Two circular coils of radii `5cm` and `10cm` carry currents of `2A`. The coils have 50 and 100 turns respectiely and are placed in such a way that their planes as well as their cetre coincide. Magnitude of magnetic field at the common centre of coils is

Options

A. `8pixx10^-4T` if currents in the coils are in same sense
B. `4pixx10^-4T` if current in the coils are in opposite sense
C. zero if currents in the coils are in opposite sense
D. `8pixx10^-4` T if currents in the coils are in opposite sense

Explanation

To find the magnitude of the magnetic field at the common center of two circular coils, we can use the formula for the magnetic field at the center of a circular coil:

\[ B = \frac{\mu_0 N I}{2R} \]

where:
- \(B\) is the magnetic field,
- \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} \, \text{T m/A}\)),
- \(N\) is the number of turns,
- \(I\) is the current in amperes,
- \(R\) is the radius of the coil in meters.

### Step 1: Calculate the magnetic field due to Coil 1

Given:
- Radius of Coil 1, \(R_1 = 5 \, \text{cm} = 0.05 \, \text{m}\)
- Number of turns of Coil 1, \(N_1 = 50\)
- Current in Coil 1, \(I = 2 \, \text{A}\)

Using the formula:

\[ B_1 = \frac{\mu_0 N_1 I}{2R_1} \]

Substituting the values:

\[ B_1 = \frac{4\pi \times 10^{-7} \times 50 \times 2}{2 \times 0.05} \]

Calculating:

\[ B_1 = \frac{4\pi \times 10^{-7} \times 100}{0.1} \]
\[ B_1 = 4\pi \times 10^{-6} \, \text{T} \]

### Step 2: Calculate the magnetic field due to Coil 2

Given:
- Radius of Coil 2, \(R_2 = 10 \, \text{cm} = 0.1 \, \text{m}\)
- Number of turns of Coil 2, \(N_2 = 100\)

Using the formula:

\[ B_2 = \frac{\mu_0 N_2 I}{2R_2} \]

Substituting the values:

\[ B_2 = \frac{4\pi \times 10^{-7} \times 100 \times 2}{2 \times 0.1} \]

Calculating:

\[ B_2 = \frac{4\pi \times 10^{-7} \times 200}{0.2} \]
\[ B_2 = 4\pi \times 10^{-6} \, \text{T} \]

### Step 3: Determine the total magnetic field at the center

Since both coils are in the same plane and their centers coincide, the total magnetic field \(B_{\text{net}}\) at the center is the sum of the magnetic fields due to both coils:

\[ B_{\text{net}} = B_1 + B_2 \]

Substituting the values:

\[ B_{\text{net}} = 4\pi \times 10^{-6} + 4\pi \times 10^{-6} \]
\[ B_{\text{net}} = 8\pi \times 10^{-6} \, \text{T} \]

### Final Answer

The magnitude of the magnetic field at the common center of the coils is:

\[ B_{\text{net}} = 8\pi \times 10^{-6} \, \text{T} \]