The Force Between Two Parallel Current

There is a magnetic field surrounding an electrical conductor when the flow of current happens through it. The strength of the magnetic field runs in proportion to the current flow. This remains so till there is the flow of current through the conductor, when the flow of current stops the magnetic field disappears. The direction of the flow of current determines the direction of the flux of the magnetic lines inside the field. One can use a magnetic compass to estimate the direction of the magnetic field.

There is an interaction between the magnetic lines of flux in the magnetic field of the two current-carrying conductors running parallel to each other.

The lines of flux between the current carrying conductors are in the same direction if the flow of currents run in opposite directions. In an attempt to lessen the flux compression, this concentration of flux lines creates a force of repulsion between the conductors.

Flux lines join to surround both conductors when the flow of current is in the same direction. In an attempt to lower the tension of the flux lines, this stretches the lines of flux, generating a force of attraction between the two conductors.

In a high flow current system, the conductors are generally fastened or attached adequately to manage their intensity of attraction or repulsion, when the current carrying conductors are placed parallel to each other. This is done to avoid excessive damage to the conductors in the event of the occurrence of short circuits.

Biot-Savart Law

The study of the relationship between current and the magnetic field it produces is given by Biot-Savart’s law.

According to the Biot-Savart law the magnetic field formed due to a small current element at a given point is directly proportional to the 

• current element’s length,
• the current being passed,
• the sine value of the angle between the direction of the current,and the line that joins the given point and
• the current element

And is inversely proportional to the square of the distance of that point.

The direction of the magnetic field formed is the same as the direction of dl cross r.

The right hand rule can also be used to figure out the direction of the magnetic field.

For example, let’s say there is a finite conductor XY that is carrying a current I.

Here the magnetic field dB due to an infinitesimal element (dl) of the conductor is to be determined at a point P at a distance of r from it.

Assuming that the angle between dl and the position vector ( r) is θ, according to Biot-Savart’s law, the magnitude of the magnetic field  dB is proportional to the current (I), the elemental length

|dl| is inversely proportional to the square of the distance (r).

Also, its direction is perpendicular to the plane, which contains dl and  r.

Therefore Biot- Savart Law can be written as,

|dB|=(μ0/4π)(IdlsinΘ/r2)———–(1)

 In this,

μ0 is the permeability of free space and is equal to 4π × 10-7TmA-1.

Force Between Two Parallel Current Carrying Conductors

We now know that there exists a magnetic field caused due to a conductor carrying current . We also know the relation between them as explained by Biot Savart’s law.

Ordinary currents create massive magnetic fields, which exert a huge force on ordinary currents.

As a result, powerful forces between current-carrying wires are possible.

The force between wires, on the other hand, may not be expected to determine the ampere.

This force may also contribute to the failure of massive circuit breakers when attempting to disturb large currents.

Keeping the above information in mind, we can calculate the force between two long straight and parallel wires separated by a distance r.

Consider two current carrying wires and the magnetic fields created by these  and the resulting forces they exert on one another. 

Let us study  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

B1=μ0I1/2πr

Because this field is uniform along wire 2 and perpendicular to it, the force F2 it exerts on wire 2 is given by

F=IlBsin⁡θ

 with sinθ=1

F2=I2lB1

Because the forces on the wires are equal in strength according to Newton’s third law, we simply write F to represent the strength of F2. (Please keep in mind that F1 = F2.) Since the wires are so long, it’s easier to think about F/l, or force/unit length. Substituting the expression for B1 into the previous equation and rearrangement of terms yields.

F/l = μ0I1I2/2πr

F/l denotes the force per unit length among the  parallel currents I1 and I2 isolated by r. If the currents are flowing towards one side, the force attracts; if they are in different directions, the force repels.The pinch effect that occurs in electric arcs and plasmas is caused by this.

Whether or not currents flow through wires, the force exists. An attraction exists that pushes currents into a smaller tube in an electric arc, where currents move in parallel.

Definition of ampere

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

F = μ0I1I2 l/2πr

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

F/l = μ0I1I2/2πr

If  I1=I2 = 1A , r = 1 m , then 

F/l = μ0/2π 1*1/1 = 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.

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.