Force Between Parallel Currents – how to derive the formula

Here, we will find out the formula of the force that acts between parallel currents.

Force Between Parallel Currents – deriving the formula

Consider two long wires kept parallel to each other such that the separation d between them is quite small as compared to their lengths.

Suppose currents I₁ and I₂ flow through the wires in the same direction (see Fig. 1).

Consider a small element dl of the wire carrying current I₂. The magnetic field B at dl due to the wire carrying current I₁ is:

(B is normal to and directed into the plane of the figure)

The magnetic force on this element is

(directed towards the wire carrying current I₁)

Thus the wire carrying current I₂ is attracted towards the wire carrying current I₁. By Newton’s third law the force acting on the wire carrying current I₁ will also be attractive. Thus the two wires are attracted towards each other.

The force per unit length on each of the wires due to the other wire will be,

Parallel currents attract each other, and antiparallel currents repel each other.