The ratio of magnetic inductions at the centre of a circular coil of radius a and on its axis at a distance equal to its radius, will be-

Options

A. `1/(sqrt(2))`
B. `(sqrt(2))/1`
C. `1/(2sqrt(2))`
D. `(2sqrt(2))/1`

Explanation

To solve the problem of finding the ratio of magnetic inductions at the center of a circular coil of radius \( a \) and on its axis at a distance equal to its radius, we will follow these steps:

### Step 1: Magnetic Field at the Center of the Coil
The magnetic field \( B_c \) at the center of a circular coil of radius \( a \) carrying a current \( I \) is given by the formula:

\[ B_c = \frac{\mu_0 I}{2a} \]

Where \( \mu_0 \) is the permeability of free space.

### Step 2: Magnetic Field on the Axis of the Coil
The magnetic field \( B_r \) at a point on the axis of the coil at a distance equal to its radius \( a \) can be calculated using the formula:

\[ B_r = \frac{\mu_0 I a^2}{2(a^2 + a^2)^{3/2}} \]

This simplifies to:

\[ B_r = \frac{\mu_0 I a^2}{2(2a^2)^{3/2}} = \frac{\mu_0 I a^2}{2 \cdot (2^{3/2} a^3)} = \frac{\mu_0 I}{4 \sqrt{2} a} \]

### Step 3: Finding the Ratio of Magnetic Fields
Now we can find the ratio of the magnetic field at the center \( B_c \) to the magnetic field on the axis \( B_r \):

\[ \text{Ratio} = \frac{B_c}{B_r} = \frac{\frac{\mu_0 I}{2a}}{\frac{\mu_0 I}{4 \sqrt{2} a}} \]

### Step 4: Simplifying the Ratio
When we simplify this expression, we get:

\[ \text{Ratio} = \frac{\frac{1}{2}}{\frac{1}{4 \sqrt{2}}} = \frac{1}{2} \cdot \frac{4 \sqrt{2}}{1} = 2 \sqrt{2} \]

### Final Answer
Thus, the ratio of the magnetic inductions at the center of the circular coil and on its axis at a distance equal to its radius is:

\[ \text{Ratio} = 2\sqrt{2} \]