A magnetic dipole of magnetic moment `6xx 10^(-2) A m^2` and moment of inertia `12 xx 10^(-6) kg m^2` performs oscillations in a magnetic field of `2 xx 10^(-2)` T. The time taken by the dipole to complete 20 oscillations is `(pi~~3)`

Options

A. 18 s
B. 6 s
C. 36 s
D. 12 s

Answer

To solve the problem, we need to find the time taken by a magnetic dipole to complete 20 oscillations in a magnetic field. We will use the formula for the time period of oscillation of a magnetic dipole in a magnetic field.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Magnetic moment, \( M = 6 \times 10^{-2} \, \text{A m}^2 \)
- Moment of inertia, \( I = 12 \times 10^{-6} \, \text{kg m}^2 \)
- Magnetic field strength, \( B = 2 \times 10^{-2} \, \text{T} \)

2. **Formula for Time Period (T):**
The time period \( T \) of oscillation for a magnetic dipole in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{MB}} \]
where \( M \) is the magnetic moment and \( B \) is the magnetic field strength.

3. **Calculate \( MB \):**
First, we need to calculate the product \( MB \):
\[ MB = M \times B = (6 \times 10^{-2}) \times (2 \times 10^{-2}) = 12 \times 10^{-4} \, \text{kg m}^2/\text{s}^2 \]

4. **Substitute Values into the Time Period Formula:**
Now, substitute the values of \( I \) and \( MB \) into the time period formula:
\[ T = 2\pi \sqrt{\frac{12 \times 10^{-6}}{12 \times 10^{-4}}} \]

5. **Simplify the Expression:**
The \( 12 \) in the numerator and denominator cancels out:
\[ T = 2\pi \sqrt{\frac{10^{-6}}{10^{-4}}} = 2\pi \sqrt{10^{-2}} = 2\pi \times 10^{-1} = \frac{2\pi}{10} = \frac{\pi}{5} \, \text{s} \]

6. **Calculate Time for 20 Oscillations:**
The time taken for 20 oscillations is:
\[ \text{Time for 20 oscillations} = 20 \times T = 20 \times \frac{\pi}{5} = 4\pi \, \text{s} \]

7. **Substituting the Value of \( \pi \):**
Given that \( \pi \approx 3 \):
\[ \text{Time for 20 oscillations} = 4 \times 3 = 12 \, \text{s} \]

### Final Answer:
The time taken by the dipole to complete 20 oscillations is \( 12 \, \text{s} \).

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