Two long conducting wires are placed parallel to each other at a distance d. Current carried by the wires is `i_(1)" and " i_(2)`. If F is the force acting between them, per unit of length, then which of the following statments is true?

Options

A. F is proportional to `i_(1)i_(2)`
B. F is inversely proportional to d
C. F will be attraction if currents are in the same direction
D. F will be repulsive if currents are in the same direction

Explanation

To solve the problem regarding the force acting between two long parallel conducting wires carrying currents \( I_1 \) and \( I_2 \), we can follow these steps:

### Step 1: Understanding the Setup
We have two infinitely long parallel wires separated by a distance \( d \). The first wire carries a current \( I_1 \) and the second wire carries a current \( I_2 \).

### Step 2: Magnetic Field Due to a Current-Carrying Wire
The magnetic field \( B \) created by a long straight wire at a distance \( d \) from it is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi d} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Magnetic Field at the Location of the Second Wire
For wire 1 carrying current \( I_1 \), the magnetic field at the location of wire 2 (distance \( d \) away) is:
\[ B_1 = \frac{\mu_0 I_1}{2 \pi d} \]
The direction of this magnetic field can be determined using the right-hand rule.

### Step 4: Force on the Second Wire
The force \( F \) on a current-carrying wire in a magnetic field is given by:
\[ F = I \cdot L \cdot B \cdot \sin(\theta) \]
For wire 2 carrying current \( I_2 \) in the magnetic field \( B_1 \), the force per unit length \( f \) is:
\[ f = \frac{F}{L} = I_2 \cdot B_1 \]
Since the angle \( \theta \) between the current direction and the magnetic field is \( 90^\circ \) (as they are perpendicular), we have \( \sin(90^\circ) = 1 \).

### Step 5: Substituting the Magnetic Field
Substituting \( B_1 \) into the equation for force per unit length:
\[ f = I_2 \cdot \frac{\mu_0 I_1}{2 \pi d} \]
Thus, we can write:
\[ f = \frac{\mu_0 I_1 I_2}{2 \pi d} \]

### Step 6: Analyzing the Nature of the Force
- If the currents \( I_1 \) and \( I_2 \) are in the **same direction**, the force \( f \) is attractive.
- If the currents are in **opposite directions**, the force is repulsive.

### Conclusion
From our analysis, we can conclude:
1. The force \( F \) is proportional to \( I_1 \) and \( I_2 \).
2. The force \( F \) is inversely proportional to the distance \( d \).
3. The force is attractive if the currents are in the same direction.
4. The force is repulsive if the currents are in opposite directions.

Thus, the correct statements are:
- \( F \) is proportional to \( I_1 \times I_2 \).
- \( F \) is inversely proportional to \( d \).
- \( F \) is attractive if currents are in the same direction.