12.5 Magnetic Force Between Current-Carrying Conductors

Introduction

The interaction between two parallel current-carrying conductors gives rise to a magnetic force. This force results directly from the magnetic fields generated by the moving charges within the wires. The direction of this force — whether attractive or repulsive — depends entirely on the relative direction of the currents in the two conductors.

Direction of the Force

The nature of the force between two parallel wires is determined by the direction of their currents.

Attractive Force: Currents in the Same Direction

When two conductors carry currents flowing in the same direction, they attract each other.

The magnetic field produced by each wire circulates around it. Between the wires, these two fields are in opposite directions, leading to a region of weaker net magnetic field. Outside the wires, the fields reinforce each other. The net force acts from the stronger external field region toward the weaker internal field region, pushing the wires together.

A neutral point exists between the conductors where the magnetic fields from each wire are equal in magnitude and opposite in direction, cancelling each other out completely. Its position is exactly midway if the currents are equal.

Repulsive Force: Currents in Opposite Directions

When two conductors carry currents flowing in opposite directions, they repel each other.

In this configuration, the magnetic fields between the wires are in the same direction, combining to create a region of stronger net magnetic field. The net force acts from this stronger internal field toward the weaker external fields, pushing the wires apart.

Mathematical Expression for the Force

Step 1 — Magnetic Field of a Single Wire

The magnetic field $B$ at a perpendicular distance $r$ from a long, straight conductor carrying current $I$ is given by Ampère's Law:

$B=\frac{{\mu }_{0}I}{2\pi r}$

where ${\mu }_0=4\pi \times 10^{-7}\text{T⋅m/A}$ is the permeability of free space.

Step 2 — Force on a Wire in an External Field

A wire of length $L$ carrying current $I_2$ placed in an external magnetic field $B_1$ experiences a force:

$F=B_1{I}_{2}L$

Step 3 — Force Per Unit Length

Combining the two results above, the force that wire 1 (current $I_1$) exerts on wire 2 (current $I_2$) separated by distance $r$ is:

$F_{12}=\left(\frac{{\mu }_0{I}_1}{2\pi r}\right)I_{2}L$

Dividing by $L$ gives the force per unit length:

$\boxed{\frac{F}{L}=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}}$

By Newton's Third Law, the force exerted by wire 2 on wire 1 is equal in magnitude and opposite in direction — they form an action–reaction pair.

Definition of the Ampere

The SI unit of current, the Ampere (A), is defined using the force between two parallel conductors:

Summary Table