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

Options

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

Explanation

To solve the problem, we need to find the distance of a point on the axis of a current-carrying circular coil from its center, given that the magnetic induction at the center is \(5\sqrt{5}\) times the magnetic induction at that point.

### Step-by-Step Solution:

1. **Identify the Given Parameters:**
- Radius of the coil, \( r = 10 \, \text{cm} \)
- Magnetic induction at the center, \( B_{\text{center}} \)
- Magnetic induction at point P on the axis, \( B_P \)
- Relationship: \( B_{\text{center}} = 5\sqrt{5} \times B_P \)

2. **Magnetic Induction at the Center of the Coil:**
The magnetic induction at the center of a circular coil is given by the formula:
\[ B_{\text{center}} = \frac{\mu_0 n I}{2r} \]
where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current.

3. **Magnetic Induction at Point P on the Axis:**
The magnetic induction at a point on the axis of the coil at a distance \( x \) from the center is given by:
\[ B_P = \frac{\mu_0 n I r^2}{2(r^2 + x^2)^{3/2}} \]

4. **Set Up the Equation:**
According to the problem, we have:
\[ \frac{\mu_0 n I}{2r} = 5\sqrt{5} \times \frac{\mu_0 n I r^2}{2(r^2 + x^2)^{3/2}} \]
Canceling \( \mu_0 n I \) from both sides, we get:
\[ \frac{1}{2r} = 5\sqrt{5} \times \frac{r^2}{2(r^2 + x^2)^{3/2}} \]

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

6. **Cube Both Sides:**
\[ r^2 + x^2 = (5\sqrt{5})^{2/3} r^2 \]
This simplifies to:
\[ r^2 + x^2 = 25r^2 \]
Rearranging gives:
\[ x^2 = 25r^2 - r^2 = 24r^2 \]

7. **Substituting the Value of r:**
Substitute \( r = 10 \, \text{cm} \):
\[ x^2 = 24 \times (10)^2 = 2400 \]
Taking the square root:
\[ x = \sqrt{2400} = 20 \, \text{cm} \]

### Final Answer:
The distance of the point from the center of the coil is \( \boxed{20 \, \text{cm}} \).