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

Origin of Forces

Conductors carrying current generate magnetic fields. When these conductors are in proximity, their magnetic fields interact. Ampere’s Law and the Biot-Savart Law describe how the magnetic field from one conductor affects another, leading to a magnetic force.

Force Calculation and Formula

Ampere’s Law enables the calculation of the force per unit length (\( F/L \)) between two parallel conductors. Consider conductors with currents \( I_1 \) and \( I_2 \) separated by distance \( d \).

The magnetic field (\( B \)) at distance \( r \) from a long, straight conductor with current \( I \) is:

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

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

For currents \( I_1 \) and \( I_2 \) separated by \( d \), the force per unit length on the second conductor due to the first conductor's magnetic field is:

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

Consequently, the formula for the force per unit length is:

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

Force Direction

The force direction is determined by the current directions. Opposite current directions result in an attractive force, while currents in the same direction lead to repulsion.