Two similar coils are kept mutually perpendicular such that their centres coincide. At the centre, find the ratio of the magnetic field due to one coil and the resultant magnetic field by both coils, if the same current is flown

Options

A. `1 : sqrt2`
B. `1 : 2`
C. `2 : 1`
D. `sqrt3 : 1`

Answer

To solve the problem, we need to find the ratio of the magnetic field due to one coil to the resultant magnetic field produced by both coils when they are placed mutually perpendicular to each other.

### Step-by-Step Solution:

1. **Identify the Magnetic Field due to One Coil**:
The magnetic field (B1) at the center of a single coil carrying current I is given by the formula:
\[ B_1 = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space, I is the current, and R is the radius of the coil.

2. **Identify the Magnetic Field due to the Second Coil**:
Since the second coil is similar and carries the same current, the magnetic field (B2) at the center due to the second coil is also:
\[ B_2 = \frac{\mu_0 I}{2R} \]
However, B2 is perpendicular to B1.

3. **Calculate the Resultant Magnetic Field**:
To find the resultant magnetic field (B) when two magnetic fields are perpendicular, we use the Pythagorean theorem:
\[ B = \sqrt{B_1^2 + B_2^2} \]
Substituting the values of B1 and B2:
\[ B = \sqrt{\left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{2R}\right)^2} \]
This simplifies to:
\[ B = \sqrt{2 \left(\frac{\mu_0 I}{2R}\right)^2} = \sqrt{2} \cdot \frac{\mu_0 I}{2R} \]

4. **Find the Ratio of the Magnetic Field of One Coil to the Resultant Magnetic Field**:
We need to find the ratio of B1 to B:
\[ \text{Ratio} = \frac{B_1}{B} = \frac{\frac{\mu_0 I}{2R}}{\sqrt{2} \cdot \frac{\mu_0 I}{2R}} \]
The \(\frac{\mu_0 I}{2R}\) terms cancel out:
\[ \text{Ratio} = \frac{1}{\sqrt{2}} \]

5. **Express the Ratio in Standard Form**:
The ratio can be expressed as:
\[ \text{Ratio} = 1 : \sqrt{2} \]

### Final Answer:
The ratio of the magnetic field due to one coil to the resultant magnetic field by both coils is:
\[ 1 : \sqrt{2} \]