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

The time period of oscillation of a dipole in a magnetic field is given by:
\(T = 2π√(\frac{I}{MB})\)
where I is the moment of inertia of the dipole about its axis of rotation.
For a given dipole, the moment of inertia and the dipole moment are constant. Therefore, the time period of oscillation is directly proportional to the square root of the magnetic field strength.
Let the time period of oscillation in the B₁ magnetic field be T₁ and in the B₂ magnetic field be T₂. Then:
\(\frac{T_1}{T_2} = √(\frac{B_2}{B_1})\)
60 oscillations in 20 seconds in the B₁ magnetic field gives:
\(T_1 = \frac{20}{60} = \frac{1}{3} seconds\)
60 oscillations in 30 seconds in the B₂ magnetic field gives:
\(T_2 = \frac{30}{60} = \frac{1}{2} seconds\)
Substituting these values in the above equation, we get:
\(\frac{\frac{1}{3} }{\frac{1}{2}} = √\frac{(B_2)}{(B_1)}\)
\(\frac{2}{3} = \sqrt{\frac{B_2}{B_1}}\)
Squaring both sides, we get:
\(\frac{4}{9} = \frac{B_2}{B_1}\)
Therefore, \( \frac{B_1}{B_2}=\frac{9}{4}\).
Answer. D

Answer 2