Points A and B are located along a line perpendicular to the axis of a small bar magnet. Line joining the points A and B is passing through the centre of bar magnet. Point A is at a distance x from the centre of magnet on one side and point B is on the other side of magnet at a distance 3x from the centre of magnet. If `B_1` and `B_2` are the magnetic field intensities at A and B respectively then find approximate value of `B_1 :B_2`

Options

A. `1:9`
B. `9:1`
C. `1:27`
D. `27:1`

Explanation

To solve the problem, we need to find the ratio of the magnetic field intensities \( B_1 \) at point A and \( B_2 \) at point B, given their distances from the center of a small bar magnet.

### Step-by-Step Solution:

1. **Identify the distances**:
- Point A is at a distance \( x \) from the center of the magnet.
- Point B is at a distance \( 3x \) from the center of the magnet.

2. **Understand the formula for magnetic field intensity**:
The magnetic field intensity \( B \) at a point along the perpendicular bisector of a small bar magnet is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{m}{d^3} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( m \) is the magnetic moment of the magnet,
- \( d \) is the distance from the center of the magnet to the point where the field is being calculated.

3. **Calculate \( B_1 \) at point A**:
For point A, the distance \( d_1 = x \):
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{m}{x^3} \]

4. **Calculate \( B_2 \) at point B**:
For point B, the distance \( d_2 = 3x \):
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{m}{(3x)^3} = \frac{\mu_0}{4\pi} \cdot \frac{m}{27x^3} \]

5. **Find the ratio \( \frac{B_1}{B_2} \)**:
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0}{4\pi} \cdot \frac{m}{x^3}}{\frac{\mu_0}{4\pi} \cdot \frac{m}{27x^3}} = \frac{27x^3}{x^3} = 27 \]

6. **Final result**:
The ratio of the magnetic field intensities at points A and B is:
\[ B_1 : B_2 = 27 : 1 \]