## Forces on currents in magnetic fields

7-13-00

Sections 20.6 and 20.9

### The magnetic force on a current-carrying wire

A magnetic field will exert a force on a single moving charge, so it follows that it will also exert a force on a current, which is a collection of moving charges.

The force experienced by a wire of length l carrying a current I in a magnetic field B is given by

Again, the right-hand rule can be used to find the direction of the force. In this case, your thumb points in the direction of the current, your fingers point in the direction of B. Your palm gives the direction of F.

### The force between two parallel wires

Parallel wires carrying currents will exert forces on each other. One wire sets up a magnetic field that influences the other wire, and vice versa. When the current goes the same way in the two wires, the force is attractive. When the currents go opposite ways, the force is repulsive. You should be able to confirm this by looking at the magnetic field set up by one current at the location of the other wire, and by applying the right-hand rule.

Here's the approach. In the picture above, both wires carry current in the same direction. To find the force on wire 1, look first at the magnetic field produced by the current in wire 2. Everywhere to the right of wire 2, the field due to that current is into the page. Everywhere to the left, the field is out of the page. Thus, wire 1 experiences a field that is out of the page.

Now apply the right hand rule to get the direction of the force experienced by wire 1. The current is up (that's your fingers) and the field is out of the page (curl your fingers that way). Your thumb should point right, towards wire 2. The same process can be used to figure out the force on wire 2, which points toward wire 1.

Reversing one of the currents reverses the direction of the forces.

The magnitude of the force in this situation is given by F = IlB. To get the force on wire 1, the current is the current in wire 1. The field comes from the other wire, and is proportional to the current in wire 2. In other words, both currents come into play. Using the expression for the field from a long straight wire, the force is given by:

Note that it is often the force per unit length, F / l, that is asked for rather than the force.

### The torque on a current loop

A very useful effect is the torque exerted on a loop by a magnetic field, which tends to make the loop rotate. Many motors are based on this effect.

The torque on a coil with N turns of area A carrying a current I is given by:

The combination NIA is usually referred to as the magnetic moment of the coil. It is a vector normal (i.e., perpendicular) to the loop. If you curl your fingers in the direction of the current around the loop, your thumb will point in the direction of the magnetic moment.