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

Options

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

Explanation

To solve the problem step by step, we need to calculate the force experienced by wire B due to the current in wire A. Here’s how we can do it:

### Step 1: Understand the Situation
We have two parallel wires, A and B. Wire A carries a current of 10 A, and wire B carries a current of 2 A in the opposite direction. The distance between the two wires is 10 cm (0.1 m), and the length of wire B is 2 m.

### Step 2: Use the Formula for Force Between Parallel Currents
The force per unit length between two parallel wires carrying currents I1 and I2 is given by the formula:
\[ F/L = \frac{\mu_0}{2\pi} \cdot \frac{I_1 I_2}{R} \]
where:
- \( F \) is the force,
- \( L \) is the length of the wire,
- \( \mu_0 \) 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 3: Substitute the Values
Here, we have:
- \( I_1 = 10 \, \text{A} \) (current in wire A),
- \( I_2 = 2 \, \text{A} \) (current in wire B),
- \( R = 0.1 \, \text{m} \) (distance between the wires),
- \( L = 2 \, \text{m} \) (length of wire B).

Now substituting these values into the formula:
\[ F/L = \frac{(4\pi \times 10^{-7})}{2\pi} \cdot \frac{10 \times 2}{0.1} \]

### Step 4: Simplify the Expression
The \( \pi \) terms cancel out:
\[ F/L = 2 \times 10^{-7} \cdot \frac{20}{0.1} \]
\[ F/L = 2 \times 10^{-7} \cdot 200 \]
\[ F/L = 4 \times 10^{-5} \, \text{N/m} \]

### Step 5: Calculate the Total Force
Now, to find the total force \( F \) on wire B, we multiply the force per unit length by the length of wire B:
\[ F = (F/L) \cdot L = (4 \times 10^{-5}) \cdot 2 \]
\[ F = 8 \times 10^{-5} \, \text{N} \]

### Step 6: Conclusion
The force on wire B is:
\[ F = 8 \times 10^{-5} \, \text{N} \]