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).

force between parallel currents
figure 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:

Force Between Parallel Currents - deriving the formula

(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,

Force Between Parallel Currents - deriving the formula

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

Force Between Parallel Currents – how to derive the formula
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