Two straight parallel wires, both carrying `10` ampere in the same direction attract each other with a force of `1xx10^(-3)N`. If both currents are doubled, the force of attraction will be

Options

A. `1xx10^(3)N`
B. `2xx10^(3)N`
C. `4xx10^(3)N`
D. `0.25xx10^(3)N`

Answer

To solve the problem, we will use the formula for the force between two parallel current-carrying wires. The force \( F \) between two parallel wires carrying currents \( I_1 \) and \( I_2 \) separated by a distance \( r \) is given by:

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

Where:
- \( F \) is the force between the wires,
- \( \mu_0 \) is the permeability of free space,
- \( I_1 \) and \( I_2 \) are the currents in the wires,
- \( L \) is the length of the wires,
- \( r \) is the distance between the wires.

### Step-by-Step Solution:

1. **Identify the initial values:**
- Given that both wires carry a current of \( I_1 = I_2 = 10 \, \text{A} \).
- The initial force \( F \) is \( 1 \times 10^{-3} \, \text{N} \).

2. **Determine the effect of doubling the currents:**
- If both currents are doubled, then:
\[ I_1' = 2 \times I_1 = 2 \times 10 \, \text{A} = 20 \, \text{A} \]
\[ I_2' = 2 \times I_2 = 2 \times 10 \, \text{A} = 20 \, \text{A} \]

3. **Substitute the new currents into the force equation:**
- The new force \( F' \) can be calculated as:
\[ F' = \frac{{\mu_0 I_1' I_2' L}}{{2 \pi r}} = \frac{{\mu_0 (20)(20) L}}{{2 \pi r}} \]

4. **Relate the new force to the original force:**
- Since the original force was:
\[ F = \frac{{\mu_0 (10)(10) L}}{{2 \pi r}} = \frac{{\mu_0 (100) L}}{{2 \pi r}} \]
- The new force can be expressed in terms of the original force:
\[ F' = \frac{{\mu_0 (20)(20) L}}{{2 \pi r}} = \frac{{\mu_0 (400) L}}{{2 \pi r}} = 4 \times \frac{{\mu_0 (100) L}}{{2 \pi r}} = 4F \]

5. **Calculate the new force:**
- Since \( F = 1 \times 10^{-3} \, \text{N} \):
\[ F' = 4 \times (1 \times 10^{-3}) = 4 \times 10^{-3} \, \text{N} \]

### Final Answer:
The new force of attraction when both currents are doubled is \( 4 \times 10^{-3} \, \text{N} \).