Magnetic force between two parallel wires
Two long, straight, parallel wires carrying steady currents exert forces on each other.
The key idea is:
one wire creates a magnetic field around it, and the other wire experiences a magnetic force in that field.
1) Magnetic field of a long straight wire
At perpendicular distance \(d\) from a long straight wire carrying current \(I\), the magnetic field magnitude is
\[
B(d)=\frac{\mu_0\,|I|}{2\pi d},
\qquad \mu_0 = 4\pi\times 10^{-7}\ \mathrm{T\cdot m/A}.
\]
The direction is tangential (azimuthal) around the wire, given by the right-hand rule.
2) Force per unit length
A wire segment of length \(L\) carrying current \(I_2\) in a magnetic field \(B\) experiences force
\(F = I_2 L B\) when the wire is perpendicular to \( \mathbf{B} \) (which is the case for parallel wires).
Therefore,
\[
\frac{F}{L}=|I_2|\,B_1(d)
=\frac{\mu_0\,|I_1|\,|I_2|}{2\pi d}.
\]
For infinite wires, the standard result is reported as \(F/L\). For a finite wire length \(L\), use \(F=(F/L)\,L\).
3) Attraction vs. repulsion
- Same direction (\(I_1 I_2>0\)) → attraction (wires pull together).
- Opposite direction (\(I_1 I_2<0\)) → repulsion (wires push apart).
This comes from the right-hand rule:
the magnetic field generated by wire 1 at the location of wire 2 combines with wire 2’s current direction to set the force direction.
4) “Current balance” and the ampere
Historically, the ampere was tied to the force between two parallel wires.
If two long wires are 1 m apart and each carries 1 A (same direction),
then
\[
\left(\frac{F}{L}\right)_{1\ \mathrm{A},\ d=1\ \mathrm{m}}
=\frac{\mu_0}{2\pi}
\approx 2\times 10^{-7}\ \mathrm{N/m}.
\]
(Modern SI defines the ampere via the elementary charge \(e\), but the force result above remains a useful benchmark.)
5) Limits of the model
- Assumes wires are very long (edge effects neglected).
- Assumes separation \(d\) is measured between wire axes.
- Finite-length wires require corrections; the calculator’s “total force” option uses \(F=(F/L)L\) as an estimate.
