Two coplanar concentric circular coils of radii r and 2r, have the same number of turns n. The smaller coil carries a clockwise current i, while the larger coil carries an anticlockwise current 2i. The magnetic field induction at the centre is

Options

A. `(3mu_(0)n i)/(4r)`
B. zero
C. `(mu_(0)n i)/(4r)`
D. `(mu_(0)n i)/(2r)`

Answer

To find the magnetic field induction at the center of two coplanar concentric circular coils with given parameters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Smaller coil (Coil 1) has radius \( r \) and carries a clockwise current \( i \).
- Larger coil (Coil 2) has radius \( 2r \) and carries an anticlockwise current \( 2i \).
- Both coils have the same number of turns \( n \).

2. **Magnetic Field Due to Coil 1**:
- The formula for the magnetic field \( B \) at the center of a circular coil is given by:
\[ B = \frac{\mu_0 n I}{2R} \]
- For Coil 1 (smaller coil):
\[ B_1 = \frac{\mu_0 n i}{2r} \]
- Since the current is clockwise, the direction of the magnetic field \( B_1 \) is directed downwards (into the plane).

3. **Magnetic Field Due to Coil 2**:
- For Coil 2 (larger coil):
\[ B_2 = \frac{\mu_0 n (2i)}{2(2r)} = \frac{\mu_0 n i}{2r} \]
- Since the current is anticlockwise, the direction of the magnetic field \( B_2 \) is directed upwards (out of the plane).

4. **Combine the Magnetic Fields**:
- The magnetic fields \( B_1 \) and \( B_2 \) act in opposite directions:
- \( B_1 \) (downwards) = \( \frac{\mu_0 n i}{2r} \)
- \( B_2 \) (upwards) = \( \frac{\mu_0 n i}{2r} \)
- Therefore, the net magnetic field \( B_{net} \) at the center is:
\[ B_{net} = B_2 - B_1 = \frac{\mu_0 n i}{2r} - \frac{\mu_0 n i}{2r} = 0 \]

5. **Conclusion**:
- The magnetic field induction at the center of the coils is zero.

### Final Answer:
The magnetic field induction at the center is \( 0 \).

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