Force Between Parallel Currents

Branch note: The force is not a direct “current attracts current” rule. Each current creates a magnetic field, and that field acts on the other current.

Overview

This branch note explains the interaction between long parallel current-carrying conductors. It is separated from the hub because students often need a full two-step explanation: one wire produces a magnetic field, and the other current-carrying wire experiences a force in that field.

Core Ideas

• A current-carrying wire produces a magnetic field around it.
• A second current-carrying wire placed in that field experiences a magnetic force.
• Same-direction parallel currents attract; opposite-direction parallel currents repel.
• The force pair is equal and opposite even if the two currents are unequal.
• The standard formula assumes long straight parallel wires with separation much smaller than their lengths.

Exam Relevance

Students are expected to explain attraction or repulsion using field-plus-force reasoning, derive the force per unit length from $B={\mu }_{0}I/(2\pi d)$ and $F=BIL$, and apply the formula only under the long-parallel-wire model assumptions.

1. Two-step causal explanation

For wire 2:

1. wire 1 produces a magnetic field ${\vec{B}}_1$ at wire 2;
2. wire 2, carrying current $I_2$, experiences ${\vec{F}}_2=I_2\vec{L}\times {\vec{B}}_1$.

Repeat with the labels exchanged to find the force on wire 1.

Figure: Each wire lies in the magnetic field produced by the other wire. Same-direction currents produce attractive forces, while opposite-direction currents produce repulsive forces.

The result is:

• same-direction parallel currents attract;
• opposite-direction parallel currents repel.

2. Derivation of the magnitude

For two long straight parallel wires in air or free space, with centre-to-centre separation $d$ and end effects neglected, the field produced by wire 1 at wire 2 is

$$B_1=\frac{{\mu }_0{I}_1}{2\pi d}.$$

The field is perpendicular to wire 2, so the force on a common interacting length $L$ is

$$F_2=B_1{I}_{2}L.$$

Therefore

$$F=\frac{{\mu }_0{I}_1{I}_{2}L}{2\pi d},$$

and

$$\frac{F}{L}=\frac{{\mu }_0{I}_1{I}_2}{2\pi d}.$$

This derived expression is model-dependent; it is not a formula for curved, short or non-parallel conductors.

3. Why the forces are equal

At each wire the field magnitude produced by the other may differ if $I_1\ne I_2$:

$$B_1\propto I_1,B_2\propto I_2.$$

But each force contains the product $I_1{I}_2$:

$$F_1=F_2=\frac{{\mu }_0{I}_1{I}_{2}L}{2\pi d}.$$

The forces are equal in magnitude and opposite in direction, forming a Newton’s-third-law pair.

4. Worked example

Two long parallel wires $0.080\mathrm{m}$ apart carry $3.0\mathrm{A}$ and $5.0\mathrm{A}$ in the same direction. For a common length of $0.60\mathrm{m}$,

$$F=\frac{(4\pi \times 10^{-7})(3.0)(5.0)(0.60)}{2\pi (0.080)}=2.25\times 10^{-5}\mathrm{N}.$$

The wires attract. Each wire experiences this magnitude, not a force proportional only to its own current.

5. Proportional reasoning

With other quantities fixed:

• double either current $\Rightarrow$ double $F/L$;
• double both currents $\Rightarrow$ quadruple $F/L$;
• double separation $\Rightarrow$ halve $F/L$;
• reversing one current changes attraction to repulsion without changing the magnitude.

6. Direction workflow

For the force on a chosen wire:

1. ignore the chosen wire temporarily;
2. use the other wire and the right-hand grip rule to find $\vec{B}$ at the chosen wire;
3. combine that $\vec{B}$ with the chosen wire’s conventional current using Fleming’s left-hand rule;
4. repeat for the other wire as a consistency check.

Do not use the chosen wire’s own field to calculate a net sideways force on itself in this model.

7. Historical enrichment

The force between ideal parallel currents was once used in the SI definition of the ampere. Since the 2019 SI revision, the ampere is defined by fixing the elementary charge $e$. The parallel-wire interaction remains an important physical result, but the old force-based statement is historical rather than the current SI definition.

Links

• Magnetic Force
• Magnetic Force Common Exam Traps
• Magnetic Fields from Currents