Question

Derive an expression for force per unit length between two parallel straight conductors carrying current in the same direction.

Hint:

The force between two parallel conductors is directly proportional to the product of the currents and inversely proportional to the distance between them.

Answer 1

Step 1: Understanding the concept.
According to Ampere’s Law, two parallel conductors carrying currents exert a force on each other. The force between the conductors is due to the magnetic field created by the current in one conductor, which acts on the other conductor.
Step 2: Magnetic field due to one conductor.
The magnetic field \( B \) at a distance \( r \) from a long straight conductor carrying a current \( I \) is given by Ampere’s law: \[ B = \dfrac{\mu_0 I}{2 \pi r} \] where: - \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4 \pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \)), - \( I \) is the current in the conductor, - \( r \) is the distance from the conductor.
Step 3: Force on the second conductor.
The force on a length \( L \) of the second conductor due to the magnetic field \( B \) produced by the first conductor is given by the formula: \[ F = I L B \] Substitute the value of \( B \) from the previous equation: \[ F = I L \left( \dfrac{\mu_0 I}{2 \pi r} \right) \] Simplifying: \[ F = \dfrac{\mu_0 I^2 L}{2 \pi r} \] Step 4: Force per unit length.
To find the force per unit length, divide both sides of the equation by \( L \): \[ \frac{F}{L} = \dfrac{\mu_0 I^2}{2 \pi r} \] This is the expression for the force per unit length between two parallel conductors carrying currents in the same direction.
Step 5: Conclusion.
The force per unit length between two parallel conductors carrying currents in the same direction is given by: \[ \frac{F}{L} = \dfrac{\mu_0 I^2}{2 \pi r} \]