Two concentric coils each of radius equal to `2pi` cm are placed at tight angles to each there. 3 A and 4 A are the currents flowing in each coil respectively.
The megnetic induction in `Wb//m^(2)` at the centre of the coil will be `(mu_(0) =4pi xx10^(-7)Wb//Am)`

Options

A. `7xx10^(-5)`
B. `5xx10^(-5)`
C. `10^(-5)`
D. `12xx10^(-5)`

Explanation

To find the magnetic induction at the center of two concentric coils placed at right angles to each other, we will follow these steps:

### Step-by-Step Solution

1. **Identify the Given Values:**
- Radius of each coil, \( r = 2\pi \) cm = \( 2\pi \times 10^{-2} \) m.
- Current in the first coil, \( I_1 = 3 \) A.
- Current in the second coil, \( I_2 = 4 \) A.
- Permeability of free space, \( \mu_0 = 4\pi \times 10^{-7} \) Wb/A·m.

2. **Calculate the Magnetic Field due to Each Coil:**
- The magnetic field at the center of a single circular coil is given by the formula:
\[ B = \frac{\mu_0 I}{2r} \]
- For the first coil (with current \( I_1 \)):
\[ B_1 = \frac{\mu_0 I_1}{2r} = \frac{4\pi \times 10^{-7} \times 3}{2 \times (2\pi \times 10^{-2})} \]
- Simplifying \( B_1 \):
\[ B_1 = \frac{4\pi \times 10^{-7} \times 3}{4\pi \times 10^{-2}} = \frac{3 \times 10^{-5}}{1} = 3 \times 10^{-5} \text{ Wb/m}^2 \]

- For the second coil (with current \( I_2 \)):
\[ B_2 = \frac{\mu_0 I_2}{2r} = \frac{4\pi \times 10^{-7} \times 4}{2 \times (2\pi \times 10^{-2})} \]
- Simplifying \( B_2 \):
\[ B_2 = \frac{4\pi \times 10^{-7} \times 4}{4\pi \times 10^{-2}} = \frac{4 \times 10^{-5}}{1} = 4 \times 10^{-5} \text{ Wb/m}^2 \]

3. **Determine the Resultant Magnetic Field:**
- Since the coils are at right angles to each other, the resultant magnetic field \( B \) can be found using the Pythagorean theorem:
\[ B = \sqrt{B_1^2 + B_2^2} \]
- Substituting the values:
\[ B = \sqrt{(3 \times 10^{-5})^2 + (4 \times 10^{-5})^2} \]
- Calculating:
\[ B = \sqrt{(9 \times 10^{-10}) + (16 \times 10^{-10})} = \sqrt{25 \times 10^{-10}} = 5 \times 10^{-5} \text{ Wb/m}^2 \]

### Final Answer
The magnetic induction at the center of the coils is:
\[ B = 5 \times 10^{-5} \text{ Wb/m}^2 \]