The magnetic induction at a point P on the axis is equal to the magnetic induction at a point Q on the equator of a short magnetic dipole. What is the ratio of the distances of P and Q from the centre of the dipole?

Options

A. `2^(1//3)`
B. `3^(1//3)`
C. `2^(2//3)`
D. `4^(1//3)`

Explanation

To solve the problem, we need to find the ratio of the distances of points P and Q from the center of a short magnetic dipole, where the magnetic induction at point P (on the axis) is equal to the magnetic induction at point Q (on the equator).

### Step-by-Step Solution:

1. **Understand the Setup**:
- Let \( d_1 \) be the distance from the center of the dipole to point P (on the axis).
- Let \( d_2 \) be the distance from the center of the dipole to point Q (on the equator).

2. **Magnetic Field on the Axis**:
- The magnetic field \( B_A \) at a point on the axis of a short magnetic dipole is given by:
\[ B_A = \frac{\mu_0}{4\pi} \cdot \frac{2M}{d_1^3} \]
where \( M \) is the magnetic moment of the dipole.

3. **Magnetic Field on the Equator**:
- The magnetic field \( B_E \) at a point on the equator of a short magnetic dipole is given by:
\[ B_E = \frac{\mu_0}{4\pi} \cdot \frac{M}{d_2^3} \]

4. **Equating the Magnetic Fields**:
- According to the problem, the magnetic induction at point P is equal to that at point Q:
\[ B_A = B_E \]
- Therefore, we can write:
\[ \frac{\mu_0}{4\pi} \cdot \frac{2M}{d_1^3} = \frac{\mu_0}{4\pi} \cdot \frac{M}{d_2^3} \]

5. **Canceling Common Terms**:
- We can cancel \( \frac{\mu_0}{4\pi} \) and \( M \) from both sides:
\[ \frac{2}{d_1^3} = \frac{1}{d_2^3} \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ 2d_2^3 = d_1^3 \]

7. **Finding the Ratio**:
- Taking the cube root on both sides:
\[ \frac{d_1}{d_2} = \sqrt[3]{2} \]

8. **Final Answer**:
- Thus, the ratio of the distances \( \frac{d_1}{d_2} \) is:
\[ \frac{d_1}{d_2} = 2^{1/3} \]