Magnetic fields at two points on the axis of a circular coil at a distance of `0.05m` and `0.2 m` from the centre are in the ratio `8:1`. The radius of the coil is

Options

A. `1.0m`
B. `0.1m`
C. `0.15m`
D. `0.2m`

Answer

To find the radius of the circular coil based on the given magnetic field ratios at two different points on its axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Magnetic Field Formula**:
The magnetic field \( B \) at a distance \( x \) from the center of a circular coil of radius \( r \) carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I r^2}{2 (r^2 + x^2)^{3/2}} \]
where \( \mu_0 \) is the permeability of free space.

2. **Define the Magnetic Fields at Two Points**:
Let \( B_1 \) be the magnetic field at a distance \( x_1 = 0.05 \, m \) and \( B_2 \) be the magnetic field at a distance \( x_2 = 0.2 \, m \).
- For \( B_1 \):
\[ B_1 = \frac{\mu_0 I r^2}{2 (r^2 + (0.05)^2)^{3/2}} \]
- For \( B_2 \):
\[ B_2 = \frac{\mu_0 I r^2}{2 (r^2 + (0.2)^2)^{3/2}} \]

3. **Set Up the Ratio**:
We are given that the ratio of the magnetic fields is \( \frac{B_1}{B_2} = \frac{8}{1} \):
\[ \frac{B_1}{B_2} = \frac{(r^2 + (0.2)^2)^{3/2}}{(r^2 + (0.05)^2)^{3/2}} = 8 \]

4. **Simplify the Ratio**:
Taking the cube root of both sides, we have:
\[ \frac{(r^2 + 0.04)}{(r^2 + 0.0025)} = 8^{2/3} = 4 \]

5. **Cross Multiply**:
Cross multiplying gives:
\[ r^2 + 0.04 = 4(r^2 + 0.0025) \]

6. **Expand and Rearrange**:
Expanding the right side:
\[ r^2 + 0.04 = 4r^2 + 0.01 \]
Rearranging gives:
\[ 0.04 - 0.01 = 4r^2 - r^2 \]
\[ 0.03 = 3r^2 \]

7. **Solve for \( r^2 \)**:
Dividing both sides by 3:
\[ r^2 = 0.01 \]

8. **Find the Radius**:
Taking the square root gives:
\[ r = \sqrt{0.01} = 0.1 \, m \]

### Final Answer:
The radius of the coil is \( 0.1 \, m \).