# The forces between currents

When a current flows in a wire a magnetic field is produced around the wire. If two conductors are placed close together then the field of one wire affects the other and a force exists between the two wires.

This may be demonstrated simply by using the apparatus shown in Figure 1, where a direct current of some 5 A is allowed to flow along two pieces of cooking foil. When the currents in the two pieces of foil are in the same direction the strips of foil attract each other and when they are in opposite directions they repel each other.

The magnetic fields for the two wires are shown in Figure 2(a) for currents in the same direction and 2(b) for currents in opposite directions. Points where there is a zero resultant field are known as neutral points.

The force between the conductors is directly proportional to the current flowing in them and this therefore forms the basis for a definition of the ampere.

The Ampere is defined as the constant current which, when flowing in two infinitely long, straight and parallel conductors of negligible cross section placed 1 m apart in a vacuum causes a force between them of 2x10-7 N per metre of their length.

## The value of mo

Consider the force of attraction between two infinitely long wires, each carrying a current I (Figure 3).
The field B at one wire due to the other wire is:

B = moI/2pr
where r is the distance between the two wires. The force F on a length L of this wire is therefore

Force (F) between two currents = BIL = moI2L/2pr

If the wires are carrying a current of 1 A and are 1 m apart in a vacuum and we consider unit length, then by the definition of the ampere we have:

r = 1m I=1A F= 2x10-7N

Therefore:
Permeability of free space (mo) = 4 px 10-7 H m-1

Example problems
Example 1
Calculate the magnetic flux density at the centre of a solenoid of 2000 turns and 50 cm long if a current of 0.5 A flows in the coil.

B = moNI/L = [4px10-7 x 2000 x 0.5]/0.5 = 2.5 x 10-3T

Example 2
What current must flow in an infinitely long straight wire to give a flux density the same as the above at 0.3 m from the wire?

B = moI/2pr
Therefore I = 2.5x10-3x2px0.3 = 3570 A!

© Keith Gibbs 2011