A circular current carrying coil has a radius R. The distance from the centre of the coil on the axis where the magnetic induction will be `(1)/(8)` th to its value at the centre of the coil, is

Options

A. `(R)/(sqrt(3))`
B. `Rsqrt(3)`
C. `2sqrt(3)R`
D. `(2)/(sqrt(3))R`

Answer

To solve the problem step by step, we need to find the distance from the center of a circular current-carrying coil on its axis where the magnetic induction (magnetic field) is \( \frac{1}{8} \) of its value at the center of the coil.

### Step 1: Understand the magnetic field at the center and on the axis of the coil

The magnetic field at the center of a circular coil carrying current \( I \) is given by the formula:
\[ B_0 = \frac{\mu_0 n I}{2R} \]
where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( R \) is the radius of the coil.

The magnetic field at a distance \( x \) along the axis of the coil is given by:
\[ B(x) = \frac{\mu_0 n I R^2}{2(R^2 + x^2)^{3/2}} \]

### Step 2: Set up the equation for the magnetic field at distance \( x \)

According to the problem, we want to find \( x \) such that:
\[ B(x) = \frac{1}{8} B_0 \]

Substituting the expressions for \( B(x) \) and \( B_0 \):
\[ \frac{\mu_0 n I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{8} \left( \frac{\mu_0 n I}{2R} \right) \]

### Step 3: Simplify the equation

Cancelling out common terms on both sides, we have:
\[ \frac{R^2}{(R^2 + x^2)^{3/2}} = \frac{1}{8R} \]

### Step 4: Cross-multiply to eliminate the fraction

Cross-multiplying gives:
\[ 8R R^2 = (R^2 + x^2)^{3/2} \]
which simplifies to:
\[ 8R^3 = (R^2 + x^2)^{3/2} \]

### Step 5: Take the cube root of both sides

Taking the cube root of both sides, we get:
\[ 2R = (R^2 + x^2)^{1/2} \]

### Step 6: Square both sides to eliminate the square root

Squaring both sides results in:
\[ 4R^2 = R^2 + x^2 \]

### Step 7: Rearranging the equation

Rearranging gives us:
\[ x^2 = 4R^2 - R^2 = 3R^2 \]

### Step 8: Solve for \( x \)

Taking the square root of both sides:
\[ x = \sqrt{3} R \]

### Final Answer

Thus, the distance from the center of the coil on the axis where the magnetic induction is \( \frac{1}{8} \) of its value at the center of the coil is:
\[ x = \sqrt{3} R \]