The radius of a circular current carrying coil is R. The distance on the axis from the centre of the coil where the intensity of magnetic field is `(1)/(2sqrt(2))` times that at the centre, will be:

Options

A. `x=2R`
B. `x=3R//2`
C. `x=R`
D. `x=R//2`

Answer

To solve the problem, we need to find the distance \( x \) from the center of a circular current-carrying coil where the intensity of the magnetic field is \( \frac{1}{2\sqrt{2}} \) times that at the center.

### Step-by-step Solution:

1. **Identify the Magnetic Field at the Center**:
The magnetic field \( B_0 \) at the center of a circular coil of radius \( R \) carrying current \( I \) is given by:
\[ B_0 = \frac{\mu_0 I}{2R} \]

2. **Magnetic Field on the Axis**:
The magnetic field \( B_P \) at a distance \( x \) on the axis of the coil is given by the formula:
\[ B_P = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} \]

3. **Set up the Equation**:
According to the problem, we have:
\[ B_P = \frac{1}{2\sqrt{2}} B_0 \]
Substituting the expression for \( B_0 \):
\[ B_P = \frac{1}{2\sqrt{2}} \cdot \frac{\mu_0 I}{2R} = \frac{\mu_0 I}{4\sqrt{2} R} \]

4. **Equate the Two Expressions**:
Now, we equate the two expressions for \( B_P \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{\mu_0 I}{4\sqrt{2} R} \]
We can cancel \( \mu_0 I \) from both sides (assuming \( I \neq 0 \)):
\[ \frac{R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{4\sqrt{2} R} \]

5. **Cross Multiply**:
Cross-multiplying gives:
\[ 4\sqrt{2} R \cdot R^2 = 2(R^2 + x^2)^{3/2} \]
Simplifying this:
\[ 4\sqrt{2} R^3 = 2(R^2 + x^2)^{3/2} \]

6. **Divide by 2**:
Dividing both sides by 2:
\[ 2\sqrt{2} R^3 = (R^2 + x^2)^{3/2} \]

7. **Cube Both Sides**:
Cubing both sides:
\[ (2\sqrt{2} R^3)^2 = R^2 + x^2 \]
This simplifies to:
\[ 8 R^6 = R^2 + x^2 \]

8. **Rearranging the Equation**:
Rearranging gives:
\[ x^2 = 8 R^6 - R^2 \]

9. **Factor Out R^2**:
Factoring out \( R^2 \):
\[ x^2 = R^2(8 R^4 - 1) \]

10. **Taking the Square Root**:
Taking the square root gives:
\[ x = R \sqrt{8 R^4 - 1} \]

11. **Final Result**:
To find the specific distance \( x \), we can substitute \( R \) with its value if provided, or we can leave it in terms of \( R \).