Two identical coils have· a common centre and their planes are at right angles to each other and carry equal currents. If the magnitude of the induction field at the centre due to one of the coil is 'B ', then the resultant magnetic induction field due to combination at their common centre is

Options

A. B
B. `sqrt2B`
C. `B//sqrt2`
D. 2B

Answer

To solve the problem, we need to find the resultant magnetic induction field at the common center of two identical coils that are oriented perpendicularly to each other and carrying equal currents.

### Step-by-Step Solution:

1. **Identify the Magnetic Field Due to One Coil**:
- Let the magnetic induction field at the center due to one coil be \( B \).

2. **Direction of Magnetic Fields**:
- For the first coil (let's call it Coil 1), the magnetic field at the center can be denoted as \( B_1 \).
- For the second coil (Coil 2), the magnetic field at the center can be denoted as \( B_2 \).
- Since the coils are perpendicular to each other, \( B_1 \) and \( B_2 \) will be at right angles (90 degrees) to each other.

3. **Magnitude of Magnetic Fields**:
- Given that both coils carry equal currents, the magnitudes of the magnetic fields will be the same:
\[ B_1 = B \quad \text{and} \quad B_2 = B \]

4. **Resultant Magnetic Field Calculation**:
- To find the resultant magnetic field \( B \) at the center due to both coils, we can use the Pythagorean theorem since the fields are perpendicular:
\[ B = \sqrt{B_1^2 + B_2^2} \]
- Substituting the values of \( B_1 \) and \( B_2 \):
\[ B = \sqrt{B^2 + B^2} \]
- This simplifies to:
\[ B = \sqrt{2B^2} = \sqrt{2}B \]

5. **Final Result**:
- The resultant magnetic induction field at the common center due to the combination of the two coils is:
\[ B = \sqrt{2}B \]

### Conclusion:
The resultant magnetic induction field at the common center of the two coils is \( \sqrt{2}B \).