Magnetic field`B_(1)` due to abar magnet at a point P on axial line is equal to magnetic field `B_(2)` due to the same magnet at point Q on equatorial line.What ios the ratio distances of point P and Q from centre?

Options

A. `2^(-1//3)`
B. 2
C. `2^(1//3)`
D. `1//2`

Explanation

To solve the problem, we need to find the ratio of distances \(D_1\) (distance from the center of the bar magnet to point P on the axial line) and \(D_2\) (distance from the center of the bar magnet to point Q on the equatorial line) given that the magnetic fields \(B_1\) and \(B_2\) at these points are equal.

### Step-by-Step Solution:

1. **Understand the Magnetic Field Formulas**:
- The magnetic field \(B_1\) at a point on the axial line of a bar magnet is given by the formula:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{D_1^3} \]
- The magnetic field \(B_2\) at a point on the equatorial line of a bar magnet is given by:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{m}{D_2^3} \]

2. **Set the Magnetic Fields Equal**:
- According to the problem, we have:
\[ B_1 = B_2 \]
- Substituting the formulas for \(B_1\) and \(B_2\):
\[ \frac{\mu_0}{4\pi} \cdot \frac{2m}{D_1^3} = \frac{\mu_0}{4\pi} \cdot \frac{m}{D_2^3} \]

3. **Cancel Common Terms**:
- We can cancel \(\frac{\mu_0}{4\pi}\) and \(m\) from both sides:
\[ \frac{2}{D_1^3} = \frac{1}{D_2^3} \]

4. **Cross-Multiply**:
- Cross-multiplying gives:
\[ 2D_2^3 = D_1^3 \]

5. **Rearranging the Equation**:
- Rearranging the equation to find the ratio \(D_1\) to \(D_2\):
\[ \frac{D_1^3}{D_2^3} = 2 \]

6. **Taking the Cube Root**:
- Taking the cube root of both sides:
\[ \frac{D_1}{D_2} = 2^{1/3} \]

### Final Result:
The ratio of distances \(D_1\) and \(D_2\) is:
\[ \frac{D_1}{D_2} = 2^{1/3} \]