Question

Show that currents in two long, straight, parallel wires exert forces on each other. Derive the expression for the force per unit length on each conductor.

Answer 1

Case I: Both wires carry current in the same direction.

1. Consider two long parallel wires separated by distance d and carrying current I₁ and I₂ respectively same direction as same as shown in the figure below:  
   
   Two long parallel wires, distance d apart
2. The magnetic field at the second wire due to the current I₁ in the first one, according to Biot - Savart’s law is B = `(mu_0"I"_1)/(2pi"d")` ….(1)
3. By the right-hand rule, the direction of this field is into the plane of the paper.
4. Force on the wire 2, because of the current I₂ and the magnetic field B due to current in wire 1, applying Lorentz force law is, 
   F = `"I"_2((mu_0"I"_1)/(2pi"d"))int"dl"` ….(2)
   The direction of this force is towards wire 1, i.e., it will be an attractive force.
5. Force of attraction per unit length of the wire will be
   `"F"/"L" = mu_0/(2pi) ("I"_1"I"_2)/"d"` ….(3)

Case II: Two wires carry current in opposite direction.

Force is of repulsive nature between antiparallel currents and the magnitude of the force of repulsion per unit length is, `|"F"/"L"| = mu_0/(2pi) ("I"_1"I"_2)/"d"`