Question

A dipole having dipole moment M is placed in two magnetic fields of strength B1 and B2 respectively. The dipole oscillates 60 times in 20 seconds in the \(B_1\) magnetic field and 60 oscillations in 30 seconds in the \(B_2\) magnetic field. Then find the \(\frac{B_1}{B_2}\)

• A. \(\frac{3}{2}\)
• B. \(\frac{2}{3}\)
• C. \(\frac{4}{9}\)
• D. \(\frac{9}{4}\)

Answer 1

The Correct Option is D

To solve the given problem, we need to determine the ratio \(\frac{B_1}{B_2}\) using the information about the oscillations of a magnetic dipole in two different magnetic fields.

The frequency of oscillation \(f\) of a magnetic dipole in a magnetic field \(B\) is given by the formula:

\(f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{\mu B}{I}}\)

where:

• \(\mu\) is the magnetic moment of the dipole.
• \(I\) is the moment of inertia of the dipole.

For the first scenario in the magnetic field \(B_1\):

• The dipole oscillates 60 times in 20 seconds. Thus, the frequency is:
• \(f_1 = \frac{60}{20} = 3\, \text{Hz}\)

Substituting into the frequency formula, we have:

\(\frac{1}{2\pi} \sqrt{\frac{\mu B_1}{I}} = 3\)

For the second scenario in the magnetic field \(B_2\):

• The dipole oscillates 60 times in 30 seconds. Thus, the frequency is:
• \(f_2 = \frac{60}{30} = 2\, \text{Hz}\)

Substituting into the frequency formula, we have:

\(\frac{1}{2\pi} \sqrt{\frac{\mu B_2}{I}} = 2\)

Divide the equation for \(f_1\) by the equation for \(f_2\):

\(\frac{3}{2} = \sqrt{\frac{B_1}{B_2}}\)

Squaring both sides to eliminate the square root:

\(\left(\frac{3}{2}\right)^2 = \frac{B_1}{B_2}\)

\(\frac{B_1}{B_2} = \frac{9}{4}\)

Thus, the correct answer is: \(\frac{9}{4}\), which matches the given correct option.