Two long parallel wires, separated by a distance R have equal current I flowing in each of them. The magnetic field of one exerts a force F on the other. The distance R is increased to 2R and the current in each wire is reduced from I to `I//2`. What is the force between them now?

Options

A. 4F
B. F
C. `F//4`
D. `F//8`

Answer

To solve the problem, we need to determine the force between two long parallel wires when the distance between them and the current flowing through them are changed.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Two long parallel wires are separated by a distance \( R \).
- Each wire carries a current \( I \).
- The force between the wires is given as \( F \).

2. **Use the Formula for Force Between Two Parallel Wires**:
The formula for the force per unit length \( F \) between two long parallel wires carrying currents \( I_1 \) and \( I_2 \) separated by a distance \( r \) is:
\[ F = \frac{\mu_0 I_1 I_2}{2\pi r} \]
Here, \( \mu_0 \) is the permeability of free space.

3. **Calculate the Initial Force**:
For the initial setup:
- \( I_1 = I \)
- \( I_2 = I \)
- \( r = R \)
Thus, the initial force \( F \) can be expressed as:
\[ F = \frac{\mu_0 I^2}{2\pi R} \]

4. **Change the Conditions**:
Now, the distance is increased to \( 2R \) and the current in each wire is reduced to \( \frac{I}{2} \).

5. **Calculate the New Force**:
For the new setup:
- \( I_1 = \frac{I}{2} \)
- \( I_2 = \frac{I}{2} \)
- \( r = 2R \)
The new force \( F' \) can be expressed as:
\[ F' = \frac{\mu_0 \left(\frac{I}{2}\right)^2}{2\pi (2R)} \]
Simplifying this:
\[ F' = \frac{\mu_0 \frac{I^2}{4}}{2\pi (2R)} = \frac{\mu_0 I^2}{8\pi R} \]

6. **Relate the New Force to the Initial Force**:
Now, we can relate \( F' \) to \( F \):
\[ F' = \frac{1}{4} \cdot \frac{\mu_0 I^2}{2\pi R} = \frac{1}{4} F \]

### Final Answer:
The new force \( F' \) between the wires when the distance is increased to \( 2R \) and the current is reduced to \( \frac{I}{2} \) is:
\[ F' = \frac{1}{4} F \]