Currents of `10A, 2A` are passed through two parallel wires `A` and `B` respectively in opposite directions. If the wire `A` is infinitely long and the length of the wire `B` is 2 metre, the force on the conductor `B`, which is situated at `10cm` distance from `A` will be

Options

A. `8xx10^(-5)N`
B. `4xx10^(-5)N`
C. `4xx10^(-7)N`
D. `8xx10^(-7)N`

Answer

To solve the problem, we will use the formula for the magnetic force per unit length between two parallel current-carrying wires. The formula is given by:

\[ F/L = \frac{\mu_0 I_1 I_2}{2 \pi r} \]

Where:
- \( F \) is the force between the wires,
- \( L \) is the length of the wire experiencing the force,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( I_1 \) and \( I_2 \) are the currents in the wires,
- \( r \) is the distance between the wires.

### Step-by-step Solution:

1. **Identify the given values**:
- Current in wire A, \( I_1 = 10 \, \text{A} \)
- Current in wire B, \( I_2 = 2 \, \text{A} \)
- Distance between the wires, \( r = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Length of wire B, \( L = 2 \, \text{m} \)

2. **Substitute the values into the formula**:
- We need to calculate the force per unit length first:
\[ F/L = \frac{\mu_0 I_1 I_2}{2 \pi r} \]
Substituting the values:
\[ F/L = \frac{(4\pi \times 10^{-7}) \times (10) \times (2)}{2 \pi \times (0.1)} \]

3. **Simplify the expression**:
- The \( \pi \) in the numerator and denominator cancels out:
\[ F/L = \frac{(4 \times 10^{-7}) \times (10) \times (2)}{2 \times (0.1)} \]
- This simplifies to:
\[ F/L = \frac{(4 \times 10^{-7}) \times 20}{0.2} \]
- Further simplifying gives:
\[ F/L = \frac{80 \times 10^{-7}}{0.2} = 400 \times 10^{-7} = 4 \times 10^{-5} \, \text{N/m} \]

4. **Calculate the total force on wire B**:
- Now, multiply the force per unit length by the length of wire B:
\[ F = (F/L) \times L = (4 \times 10^{-5}) \times 2 = 8 \times 10^{-5} \, \text{N} \]

5. **Final Result**:
- The force on wire B is:
\[ F = 8 \times 10^{-5} \, \text{N} \]