Two concentric circular coils of ten turns each are situated in the same plane. Their radii are 20 and 40 cm and they carry respectively 0.2 and 0.4 ampere current in opposite direction, The magnetic field in weber / `m^2` at tteh centre is

Options

A. `mu_0//80`
B. `7mu_0//80`
C. `(5//4)mu_0`
D. zero

Explanation

To find the magnetic field at the center of two concentric circular coils carrying currents in opposite directions, we can follow these steps:

### Step 1: Identify the parameters
- Coil 1 (smaller coil):
- Number of turns (N1) = 10
- Radius (R1) = 20 cm = 0.2 m
- Current (I1) = 0.2 A
- Coil 2 (larger coil):
- Number of turns (N2) = 10
- Radius (R2) = 40 cm = 0.4 m
- Current (I2) = 0.4 A

### Step 2: Calculate the magnetic field due to Coil 1 (B1)
The formula for the magnetic field at the center of a circular coil is given by:
\[ B = \frac{\mu_0 N I}{2R} \]
Where:
- \( \mu_0 \) (permeability of free space) = \( 4\pi \times 10^{-7} \, \text{T m/A} \)

For Coil 1:
\[ B_1 = \frac{\mu_0 N_1 I_1}{2R_1} = \frac{4\pi \times 10^{-7} \times 10 \times 0.2}{2 \times 0.2} \]
\[ B_1 = \frac{4\pi \times 10^{-7} \times 2}{0.4} = 10\pi \times 10^{-7} \, \text{T} \]

### Step 3: Calculate the magnetic field due to Coil 2 (B2)
For Coil 2:
\[ B_2 = \frac{\mu_0 N_2 I_2}{2R_2} = \frac{4\pi \times 10^{-7} \times 10 \times 0.4}{2 \times 0.4} \]
\[ B_2 = \frac{4\pi \times 10^{-7} \times 4}{0.8} = 20\pi \times 10^{-7} \, \text{T} \]

### Step 4: Determine the direction of the magnetic fields
- For Coil 1, the current flows in one direction, creating a magnetic field \( B_1 \) directed downwards.
- For Coil 2, the current flows in the opposite direction, creating a magnetic field \( B_2 \) directed upwards.

### Step 5: Calculate the net magnetic field at the center
Since the magnetic fields are in opposite directions, we subtract them:
\[ B_{\text{net}} = B_2 - B_1 = 20\pi \times 10^{-7} - 10\pi \times 10^{-7} \]
\[ B_{\text{net}} = (20\pi - 10\pi) \times 10^{-7} = 10\pi \times 10^{-7} \, \text{T} \]

### Step 6: Convert to Weber/m²
Since \( 1 \, \text{T} = 1 \, \text{Wb/m}^2 \):
\[ B_{\text{net}} = 10\pi \times 10^{-7} \, \text{Wb/m}^2 \]

### Conclusion
The magnetic field at the center of the two concentric coils is:
\[ B_{\text{net}} = 10\pi \times 10^{-7} \, \text{Wb/m}^2 \]