The period of oscillations of a magnet is 2 sec. When it is remagnetised so that the pole strength is 4 times its period will be

Options

A. (a)1 sec
B. (b)2 sec
C. (c)4 sec
D. (d)8 sec

Answer

To solve the problem, we need to understand the relationship between the period of oscillation of a magnet and its pole strength.

### Step-by-Step Solution:

1. **Understand the Formula for Time Period (T):**
The time period of oscillation (T) of a magnet is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M \cdot B}} \]
where:
- \(I\) = moment of inertia,
- \(M\) = magnetic moment,
- \(B\) = magnetic field.

2. **Relationship Between Time Period and Magnetic Moment:**
The time period \(T\) is inversely proportional to the square root of the magnetic moment \(M\):
\[ T \propto \frac{1}{\sqrt{M}} \]

3. **Expression for Magnetic Moment:**
The magnetic moment \(M\) can be expressed in terms of pole strength \(m\) and length \(l\):
\[ M = m \cdot l \]
Thus, if the pole strength is increased, the magnetic moment will also increase.

4. **Change in Pole Strength:**
If the pole strength becomes 4 times its original value, we can denote the new pole strength as:
\[ m' = 4m \]

5. **New Magnetic Moment:**
The new magnetic moment \(M'\) will be:
\[ M' = m' \cdot l = 4m \cdot l = 4M \]

6. **New Time Period Calculation:**
Since the time period is inversely proportional to the square root of the magnetic moment, we can write:
\[ T' \propto \frac{1}{\sqrt{M'}} \]
Substituting \(M' = 4M\):
\[ T' \propto \frac{1}{\sqrt{4M}} = \frac{1}{2\sqrt{M}} \]

7. **Relating New Time Period to Original Time Period:**
Since \(T \propto \frac{1}{\sqrt{M}}\), we can express the new time period \(T'\) in terms of the original time period \(T\):
\[ T' = \frac{T}{2} \]

8. **Substituting the Original Time Period:**
Given that the original time period \(T\) is 2 seconds:
\[ T' = \frac{2 \text{ seconds}}{2} = 1 \text{ second} \]

### Conclusion:
The new period of oscillation when the pole strength is increased to 4 times its original value will be **1 second**.