The ratio of the magnetic field at the centre of a current carrying coil of the radius `a` and at distance `'a'` from centre of the coil and perpendicular to the axis of coil is

Options

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

Explanation

To find the ratio of the magnetic field at the center of a current-carrying coil of radius \( a \) (denoted as \( B_2 \)) and at a distance \( a \) from the center of the coil, perpendicular to the axis of the coil (denoted as \( B_1 \)), we can follow these steps:

### Step 1: Magnetic Field at the Center of the Coil
The magnetic field at the center of a circular coil of radius \( r \) carrying a current \( I \) is given by the formula:
\[ B_2 = \frac{\mu_0 I}{2r} \]
For our case, where the radius \( r = a \):
\[ B_2 = \frac{\mu_0 I}{2a} \]

### Step 2: Magnetic Field at a Distance \( a \) from the Center
The magnetic field at a distance \( x \) from the center of a circular coil, along the axis, is given by:
\[ B_1 = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}} \]
Substituting \( r = a \) and \( x = a \):
\[ B_1 = \frac{\mu_0 I a^2}{2(a^2 + a^2)^{3/2}} = \frac{\mu_0 I a^2}{2(2a^2)^{3/2}} = \frac{\mu_0 I a^2}{2(2^{3/2} a^3)} = \frac{\mu_0 I a^2}{2 \cdot 2\sqrt{2} a^3} = \frac{\mu_0 I}{4\sqrt{2} a} \]

### Step 3: Finding the Ratio \( \frac{B_2}{B_1} \)
Now we can find the ratio of the magnetic fields:
\[ \frac{B_2}{B_1} = \frac{\frac{\mu_0 I}{2a}}{\frac{\mu_0 I}{4\sqrt{2} a}} = \frac{\frac{1}{2}}{\frac{1}{4\sqrt{2}}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \]

### Final Answer
The ratio of the magnetic field at the center of the coil to the magnetic field at a distance \( a \) from the center of the coil is:
\[ \frac{B_2}{B_1} = 2\sqrt{2} \]

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