# 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 I_{1} and I_{2} respectively. (Fig 1).

The magnetic field due to current I

_{1}in AB at a distance r is

B_{1}= μ_{0}I_{1}/(2πr) …………… (1)

This magnetic field acts perpendicular to the plane of the paper and inwards. The conductor CD with current I_{2} is situated in this magnetic field. Hence, force on a segment of length L of CD due to magnetic field B_{1} is F = B_{1} I_{2} L

Substituting equation (1) F= [μ_{0}I_{1}/(2πr)] I_{2} L

=> F = (μ

_{0}I_{1}I_{2}L) / (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 I

_{2}flowing in CD at a distance r is

B_{2}= μ_{0}I_{2}/(2πr) …………… (3)

This magnetic field acts perpendicular to the plane of the paper and outwards. The conductor AB with current I_{1} is situated in this field. Hence force on a segment of length L of AB due to magnetic field B_{2} is F = B_{2} I_{1} L

substituting equation (3)

F = (μ

_{0}I_{1}I_{2}L) / (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 wire*s* carry currents in the opposite direction*, they repel each other.

Also read: Definition of ampere