The problem involves finding the force per unit length between two infinitely long parallel wires carrying currents $I_1$ and $I_2$, separated by a distance $d$.

Magnetic Force Between Two Parallel Wires

Two parallel wires carrying currents exert a magnetic force on each other. This force can be attractive or repulsive depending on the direction of the currents:

1. If the currents are in the same direction: The force is attractive.
2. If the currents are in opposite directions: The force is repulsive.

The magnitude of the force per unit length $F/L$ between the two wires is given by Ampère's force law:

$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi d}$

where:

• $\mu_0$ is the permeability of free space ($\mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}$),
• $I_1$ and $I_2$ are the currents in the wires,
• $d$ is the distance between the two wires.

Final Answer

The force per unit length between the two infinitely long parallel wires is:

$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi d}$

This force is:

• Attractive if $I_1$ and $I_2$ flow in the same direction.
• Repulsive if $I_1$ and $I_2$ flow in opposite directions.

Would you like more details or have any questions?

Here are some related questions you might be interested in:

1. How does the force change if the distance $d$ between the wires is halved?
2. What happens to the force if one of the currents is reversed?
3. Can this formula be applied if the wires are not infinitely long? How would the result change?
4. How would you calculate the total force between two finite segments of wire?
5. What is the physical interpretation of $\mu_0$ in this context?

Tip: The direction of the force between two current-carrying wires can be easily remembered using the right-hand rule: if your right hand's thumb points in the direction of the current in one wire, your fingers curl in the direction of the magnetic field it produces. The force on the second wire will then be in the direction your palm pushes.