Two long conductors, separated by a distance `d` carry current `I_(1) and I_(2)` in the same direction . They exert a force `F` on each other. Now the current in one of them is increased to two times and its direction is reversed . The distance is also increased to `3d`. The new value of the force between them is

Options

A. `-( 2 F)/(3)
B. `( F)/(3)`
C. ` - 2 F`
D. `- (F)/(3)`

Explanation

To solve the problem, we will use the formula for the force between two parallel conductors carrying currents. The force per unit length \( F \) between two long parallel conductors carrying currents \( I_1 \) and \( I_2 \) separated by a distance \( d \) is given by:

\[ F = \frac{\mu_0}{4\pi} \cdot \frac{I_1 I_2}{d} \cdot L \]

where \( \mu_0 \) is the permeability of free space and \( L \) is the length of the conductors.

### Step 1: Identify the initial conditions
- Currents: \( I_1 \) and \( I_2 \)
- Distance: \( d \)
- Force: \( F \)

The initial force can be expressed as:
\[ F = \frac{\mu_0}{4\pi} \cdot \frac{I_1 I_2}{d} \cdot L \]

### Step 2: Modify the conditions based on the problem statement
- The current in one conductor is doubled and reversed, so:
- New current \( I_1' = -2I_1 \) (the negative sign indicates the direction is reversed)
- The distance is increased to \( 3d \).

### Step 3: Write the new force expression
The new force \( F' \) between the conductors can be calculated using the modified values:
\[ F' = \frac{\mu_0}{4\pi} \cdot \frac{(-2I_1) I_2}{3d} \cdot L \]

### Step 4: Simplify the new force expression
Substituting the values into the equation:
\[ F' = \frac{\mu_0}{4\pi} \cdot \frac{-2 I_1 I_2}{3d} \cdot L \]

### Step 5: Relate the new force to the original force
We can express \( F' \) in terms of \( F \):
\[ F' = -\frac{2}{3} \cdot \frac{\mu_0}{4\pi} \cdot \frac{I_1 I_2}{d} \cdot L \]
This can be rewritten as:
\[ F' = -\frac{2}{3} F \]

### Conclusion
The new value of the force between the conductors is:
\[ F' = -\frac{2}{3} F \]
This indicates that the force is repulsive due to the reversal of the current direction.