Current Carrying Wires

Problem

Consider two current-carrying wires, one with current $I_1$, and one with current $I_2$. Using the right-hand rule, one can determine the direction of the magnetic fields across each wire. The right-hand rule is also used to determine the direction of the magnetic forces exerted on each current-carrying wire. Try it!

Part 1: If the currents through each wire are travelling in the same direction, will the wires attract or repel?

Part 2: Determine the magnitude of the magnetic force density (magnetic force per unit length) exerted on each wire.

Solution

Part 1

Refer to the diagram on the left. Using the right-hand rule, if the two currents are travelling in the same direction, the magnetic fields will be curling in the same direction (counter-clockwise in the diagram because both currents point up). Thus the generated forces are pointing towards each other, which means the wires will attract.

Part 2

First determine the magnitude of the magnetic field for each wire. One can use Ampere's Law:

$$\oint_C \mathbf{B} \cdot d\boldsymbol{l} = \mu_0 I$$

$$|\mathbf{B_1}| 2\pi r = \mu_0 I_1$$

$$|\mathbf{B_1}| = \frac{\mu_0 I_1}{2\pi r}.$$

Similarly,

$$|\mathbf{B_2}| = \frac{\mu_0 I_2}{2\pi r}.$$

The magnetic force of a current-carrying wire is

$$\mathbf{F} = I\boldsymbol{l} \times \mathbf{B}.$$

The magnitude of the magnetic force is thus

$$|\mathbf{F}| = |Il||B|\sin \theta.$$

Therefore, the magnetic force is

$$|\mathbf{F}| = |I_1 l|\left|\frac{\mu_0 I_2}{2\pi r} \right|\sin {90}^{\circ}$$

and the magnetic force density (magnetic force per unit length) is

$$\frac{|\mathbf{F}|}{l} = \frac{\mu_0 I_1 I_2}{2\pi r}.$$