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

Answer

To solve the problem step by step, we will use the formula for the force between two long parallel conductors carrying currents.

### Step 1: Understand the initial conditions
Initially, we have two long conductors separated by a distance \( d \) carrying currents \( I_1 \) and \( I_2 \) in the same direction. The force \( F \) between them is given by the formula:
\[ F = \frac{\mu_0}{2\pi} \cdot \frac{I_1 I_2}{d} \]

### Step 2: Apply the changes to the currents and distance
Now, the problem states that the current in one of the conductors (let's say \( I_2 \)) is increased to twice its original value and its direction is reversed. Thus, the new current \( I_2' \) becomes:
\[ I_2' = -2I_2 \]
The distance is also increased to \( 3d \).

### Step 3: Write the new force equation
We need to find the new force \( F' \) between the conductors after these changes. The formula for the force between two conductors is still applicable:
\[ F' = \frac{\mu_0}{2\pi} \cdot \frac{I_1 I_2'}{d'} \]
Substituting \( I_2' \) and \( d' \):
\[ F' = \frac{\mu_0}{2\pi} \cdot \frac{I_1 (-2I_2)}{3d} \]

### Step 4: Simplify the expression for \( F' \)
Now we can simplify the expression:
\[ F' = \frac{\mu_0}{2\pi} \cdot \frac{-2 I_1 I_2}{3d} \]

### Step 5: Relate \( F' \) to \( F \)
We know that the initial force \( F \) is:
\[ F = \frac{\mu_0}{2\pi} \cdot \frac{I_1 I_2}{d} \]
Now we can express \( F' \) in terms of \( F \):
\[ F' = -\frac{2}{3} \cdot F \]

### Conclusion
Thus, the new value of the force between the conductors after the changes is:
\[ F' = -\frac{2}{3} F \]