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:sqrt(2)`
B. `1:2`
C. `2:1`
D. `sqrt(3):1`

Answer

To solve the problem step by step, we will analyze the magnetic fields produced by the two coils and find the required ratio.

### Step 1: Understand the Configuration
We have two identical coils placed perpendicularly at their centers. Let's denote:
- The current flowing through each coil as \( I \).
- The radius of each coil as \( R \).
- The number of turns in each coil as \( N \).

### Step 2: Calculate the Magnetic Field from One Coil
The magnetic field at the center of a single coil is given by the formula:
\[ B = \frac{\mu_0 N I}{2R} \]
Since both coils are identical, we can denote the magnetic field due to the first coil as \( B_1 \):
\[ B_1 = \frac{\mu_0 N I}{2R} \]

### Step 3: Calculate the Magnetic Field from the Second Coil
Similarly, the magnetic field at the center due to the second coil (which is perpendicular to the first) is also:
\[ B_2 = \frac{\mu_0 N I}{2R} \]
Since both coils are identical, we have:
\[ B_2 = B_1 \]

### Step 4: Determine the Direction of the Magnetic Fields
- Let’s assume the direction of \( B_1 \) (from the first coil) is upwards.
- The direction of \( B_2 \) (from the second coil) will be to the right.

### Step 5: Calculate the Resultant Magnetic Field
Since \( B_1 \) and \( B_2 \) are perpendicular to each other, we can use the Pythagorean theorem to find the resultant magnetic field \( B_{net} \):
\[ B_{net} = \sqrt{B_1^2 + B_2^2} \]
Substituting \( B_2 = B_1 \):
\[ B_{net} = \sqrt{B_1^2 + B_1^2} = \sqrt{2B_1^2} = B_1 \sqrt{2} \]

### Step 6: Find the Ratio of the Magnetic Fields
We need to find the ratio of the magnetic field due to one coil to the resultant magnetic field:
\[ \text{Ratio} = \frac{B_1}{B_{net}} = \frac{B_1}{B_1 \sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the ratio of the magnetic field due to one coil to the resultant magnetic field by both coils is:
\[ \text{Ratio} = \frac{1}{\sqrt{2}} \quad \text{or} \quad 1 : \sqrt{2} \]