Question

Two long straight parallel conductors carry steady current I₁ and I₂ separated by a distance d. If the currents are flowing in the same direction, show how the magnetic field set up in one produces an attractive force on the other. Obtain the expression for this force. Hence define one ampere.

OR

Derive an expression for the force per unit length between two long straight parallel current carrying conductors. Hence define SI unit of current (ampere).

Answer 1

Suppose two long thin straight conductors (or wires) PQ and RS are placed parallel to each other in vacuum (or air) carrying currents I₁ and I₂ respectively. It has been observed experimentally that when the currents in the wire are in the same direction, they experience an attractive force (fig. a) and when they carry currents in opposite directions, they experience a repulsive force (fig. b).

Let the conductors PQ and RS carry currents I₁ and I₂ in same direction and placed at separation r.

Consider a current–element ‘ab’ of length ΔL of wire RS. The magnetic field produced by current-carrying conductor PQ at the location of other wire RS

According to Maxwell’s right hand rule or right hand palm rule number 1, the direction of B1 will be perpendicular to the plane of paper and directed downward. Due to this magnetic field, each element of other wire experiences a force. The direction of current element is perpendicular to the magnetic field; therefore the magnetic force on element ab of length ΔL

∴ Force acting per unit length of conductor

According to Fleming’s left hand rule, the direction of magnetic force will be towards PQ i.e. the force will be attractive.

On the other hand if the currents I₁ and I₂ in wires are in opposite directions, the force will be repulsive. The magnitude of force in each case remains the same.

Definition of SI unit of Current (ampere): In SI system of fundamental unit of current ‘ampere' is defined assuming the force between the two current carrying wires as standard.

The force between two parallel current carrying conductors of separation r is

Thus 1 ampere is the current which when flowing in each of parallel conductors placed at separation 1 m in vacuum exert a force of 2 × 10^–7 on 1 m length of either wire.

Answer 2

Two long parallel conductors a and b separated by a distance l and carrying currents I_a and I_b respectively are shown below.

By Ampere’s circuital law, we have

\(B_a = \frac{\mu_0I_a}{2\pi d}\)

Conductor b will experience a sideways force because of conductor a. Let this force be F_ba.

\(F_{b_a} = I_bLB_a\)     (F = ILB)

\(= \frac{\mu _0I_a I_bL}{2\pi d}\)

By symmetry,

F_ba = − F_ab

1 ampere is the value of that steady current which when maintained in each of the two very long, straight, parallel conductors of negligible cross section and placed one metre apart in vacuum, would produce on each of these conductors a force equal to 2 × 10⁻⁷ Newton per metre of length.