To derive the expression for the force per unit length between two parallel current-carrying conductors, we'll proceed step by step using the principles of electromagnetism, particularly Ampère's law and the Biot-Savart law.

1. Magnetic Field Due to a Long Straight Conductor

Consider an infinitely long straight conductor carrying a current $I_1$. According to the Biot-Savart law, the magnetic field $B$ at a distance $r$ from the wire is given by:

$B = \frac{\mu_0 I_1}{2 \pi r}$

where:

• $\mu_0$ is the permeability of free space ($\mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}$),
• $I_1$ is the current in the wire,
• $r$ is the radial distance from the wire.

2. Magnetic Force on a Second Conductor

Now, consider a second parallel conductor placed at a distance $d$ from the first conductor. This second wire carries a current $I_2$.

The magnetic field $B_1$ produced by the first wire at the location of the second wire (a distance $d$ away) is:

$B_1 = \frac{\mu_0 I_1}{2 \pi d}$

This magnetic field exerts a force on the second wire. The force $F$ on a small length $\Delta L$ of the second wire is given by the Lorentz force law:

$\Delta F = I_2 \Delta L B_1$

Substituting $B_1$ into the equation:

$\Delta F = I_2 \Delta L \left(\frac{\mu_0 I_1}{2 \pi d}\right)$

3. Force Per Unit Length

The force per unit length $\frac{F}{L}$ between the two wires is:

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

4. Nature of the Force

• Attractive Force: If the currents $I_1$ and $I_2$ flow in the same direction, the force is attractive.
• Repulsive Force: If the currents flow in opposite directions, the force is repulsive.

5. Final Expression

The force per unit length between two parallel current-carrying conductors separated by a distance $d$ is:

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

This is the expression that describes the magnetic force per unit length between two parallel conductors carrying currents $I_1$ and $I_2$.

Would you like more details or have any questions?

Here are some related questions you might be interested in:

1. How does the direction of current affect the force between the conductors?
2. What assumptions are made in deriving the force between two conductors?
3. How does the force change if the conductors are not infinitely long?
4. Can this formula be used in non-vacuum environments, and how would it change?
5. What would be the impact if one of the wires were replaced with a conductor carrying alternating current (AC)?

Tip: When dealing with electromagnetic forces, always remember that the magnetic field created by a current-carrying conductor forms concentric circles around the wire, with the direction given by the right-hand rule.