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The force of attraction or repulsion between two straight and parallel current-carrying conductors or wires is often called Ampère’s force law. The physical origin of this force is given by the idea that every current-carrying wire generates some magnetic field, following the Biot-Savart law, and the other wire experiences some magnetic force as a consequence, Lorentz force law is obeyed. It might help in learning that this force has something to do with why large circuit breakers blow up when they attempt to interrupt high currents. So now we will learn some further details about this case.
Force on current-carrying conductors
When two long straight and parallel current-carrying conductors are placed nearby separated by the distance r, then the amount of force between them can be found by applying what we have developed in the preceding sections. Let us consider the field produced by wire 1 and the force it exerts on wire 2 (call the force 𝐹2 ). The field due to 𝐼1 at a distance 𝑟 is given to be:
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.
