To tackle this question, we need to understand the relationship between a magnet's magnetic moment and its time period of oscillation in a uniform magnetic field. The period of oscillation for a magnetic dipole in a magnetic field can be derived from the principles of torque and moment of inertia. Let's break this down step by step.
Understanding Magnetic Moments and Oscillation
The magnetic moment (M) of a magnet is a measure of its strength and orientation in a magnetic field. When a magnet is placed in a uniform magnetic field (B), it experiences a torque that tends to align it with the field. If displaced from this equilibrium position, the magnet will oscillate back and forth.
Formula for Time Period of Oscillation
The time period of oscillation (T) for a magnetic dipole in a uniform magnetic field can be calculated using the formula:
T = 2π √(I / (M * B))
Here, I represents the moment of inertia of the magnet, M is the magnetic moment, and B is the magnetic field strength. From this formula, we can see that the time period T is proportional to the square root of the moment of inertia divided by the product of the magnetic moment and the magnetic field strength.
Comparing Two Magnets
Now, we have two magnets. The first magnet has a magnetic moment M and oscillates with a time period T. The second magnet has the same dimensions (length and breadth) but a magnetic moment of M/4. Since the length and breadth are the same, we can assume that the moment of inertia (I) remains unchanged for both magnets.
Let’s denote the time period of the second magnet as T2. We can express T2 in terms of T:
T2 = 2π √(I / ((M/4) * B))
Rearranging this, we can see:
T2 = 2π √(4I / (M * B))
Notice that we have replaced M with M/4, which introduces a factor of 4 in the numerator. Therefore, the expression simplifies to:
T2 = 2 √(I / (M * B)) * 2π
Since T = 2π √(I / (M * B)), we can rewrite T2 as:
T2 = 2T
Final Answer
Thus, when comparing the oscillation time periods of both magnets, we find that the second magnet—having a magnetic moment of M/4—will oscillate with a time period of 2T. Therefore, the correct option is (b) 2T.
