AcademySources of Magnetic Fields
Academy
Forces Between Parallel Currents
Level 1 - Physics topic page in Sources of Magnetic Fields.
Principle
Parallel currents exert magnetic forces on each other. Currents in the same direction attract; currents in opposite directions repel.
Notation
\(I_1, I_2\)
currents in the two wires
\(\mathrm{A}\)
\(d\)
separation between the wires
\(\mathrm{m}\)
\(L\)
length of wire considered
\(\mathrm{m}\)
\(F\)
magnetic force
\(\mathrm{N}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)
Method
Derivation 1: Find the field from one wire
Wire 1 produces a magnetic field at wire 2.
Field from wire 1 at wire 2
\[B_1=\frac{\mu_0 I_1}{2\pi d}\]
Force on a current length
\[F_2=I_2LB_1\]
Derivation 2: Combine the equations
Substituting the straight-wire field into the conductor-force equation gives the force per unit length.
Force per length
\[\frac{F}{L}=\frac{\mu_0 I_1I_2}{2\pi d}\]
Derivation 3: Decide attraction or repulsion
Use the right-hand rule for the field from one wire and \(\vec F=I\vec L\times\vec B\) for the other. Same-direction currents pull together; opposite-direction currents push apart.
Rules
Parallel wire force per length
\[\frac{F}{L}=\frac{\mu_0 I_1I_2}{2\pi d}\]
Same direction
\[I_1\parallel I_2\Rightarrow \text{attraction}\]
Opposite direction
\[I_1\text{ antiparallel to }I_2\Rightarrow \text{repulsion}\]
Examples
Question
Two long parallel wires carry currents in the same direction. Do they attract or repel?
Answer
They attract. Each wire lies in the magnetic field produced by the other, and the right-hand rule gives a force toward the other wire.
Checks
- The force is mutual: each wire feels the same force magnitude per length.
- The result assumes long straight parallel wires.
- The direction depends on current direction, not on charge-carrier drift direction.
- The force weakens as separation increases.