Force between two long parallel current-carrying conductors

Last updated on August 24th, 2022 at 03:00 pm

In this post, we will obtain the formula of the Force between two long parallel current-carrying conductors.

AB and CD are two straight very long parallel conductors placed in the air at a distance r. They carry currents I₁ and I₂ respectively. (Fig 1).

The magnetic field due to current I₁ in AB at a distance r is
B₁ = μ₀I₁/(2πr) …………… (1)

This magnetic field acts perpendicular to the plane of the paper and inwards. The conductor CD with current I₂ is situated in this magnetic field. Hence, force on a segment of length L of CD due to magnetic field B₁ is F = B₁ I₂ L

Substituting equation (1) F= [μ₀I₁/(2πr)] I₂ L

=> F = (μ₀I₁I₂L) / (2πr) ……………. (2)

[ Read this as reference: force on a current-carrying conductor in a magnetic field ]

By Fleming’s Left Hand Rule, F acts towards the left. Similarly, the magnetic induction due to current I₂ flowing in CD at a distance r is
B₂ = μ₀I₂/(2πr) …………… (3)

This magnetic field acts perpendicular to the plane of the paper and outwards. The conductor AB with current I₁ is situated in this field. Hence force on a segment of length L of AB due to magnetic field B₂ is F = B₂ I₁ L

substituting equation (3)

F = (μ₀I₁I₂L) / (2πr) ……………. (4)

By Fleming’s left-hand rule, this force acts towards the right. These two forces given in equations (2) and (4) attract each other.

Hence, two parallel wires carrying currents in the same direction attract each other.
Similarly, it can be shown that if two parallel wires carry currents in the opposite direction, they repel each other.