] Thus, the expression for the torque acting on the rectangular current loop is: \[ \tau = I A B \sin \theta \]

Answer 2

1. Torque on a Current Loop in a Magnetic Field: 

Consider a rectangular current loop of length \( l \) and width \( w \) carrying a current \( I \). The loop is placed in a uniform magnetic field \( \vec{B} \), and we are to derive an expression for the torque acting on the loop.

2. Magnetic Force on Each Segment:

Let the magnetic field \( \vec{B} \) be uniform and directed along the \( x \)-axis, and the current loop is lying in the \( yz \)-plane. The force on a current-carrying segment in a magnetic field is given by the Lorentz force law:

\[ \vec{F} = I \left( \vec{l} \times \vec{B} \right) \]

Where:

• \( \vec{l} \) is the length vector of the segment (magnitude of the segment and direction),
• \( \vec{B} \) is the magnetic field vector, and
• \( I \) is the current.

 

For a rectangular loop, there are four segments. The force on each segment will create a torque that acts about the center of the loop. The total torque is the sum of the individual torques from each segment of the loop.

3. Torque on the Loop:

The torque \( \tau \) acting on the loop is given by the cross product of the position vector \( \vec{r} \) (measured from the axis of rotation) and the force \( \vec{F} \) acting on the segment:

\[ \vec{\tau} = \vec{r} \times \vec{F} \]

4. Net Torque on the Current Loop:

The total torque acting on the rectangular current loop can be expressed as:

\[ \tau = I \cdot A \cdot B \cdot \sin \theta \]

Where:

• \( A = l \times w \) is the area of the rectangular loop,
• \( \theta \) is the angle between the magnetic field and the normal to the plane of the loop (which is the axis of the loop),
• \( B \) is the magnetic field strength.

 

Therefore, the torque is the product of the current, the area of the loop, the magnetic field strength, and the sine of the angle between the magnetic field and the normal to the loop.

5. Conclusion:

• The torque acting on a rectangular current loop suspended in a uniform magnetic field is:
• \( \tau = I \cdot A \cdot B \cdot \sin \theta \),
• Where \( I \) is the current, \( A \) is the area of the loop, \( B \) is the magnetic field, and \( \theta \) is the angle between the normal to the loop and the magnetic field.