Two identical coils carry equal currents have a common centre and their planes are at right angles to each other. The ratio of the magnitude of the resulatant magnetic field at the centre and the field due to one coil is

Options

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

Answer

To solve the problem, we need to find the ratio of the resultant magnetic field at the center of two identical coils that carry equal currents and are positioned at right angles to each other, compared to the magnetic field due to one coil.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two identical coils with equal currents (I) and a common center.
- One coil lies in the xy-plane, while the other is perpendicular to it (let's say it lies in the xz-plane).

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

3. **Direction of the Magnetic Field**:
- For the coil in the xy-plane, the magnetic field direction (B1) is along the z-axis (k-cap).
- For the coil in the xz-plane, the magnetic field direction (B2) is along the y-axis (i-cap).

4. **Magnetic Field Due to Both Coils**:
- The magnetic field due to the first coil (B1) is:
\[ B_1 = \frac{\mu_0 I}{2R} \hat{k} \]
- The magnetic field due to the second coil (B2) is:
\[ B_2 = \frac{\mu_0 I}{2R} \hat{i} \]

5. **Resultant Magnetic Field**:
- The resultant magnetic field (B0) at the center is the vector sum of B1 and B2:
\[ \mathbf{B_0} = \mathbf{B_1} + \mathbf{B_2} = \frac{\mu_0 I}{2R} \hat{k} + \frac{\mu_0 I}{2R} \hat{i} \]
- To find the magnitude of the resultant magnetic field:
\[ |\mathbf{B_0}| = \sqrt{B_1^2 + B_2^2} = \sqrt{\left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{2R}\right)^2} \]
- This simplifies to:
\[ |\mathbf{B_0}| = \sqrt{2 \left(\frac{\mu_0 I}{2R}\right)^2} = \frac{\mu_0 I}{2R} \sqrt{2} \]

6. **Finding the Ratio**:
- Now, we need to find the ratio of the resultant magnetic field (B0) to the magnetic field due to one coil (B1):
\[ \text{Ratio} = \frac{|\mathbf{B_0}|}{B_1} = \frac{\frac{\mu_0 I}{2R} \sqrt{2}}{\frac{\mu_0 I}{2R}} = \sqrt{2} \]

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