22.10 Magnetic Force between Two Parallel Conductors

Magnetic Force between Parallel Conductors

When two wires carry current near each other, each wire's magnetic field exerts a force on the other wire. This interaction is the basis for how the ampere (the SI unit of current) is formally defined, and it shows up in real engineering problems from power line design to plasma confinement.

How the Force Depends on Current and Distance

The magnetic force between two parallel conductors follows a straightforward pattern:

• Proportional to both currents. Doubling the current in one wire doubles the force. Doubling the current in both wires quadruples it (2 × 2 = 4).
• Inversely proportional to distance. Move the wires twice as far apart and the force drops by half. Bring them twice as close and the force doubles.

The direction of the force depends on whether the currents flow the same way or opposite ways:

• Same direction → attraction. The wires pull toward each other.
• Opposite directions → repulsion. The wires push apart.

You can verify the direction using the right-hand rule: point your thumb in the direction of current in one wire to find its magnetic field at the location of the other wire, then use the force rule ($\vec{F} = I\vec{L} \times \vec{B}$) to find which way the second wire gets pushed.

Force Calculation for Current-Carrying Wires

The force per unit length between two long, parallel wires is:

$F/L = \frac{\mu_0}{2\pi d} I_1 I_2$

where:

• $F/L$ = force per unit length (N/m)
• $\mu_0$ = permeability of free space = $4\pi \times 10^{-7}$ T·m/A
• $d$ = distance between the wires (m)
• $I_1$, $I_2$ = currents in the two wires (A)

To find the total force on a wire of length $L$, multiply: $F = (F/L) \times L$.

Example calculation:

Two parallel wires are each 1.0 m long and separated by 10 cm (0.10 m). Wire 1 carries 5.0 A and wire 2 carries 10.0 A. Find the force between them.

1. Plug into the formula: $F/L = \frac{(4\pi \times 10^{-7})(5.0)(10.0)}{2\pi (0.10)}$
2. Simplify: $F/L = \frac{(4\pi \times 10^{-7})(50)}{2\pi (0.10)} = \frac{2 \times 10^{-5}}{0.10}$
3. Cancel the $\pi$ terms and compute: $F/L = 1.0 \times 10^{-4}$ N/m
4. Since $L = 1.0$ m, the total force is $F = 1.0 \times 10^{-4}$ N

That's a tiny force, which makes sense. You need very large currents or very small separations before the magnetic force between wires becomes significant in everyday situations.

Where This Formula Comes From

The formula combines two ideas you've already seen:

1. A long straight wire carrying current $I_1$ creates a magnetic field at distance $d$: $B = \frac{\mu_0 I_1}{2\pi d}$
2. A second wire carrying current $I_2$ sitting in that field experiences a force per unit length: $F/L = I_2 B$

Substituting the first expression into the second gives you the full formula.

Pinch Effect in Arcs and Plasmas

The force between parallel currents doesn't just apply to separate wires. When current flows through a conducting fluid like a plasma, different parts of the current effectively act as parallel conductors. Since all the current flows the same direction, the resulting force is attractive, squeezing the plasma inward. This is the pinch effect.

The pinch effect has several practical applications:

• Electric arcs: The inward magnetic force helps confine and stabilize the plasma channel, allowing higher current densities and temperatures. This is used in welding, plasma cutting, and electric arc furnaces for steel production.
• Plasma confinement: Devices like Z-pinch machines and tokamaks use magnetic pinch forces (along with other field configurations) to confine hot plasma for fusion research.
• Electromagnetic forming: Short, intense current pulses through a workpiece or nearby coil create strong pinch forces that can shape metal into complex forms, useful in automotive and aerospace manufacturing.