# 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 I_{1} and I_{2} flow through the wires in the same direction (see Fig. 1).

Consider a small element dl of the wire carrying current I_{2}. The magnetic field B at dl due to the wire carrying current I_{1} 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 I_{1})

Thus the wire carrying current I_{2} is attracted towards the wire carrying current I_{1}. By Newton’s third law the force acting on the wire carrying current I_{1} 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.