Force between two parallel current-carrying conductor

In two parallel wires, both with the direction of current flow is upwards. The magnetic field lines of the first current-carrying wire occur as concentric circles centred on wire 1 and in a plane perpendicular to the wires. The magnetic field is in the counterclockwise direction as viewed from above. Around the wire, one circle is representing a magnetic field line due to that wire. The magnetic field is passing directly through wire two. The magnetic field is in the anti-clockwise direction. The force on the second current-carrying conductor is to the left, toward wire one.

The magnetic field produced by a long straight carrying conductor is perpendicular to a parallel conductor, A view from above of the two wires, with a magnetic field line are shown for each wire. Here we assume the effect of the earth magnetic field to be nil.

This field is uniform along wire 2 and at right angle to it, and so the force  𝐹2  it exerts on wire 2 is given by the equation   𝐹=𝐼𝑙𝐵𝑠𝑖𝑛𝜃  with  𝑠𝑖𝑛𝜃 =1 :

𝐹2=𝐼2𝑙𝐵1

By newton’s third law we know that the forces on the wires should be  equal in magnitude, and can write  𝐹  for the magnitude of  𝐹2. (Note that  𝐹1=−𝐹2 .) Since the wires are very long and parallel , it is convenient to think  about it in terms of  𝐹/𝑙 , i.e. the force per unit length. By putting the expression of  𝐵1  in  last equation and after  applying mathematical tools on terms we get force per unit length.

                                            fl =μoI1I22𝜋𝑟.              

𝐹/𝑙  is the force per unit length between two parallel currents carrying wires  𝐼1  and  𝐼2  respectively separated by a distance  𝑟.

 Force due to the current carrying wires is responsible for the pinch effect in electric arcs and plasmas.In an electric arc, where many currents are moving parallel to one another,this produces the attraction force which squeezes currents into a smaller tube. In very  large circuit breakers,  those were used in neighbourhoods of  power distribution systems, the pinch effect can be concentrated in an arc between plates of a switch trying to break a large current, burning of  holes, and sometimes  even ignite the equipment. Another example of this  is found in the solar plasma, where jets of ionised material, like solar flares, are shaped by magnetic forces.

• Definition of Ampere 

The following is the force between two parallel wires carrying currents on a segment of length l :-

F=μ0I1I2l/2πr

So, the force/unit length of the conductor is,

F/l=μ0I1I2/2πr

If  I1 = I2 = I A , a = 1 m , then 

F/l = μ0/2π = 4 π × 10-7 / 2π = 2 × 10-7 Nm-1

The above derived conditions , form the following definition of Ampere

Ampere is defined as the constant current that, when flowing through two parallel infinitely long straight conductors of negligible cross-section placed one metre apart in air or vacuum, experiences a force of 2 ×10-7 Newton per unit length of the conductor.

Direction of force

The force between two current-carrying wires is a good system to understand how any one of a physical quantity is independent of our choice of the right-hand that defines cross-products. As we studied earlier, any physical quantity, such as the direction of the force exerted on a wire, will always depend on two successive times of use of the right hand. In this, we first used the right-hand rule for all axial vectors to determine the direction of the magnetic field from one among two-wire. Then after we use the right-hand rule to determine the direction of the cross-product to determine the direction of the force onto the other wire. We can verify that we get the same answer if we, instead, use our left hand for defining the direction of the magnetic field (which will be in the opposite direction), and then again using the cross-product.

Conclusion

The force between two parallel current-carrying conductors is determined by the Ampere. When currents flow in the same direction, the magnetic field is polarised and the wires attract. Due to relative length contraction, if the currents flow in opposite directions, the electrons have an increased density of electrons in the other wire. Therefore the wires will repel each other.