At a place where angle of dip is `45^(@)`, a magnet oscillates with 30 oscillations per minute. At another place where angle of dip is `60^(@)`, the same magnet makes 20 oscillations per minute The ratio of total magnetic fields at the two places is

Options

A. `3sqrt(2) : 2`
B. `9:4`
C. `4:9`
D. `9sqrt(2) : 4`

Answer

To solve the problem, we need to find the ratio of the total magnetic fields at two places where a magnet oscillates at different frequencies due to different angles of dip.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two locations with different angles of dip (θ1 = 45° and θ2 = 60°) and corresponding frequencies of oscillation (f1 = 30 oscillations/min and f2 = 20 oscillations/min). We need to find the ratio of the total magnetic fields (B1 and B2) at these two locations.

2. **Using the Formula for Frequency of Oscillation**:
The frequency of oscillation of a magnet in a magnetic field is given by:
\[ f = \frac{1}{2\pi} \sqrt{\frac{B \cos \theta}{I}} \]
where:
- \( f \) is the frequency of oscillation,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle of dip,
- \( I \) is the moment of inertia of the magnet.

3. **Setting Up the Equations**:
For the first location (θ1 = 45°):
\[ f_1 = \frac{1}{2\pi} \sqrt{\frac{B_1 \cos 45°}{I}} \]
For the second location (θ2 = 60°):
\[ f_2 = \frac{1}{2\pi} \sqrt{\frac{B_2 \cos 60°}{I}} \]

4. **Substituting the Values**:
We know:
- \( f_1 = 30 \) oscillations/min,
- \( f_2 = 20 \) oscillations/min,
- \( \cos 45° = \frac{1}{\sqrt{2}} \),
- \( \cos 60° = \frac{1}{2} \).

5. **Dividing the Equations**:
Dividing the first equation by the second:
\[ \frac{f_1}{f_2} = \frac{\sqrt{B_1 \cos 45°}}{\sqrt{B_2 \cos 60°}} \]

6. **Substituting the Frequencies**:
\[ \frac{30}{20} = \frac{\sqrt{B_1 \cdot \frac{1}{\sqrt{2}}}}{\sqrt{B_2 \cdot \frac{1}{2}}} \]
Simplifying gives:
\[ \frac{3}{2} = \frac{\sqrt{B_1 \cdot \frac{1}{\sqrt{2}}}}{\sqrt{B_2 \cdot \frac{1}{2}}} \]

7. **Squaring Both Sides**:
\[ \left(\frac{3}{2}\right)^2 = \frac{B_1 \cdot \frac{1}{\sqrt{2}}}{B_2 \cdot \frac{1}{2}} \]
\[ \frac{9}{4} = \frac{B_1 \cdot 2}{B_2 \cdot \sqrt{2}} \]

8. **Rearranging the Equation**:
\[ \frac{B_1}{B_2} = \frac{9 \cdot \sqrt{2}}{8} \]

9. **Final Ratio**:
Thus, the ratio of the total magnetic fields at the two places is:
\[ \frac{B_1}{B_2} = \frac{9\sqrt{2}}{8} \]