Question

Two long straight parallel conductors carrying currents exert a force on each other. Why? Derive an expression for the force per unit length between two long straight parallel conductors carrying currents in opposite directions. Explain the nature of the force between these conductors.

Hint:

This principle is fundamental in many electrical devices, including electromagnets and electric motors, where magnetic forces are used to perform work.

Answer 1

Why Forces Are Exerted

Current-carrying conductors generate magnetic fields around them. When two conductors are close to each other, their magnetic fields interact. According to Ampere’s Law and the Biot-Savart Law, the magnetic field produced by one conductor influences the other, resulting in a magnetic force.

Correct Answer and Derivation

The force per unit length (\( F/L \)) between two parallel conductors can be calculated using Ampere’s Law. Let currents \( I_1 \) and \( I_2 \) flow through the conductors, and let them be separated by a distance \( d \).

The magnetic field (\( B \)) at a distance \( r \) from a long straight conductor carrying current \( I \) is given by:

\[ B = \frac{\mu_0 I}{2\pi r} \]

where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, T} \cdot m/A} \)).

If \( I_1 \) and \( I_2 \) are the currents in the conductors and they are separated by distance \( d \), the force per unit length on the second conductor due to the magnetic field produced by the first is:

\[ \frac{F}{L} = I_2 B = I_2 \frac{\mu_0 I_1}{2\pi d} \]

Thus, the expression for the force per unit length between them is:

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

Nature of the Force

The direction of the force depends on the direction of the currents. If the currents are in opposite directions, the magnetic fields around the conductors will attract each other, resulting in an attractive force. If the currents are in the same direction, the conductors will repel each other.