The magnetic induction at the centre of a circular coil of radius 10 cm is `5 sqrt(5)` times the magnetic induction at a point on its axis. The distance of the point from the centre of the coil, in cm is:

Options

A. 5
B. 10
C. 20
D. 25

Answer

To solve the problem, we need to find the distance \( t \) from the center of a circular coil where the magnetic induction is given in relation to the magnetic induction at the center of the coil. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field at the Center and on the Axis:**
- The magnetic induction (magnetic field) at the center of a circular coil of radius \( r \) carrying current \( i \) is given by:
\[ B_{\text{center}} = \frac{\mu_0 i}{2r} \]
- The magnetic induction at a point on the axis at a distance \( x \) from the center is given by:
\[ B_{\text{axis}} = \frac{\mu_0 i r^2}{2(r^2 + x^2)^{3/2}} \]

2. **Setting Up the Relationship:**
- According to the problem, the magnetic induction at the center is \( 5\sqrt{5} \) times the magnetic induction at a point on its axis:
\[ B_{\text{center}} = 5\sqrt{5} \cdot B_{\text{axis}} \]

3. **Substituting the Expressions:**
- Substituting the expressions for \( B_{\text{center}} \) and \( B_{\text{axis}} \):
\[ \frac{\mu_0 i}{2r} = 5\sqrt{5} \cdot \frac{\mu_0 i r^2}{2(r^2 + t^2)^{3/2}} \]

4. **Canceling Common Terms:**
- Cancel \( \mu_0 i \) from both sides:
\[ \frac{1}{2r} = 5\sqrt{5} \cdot \frac{r^2}{2(r^2 + t^2)^{3/2}} \]

5. **Simplifying the Equation:**
- Multiply both sides by \( 2r(r^2 + t^2)^{3/2} \):
\[ (r^2 + t^2)^{3/2} = 5\sqrt{5} r^3 \]

6. **Cubing Both Sides:**
- Cube both sides to eliminate the exponent:
\[ r^2 + t^2 = (5\sqrt{5} r^{3/2})^2 \]
\[ r^2 + t^2 = 125 r^2 \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ t^2 = 125r^2 - r^2 = 124r^2 \]

8. **Taking the Square Root:**
- Taking the square root of both sides:
\[ t = \sqrt{124} r = \sqrt{124} \cdot 10 \text{ cm} \]

9. **Calculating the Final Distance:**
- Since \( \sqrt{124} = 2\sqrt{31} \):
\[ t = 2\sqrt{31} \cdot 10 \approx 20 \text{ cm} \]

### Final Answer:
The distance \( t \) from the center of the coil is approximately **20 cm**.