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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.
Solution
Case I: Both wires carry current in the same direction.
- 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 - 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)
- By the right-hand rule, the direction of this field is into the plane of the paper.
- 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. - 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"`
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