Two long straight parallel wires, carrying (adjustable) currents `I_(1)` and `I_(2)` are kept at a distance d apart. If the force F between the two wires is taken as positive when the wire repel each other and negative when the wires attract each other, the graph showing the dependence of F, on the product `I_(1)I_(2)` would be:

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between the force \( F \) between two parallel wires carrying currents \( I_1 \) and \( I_2 \), and the product \( I_1 I_2 \).

### Step-by-Step Solution:

1. **Understanding the Force Between Wires**:
The force per unit length \( F/L \) between two long parallel wires carrying currents \( I_1 \) and \( I_2 \) is given by the formula:
\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi d} \]
where \( \mu_0 \) is the permeability of free space and \( d \) is the distance between the wires.

2. **Direction of the Force**:
- If the currents \( I_1 \) and \( I_2 \) are in the same direction, the wires will repel each other, leading to a positive force \( F \).
- If the currents are in opposite directions, the wires will attract each other, resulting in a negative force \( F \).

3. **Expressing the Force**:
Based on the previous understanding, we can express the force \( F \) as:
\[ F = \frac{\mu_0 I_1 I_2}{2 \pi d} L \]
Here, \( L \) is the length of the wires, and we can see that \( F \) is directly proportional to the product \( I_1 I_2 \).

4. **Considering the Sign**:
- When \( I_1 I_2 > 0 \) (currents in the same direction), \( F \) is positive (repulsion).
- When \( I_1 I_2 < 0 \) (currents in opposite directions), \( F \) is negative (attraction).

5. **Graphing the Relationship**:
- The graph of \( F \) versus \( I_1 I_2 \) will be a straight line.
- The slope of the line will be positive when \( I_1 I_2 > 0 \) and negative when \( I_1 I_2 < 0 \).
- Therefore, the graph will cross the origin and will have a negative slope in the region where \( I_1 I_2 < 0 \).

6. **Conclusion**:
The correct graph showing the dependence of \( F \) on the product \( I_1 I_2 \) will be a straight line that is positive for \( I_1 I_2 > 0 \) and negative for \( I_1 I_2 < 0 \).

### Final Answer:
The graph will be a straight line with a negative slope, indicating that \( F \) is directly proportional to \( I_1 I_2 \) but with a sign change depending on the direction of the currents.