Question

A circular current loop is placed in an external magnetic field. How is the torque related to the radius of the loop?

Answer 1

Torque is directly proportional to r².

Given:

• Circular loop is placed in an external magnetic field.

To find:

Relationship between torque and radius of loop?

Calculation:

• Let the magnetic field intensity be B, the current flowing through the loop be I and the radius of loop be r.

• Let angle between area vector of loop and magnetic field intensity be $\theta$.

Now, the net torque is given as :

$\vec{ \tau} = \vec{M} \times \vec{B}$

$\implies | \vec{ \tau} | = M \times B \times \sin( \theta)$

• M is the magnetic moment of the loop.

$\implies | \vec{ \tau} | = (I \times area) \times B \times \sin( \theta)$

$\implies | \vec{ \tau} | = (I \times \pi {r}^{2} ) \times B \times \sin( \theta)$

• For constant I , B and $\theta$, we can say that:

$\implies | \vec{ \tau} | \propto {r}^{2}$

So, the torque is directly proportional to square of radius of loop.