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Force Between Two Parallel Current-Carrying Wires

Two long parallel current-carrying wires exert magnetic forces on each other because each wire lies in the magnetic field produced by the other. The force per unit length is F/l = mu0 I1 I2/(2pi d).

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Student-friendly explanation

How to write this in exams

1
Start with the exact idea
Two long parallel current-carrying wires exert magnetic forces on each other because each wire lies in the magnetic field produced by the other. The force per unit length is F/l = mu0 I₁ I2/(2pi d).
2
Then show how to use it
Find I₁, I₂, and d in SI units. Substitute in F/l = mu0 I₁ I2/(2pi d). Simplify powers of ten carefully. Decide attraction or repulsion from current directions. State the final value with N m^-1.
3
Add one concrete example
Two overhead conductors carrying currents in the same direction attract each other magnetically. If the currents are reversed relative to each other, the force becomes repulsive.
4
Avoid this incomplete answer
Writing force in newton without multiplying by length when only F/l is calculated is incomplete for questions asking total force on a given length.

Definition

Example

Two overhead conductors carrying currents in the same direction attract each other magnetically. If the currents are reversed relative to each other, the force becomes repulsive.

Rule to remember

F/l = mu0 I₁ I2/(2pi d). I₁ and I₂ are currents in ampere, d is separation in metre, F/l is in N m^-1. Same-direction currents attract; opposite-direction currents repel. Assumes long, straight, parallel wires separated by distance d in air or vacuum approximation.

Memory hook

Parallel currents act like partners when same, opponents when opposite.

Examples and method

Worked example

Two long parallel wires 0.05 m apart carry 10 A and 20 A in the same direction. F/l = (4pi x 10^-7 x 10 x 20)/(2pi x 0.05) = 8.0 x 10^-4 N m^-1. Since currents are in the same direction, the force is attractive.

Method to apply

Find I₁, I₂, and d in SI units. Substitute in F/l = mu0 I₁ I2/(2pi d). Simplify powers of ten carefully. Decide attraction or repulsion from current directions. State the final value with N m^-1.

Diagram support

A simple two-wire diagram should show currents I₁ and I₂, separation d, magnetic field due to one wire at the other, and force arrows. Though not always mandatory, it prevents direction errors.

How CBSE asks it

Asked as a direct numerical, a reason for attraction or repulsion, or a definition-based question related to ampere.

Avoid common mistakes

Common confusion

Students often write only the magnitude formula and forget to state whether the force is attractive or repulsive.

Common wrong answer

Writing force in newton without multiplying by length when only F/l is calculated is incomplete for questions asking total force on a given length.

Exam tip

After calculating force per unit length, always add the nature of force. Direction carries marks in conceptual and assertion-reason questions.

Quick check

What happens to the force between two parallel wires if both currents are doubled and distance is unchanged?

The force per unit length becomes four times because F/l is directly proportional to I₁ I₂. Doubling both currents changes the product from I₁ I₂ to 4I₁ I₂.

Answer writing and exam use

1-mark answer

2-mark answer

Two long parallel current-carrying wires exert magnetic forces on each other because each wire lies in the magnetic field produced by the other. The force per unit length is F/l = mu0 I₁ I2/(2pi d). F/l = mu0 I₁ I2/(2pi d). I₁ and I₂ are currents in ampere, d is separation in metre, F/l is in N m^-1. Same-direction currents attract; opposite-direction currents repel. Assumes long, straight, parallel wires separated by distance d in air or vacuum approximation. Two overhead conductors carrying currents in the same direction attract each other magnetically. If the currents are reversed relative to each other, the force becomes repulsive.

3-mark answer

A current in the first wire produces a magnetic field at the position of the second wire. The second wire carrying current in this field experiences a force. Currents in the same direction attract, while currents in opposite directions repel. This interaction is also connected with the SI definition of ampere in the school-level treatment. F/l = mu0 I₁ I2/(2pi d). I₁ and I₂ are currents in ampere, d is separation in metre, F/l is in N m^-1. Same-direction currents attract; opposite-direction currents repel. Assumes long, straight, parallel wires separated by distance d in air or vacuum approximation. Two long parallel wires 0.05 m apart carry 10 A and 20 A in the same direction. F/l = (4pi x 10^-7 x 10 x 20)/(2pi x 0.05) = 8.0 x 10^-4 N m^-1. Since currents are in the same direction, the force is attractive. Asked as a direct numerical, a reason for attraction or repulsion, or a definition-based question related to ampere. Writing force in newton without multiplying by length when only F/l is calculated is incomplete for questions asking total force on a given length.

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