Force Between Parallel Currents

Overview

Two parallel current-carrying conductors exert magnetic forces on each other.

This happens because:

each wire produces a magnetic field
the other wire lies inside that magnetic field
a current-carrying conductor in a magnetic field experiences force

This topic combines ideas from:

Definition

The force between parallel currents is the magnetic interaction between two nearby current-carrying conductors due to the magnetic fields they create.

Why It Matters

This idea explains:

attraction and repulsion of current-carrying wires
mechanical forces in power systems and busbars
why magnetism and current are deeply linked

Core Physical Idea

A current creates a magnetic field.

That magnetic field can act on another nearby current.

So two wires can:

attract each other
repel each other

without physical contact.

Key Representations

Why Parallel Currents Exert Force

Consider two long straight parallel wires separated by distance $r$ :

Wire 1 carries current $I_{1}$
Wire 2 carries current $I_{2}$

Wire 1 creates a magnetic field at Wire 2.

Wire 2, carrying current, experiences force:

F = B I L

Similarly, Wire 2 creates a field acting on Wire 1.

Hence both wires exert forces on each other.

Same-Direction Currents: Attraction

If both currents flow in the same direction:

each wire is pulled toward the other

Result

Parallel currents in the same direction attract.

Opposite-Direction Currents: Repulsion

If currents flow in opposite directions:

each wire is pushed away from the other

Result

Parallel currents in opposite directions repel.

Figure: Same-direction currents attract, while opposite-direction currents repel.

Direction Logic

Use two steps:

Step 1: Find Magnetic Field of One Wire

Use the right-hand grip rule.

Step 2: Apply Force on the Other Wire

Use Fleming’s left-hand rule.

This determines whether the second wire moves toward or away.

Derivation of Force Formula

Magnetic field due to Wire 1 at distance $r$ :

B = \frac{μ _{0} I _{1}}{2 π r}

Force on Wire 2 of length $L$ :

F = B I_{2} L

Substitute:

F = \frac{μ _{0} I _{1} I _{2} L}{2 π r}

Final Formula

F = \frac{μ _{0} I _{1} I _{2} L}{2 π r}

where:

$F$ = magnitude of force
$I_{1}, I_{2}$ = currents
$L$ = interacting length
$r$ = separation
$μ_{0}$ = permeability of free space

Force Per Unit Length

Often useful:

\frac{F}{L} = \frac{μ _{0} I _{1} I _{2}}{2 π r}

Meaning of the Formula

Force increases when:

currents are larger
interacting length is larger

Force decreases when:

separation $r$ increases

Newton’s Third-Law Pair

The two wires exert equal and opposite forces.

So:

force on Wire 1 equals force on Wire 2 in magnitude
directions are opposite

This is a Newton’s third-law pair.

Short Worked Examples

Example 1: Doubling One Current

If $I_{1}$ doubles:

F doubles

Example 2: Doubling Separation

If $r$ doubles:

F halves

Example 3: Reversing One Current

Attraction becomes repulsion.

Practical Applications

1. Definition of Ampere Historically

Force between parallel currents was historically used in defining current.

2. Busbars and Power Systems

Large currents in nearby conductors can create significant mechanical forces.

3. Electromagnetic Devices

Parallel conductors inside motors and actuators experience forces.

Visual Memory Rule

Same Direction

Attract.

Opposite Direction

Repel.

Common Mistakes

Mixing attraction and repulsion
Using the wrong distance
Forgetting both wires feel force
Using electric-force logic
Ignoring length $L$

Summary

Field of a Wire

B = \frac{μ _{0} I}{2 π r}