An electric current is flowing in a circular wire of radius `10 cm `. At what distance (in cm ) from the centre on the axis of the circular wire will the magnetic field be `(1)/(8)` th of its value at the centre? (Take `sqrt(2)=1.41, sqrt(3)=1.73)`

Answer

To solve the problem, we need to find the distance from the center of a circular wire along its axis where the magnetic field is \( \frac{1}{8} \) of its value at the center.

### Step-by-Step Solution:

1. **Understand the Magnetic Field at the Center**:
The magnetic field \( B \) at the center of a circular loop of radius \( R \) carrying a current \( I \) is given by the formula:
\[ B_{\text{center}} = \frac{\mu_0 I}{4\pi R} \]
where \( \mu_0 \) is the permeability of free space.

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

3. **Set Up the Equation**:
We need to find \( x \) such that:
\[ B(x) = \frac{1}{8} B_{\text{center}} \]
Substituting the expressions for \( B(x) \) and \( B_{\text{center}} \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{8} \left(\frac{\mu_0 I}{4\pi R}\right) \]

4. **Simplify the Equation**:
Cancel \( \mu_0 I \) from both sides:
\[ \frac{R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{32\pi R} \]
Rearranging gives:
\[ 32\pi R^3 = 2R^2(R^2 + x^2)^{3/2} \]
Simplifying further:
\[ 16\pi R^3 = R^2(R^2 + x^2)^{3/2} \]

5. **Divide by \( R^2 \)**:
\[ 16\pi R = (R^2 + x^2)^{3/2} \]

6. **Cube Both Sides**:
\[ (16\pi R)^2 = R^2 + x^2)^3 \]
Expanding gives:
\[ 256\pi^2 R^2 = (R^2 + x^2)^3 \]

7. **Substitute \( R = 10 \, \text{cm} \)**:
\[ 256\pi^2 (10)^2 = (100 + x^2)^3 \]
\[ 25600\pi^2 = (100 + x^2)^3 \]

8. **Take the Cube Root**:
\[ 100 + x^2 = \sqrt[3]{25600\pi^2} \]
Using \( \pi \approx 3.14 \):
\[ 25600 \times (3.14)^2 \approx 25600 \times 9.8596 \approx 252000 \]
Thus:
\[ 100 + x^2 \approx \sqrt[3]{252000} \]

9. **Calculate \( x^2 \)**:
\[ x^2 \approx \sqrt[3]{252000} - 100 \]
Approximating gives \( x \approx 10 \, \text{cm} \).

### Final Answer:
The distance from the center on the axis of the circular wire where the magnetic field is \( \frac{1}{8} \) of its value at the center is approximately \( 10 \, \text{cm} \).