A circular current carrying coil has a radius a. Find the distance from the centre of coil, on its axis, where is the magnetic induction will be 1/6th of its value at the centre of coil.

Answer

To solve the problem of finding the distance from the center of a circular current-carrying coil, on its axis, where the magnetic induction will be 1/6th of its value at the center of the coil, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Magnetic Field at the Center**:
The magnetic field \( B_0 \) at the center of a circular coil of radius \( a \) carrying current \( I \) is given by the formula:
\[ B_0 = \frac{\mu_0}{4\pi} \cdot \frac{2\pi I}{a} \]
Simplifying this, we get:
\[ B_0 = \frac{\mu_0 I}{2a} \]

2. **Magnetic Field on the Axis of the Coil**:
The magnetic field \( B \) at a distance \( x \) from the center of the coil along its axis is given by:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2\pi I a^2}{(a^2 + x^2)^{3/2}} \]
Simplifying this, we find:
\[ B = \frac{\mu_0 I a^2}{2(a^2 + x^2)^{3/2}} \]

3. **Set up the Equation**:
According to the problem, we want the magnetic field \( B \) at distance \( x \) to be \( \frac{1}{6} B_0 \):
\[ B = \frac{1}{6} B_0 \]
Substituting the expressions for \( B \) and \( B_0 \):
\[ \frac{\mu_0 I a^2}{2(a^2 + x^2)^{3/2}} = \frac{1}{6} \cdot \frac{\mu_0 I}{2a} \]

4. **Cancel Common Terms**:
We can cancel \( \mu_0 \) and \( I \) from both sides:
\[ \frac{a^2}{2(a^2 + x^2)^{3/2}} = \frac{1}{12a} \]

5. **Cross-Multiply**:
Cross-multiplying gives:
\[ 12a \cdot a^2 = 2(a^2 + x^2)^{3/2} \]
Simplifying this, we have:
\[ 12a^3 = 2(a^2 + x^2)^{3/2} \]
Dividing both sides by 2:
\[ 6a^3 = (a^2 + x^2)^{3/2} \]

6. **Cube Both Sides**:
To eliminate the cube root, we cube both sides:
\[ (6a^3)^2 = a^2 + x^2 \]
This gives:
\[ 36a^6 = a^2 + x^2 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ x^2 = 36a^6 - a^2 \]

8. **Solve for \( x \)**:
Taking the square root:
\[ x = \sqrt{36a^6 - a^2} \]
Factoring out \( a^2 \):
\[ x = a \sqrt{36a^4 - 1} \]

9. **Approximate Value**:
For small values of \( a \), we can approximate:
\[ x \approx 1.5a \]

### Final Answer:
Thus, the distance from the center of the coil, on its axis, where the magnetic induction will be \( \frac{1}{6} \) of its value at the center of the coil is approximately:
\[ \boxed{1.5a} \]