The magnetic induction at a point P on the axis is 16 times the magnetic induction at a point Q on the equator of a short magnetic dipole. The point P is at a distance of 10 cm from the centre of the dipole, what is the distance of the point Q from its centre?

Options

A. 12.5 cm
B. 15 cm
C. 17.5 cm
D. 20 cm

Explanation

To solve the problem, we need to understand the relationship between the magnetic induction at points on the axis and equator of a short magnetic dipole.

### Step-by-Step Solution:

1. **Understanding Magnetic Induction**:
- The magnetic induction \( B \) at a point on the axis of a short magnetic dipole is given by the formula:
\[ B_{\text{axis}} = \frac{\mu_0}{4\pi} \cdot \frac{2m}{r^3} \]
where \( m \) is the magnetic moment of the dipole and \( r \) is the distance from the center of the dipole.

- The magnetic induction \( B \) at a point on the equator of the dipole is given by the formula:
\[ B_{\text{equator}} = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3} \]

2. **Setting Up the Relationship**:
- According to the problem, the magnetic induction at point P on the axis is 16 times that at point Q on the equator:
\[ B_{\text{axis}} = 16 B_{\text{equator}} \]

3. **Substituting the Formulas**:
- Substitute the formulas for \( B_{\text{axis}} \) and \( B_{\text{equator}} \):
\[ \frac{\mu_0}{4\pi} \cdot \frac{2m}{(10 \text{ cm})^3} = 16 \left( \frac{\mu_0}{4\pi} \cdot \frac{m}{r_Q^3} \right) \]
- Here, \( r_Q \) is the distance from the center of the dipole to point Q.

4. **Simplifying the Equation**:
- Cancel out the common terms \( \frac{\mu_0}{4\pi} \) and \( m \) (assuming \( m \neq 0 \)):
\[ \frac{2}{(10)^3} = 16 \cdot \frac{1}{r_Q^3} \]
- This simplifies to:
\[ \frac{2}{1000} = \frac{16}{r_Q^3} \]

5. **Cross-Multiplying**:
- Cross-multiply to solve for \( r_Q^3 \):
\[ 2 r_Q^3 = 16 \cdot 1000 \]
\[ 2 r_Q^3 = 16000 \]
\[ r_Q^3 = 8000 \]

6. **Finding \( r_Q \)**:
- Take the cube root of both sides:
\[ r_Q = \sqrt[3]{8000} = 20 \text{ cm} \]

### Final Answer:
The distance of point Q from the center of the dipole is **20 cm**.