Force on Wires

Current in a wire is just a large number of moving charges traveling together through a conductor. Since a magnetic field exerts a force on each moving charge, it also exerts a force on the wire as a whole.

This is the principle that makes electric motors spin, speakers vibrate, and MRI machines work. Every time you turn on a fan, hear sound from headphones, or watch an electric car accelerate, the magnetic-force-on-a-current-carrying-wire equation is doing its work. The physics is a straightforward extension of the single-charge Lorentz force.

Force on a Straight Current-Carrying Wire

This equation comes directly from F = qvB sin θ, applied to all the moving charges in the wire at once. The product of charge and velocity for all the charges becomes current times length (IL replaces qv).

Direction: Right-Hand Rule (FBI)

The direction of the force on the wire follows the same right-hand rule as for a single moving charge:

1. Point your fingers in the direction of conventional current (I).
2. Curl your fingers toward the magnetic field (B).
3. Your thumb points in the direction of the force (F).

The Motor Principle

An electric motor is simply a current-carrying loop of wire placed in a magnetic field. The magnetic force pushes one side of the loop up and the other side down, creating a torque that spins the loop. A commutator reverses the current direction every half-turn so the torque always pushes in the same rotational direction.

The torque on a rectangular current loop in a magnetic field is:

$\displaystyle \tau = nIAB \sin\theta$

where $n$ = number of turns, $I$ = current, $A$ = area of the loop, $B$ = magnetic field, and $\theta$ = angle between the field and the normal to the loop. This is the same $\sin\theta$ pattern as torque on a dipole.

Forces Between Parallel Wires

Two parallel wires carrying current exert forces on each other through their magnetic fields. Each wire creates a magnetic field, and the other wire sits in that field and experiences a force.

Same-direction currents attract. The magnetic field from wire 1 pushes wire 2 toward wire 1, and vice versa.

Opposite-direction currents repel. The magnetic field from wire 1 pushes wire 2 away, and vice versa.

This might seem counterintuitive (same-direction currents attract, while same-sign charges repel), but you can verify it with the right-hand rule. First, find the direction of the magnetic field from wire 1 at the location of wire 2. Then find the force on wire 2 using F = BIL.

A Note on Units

The tesla (T) is the SI unit of magnetic field. It can be expressed in several equivalent ways:

$1 \text{ T} = 1 \text{ kg/(A·s}^2\text{)} = 1 \text{ N/(A·m)} = 1 \text{ V·s/m}^2$

The most useful equivalence for problem-solving is 1 T = 1 N/(A·m), which you can verify by checking that $F = BIL$ gives newtons when $B$ is in tesla, $I$ is in amps, and $L$ is in meters.