Force between two long parallel current-carrying conductors

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).

Force between two long parallel current-carrying conductors
figure 1: Force between two long parallel current-carrying conductors

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 and if they carry currents in the opposite direction, repel each other.

Also read: Definition of ampere

Force between two long parallel current-carrying conductors
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