Force between two parallel current-carrying conductors

Introduction

Experimentally it has been observed that when two current-carrying conductors are placed in the vicinity when having the opposite direction of current flow, there is an observable force between them that pulls both the current carrying conductors closer. Also, the conductors repel each other when the direction of the current flow is the same. We’ve learned that a magnetic field exists as a result of a conductor carrying a current that obeys the Biot-Savart law. We’ve also discovered that an external magnetic field exerts a force on a current-carrying conductor. This is a result of the Lorentz force formula. As a result, it stands to reason that two current-carrying conductors placed close to each other will exert (magnetic) forces on each other.

The force between two parallel current-carrying conductors

Ampere studied the nature of this magnetic force and its dependence on the magnitude of the current, the shape and size of the conductors and the distances between the conductors between 1820 and 1825. He was a hardworking scientist who was dazzled by the magic of electromagnetism. In this section, we will use the simple example of two parallel current-carrying conductors to better understand Ampere’s laborious work.

The figure depicts two long parallel conductors a and b separated by d and carrying (parallel) currents Ia and Ib, respectively. The same magnetic field Ba is produced by the conductor ‘a’ at all points along the conductor ‘b’. According to the right-hand rule, the direction of this field is downward (when the conductors are placed horizontally). Ampere’s circuital law determines its magnitude:

Ba=0Ia2𝜋d

Due to the field Ba, the conductor ‘b’ carrying a current Ib will experience a sideways force. This force is directed towards conductor ‘a’. This force is known as Fba or the force on a segment L of b caused by a. This force’s magnitude is given by: 

Fba = Ib x L x Ba 

= 0IaIb2𝜋dL

We can calculate the force Fab on a segment of length L of ‘a’ due to the current in ‘b’ using the same reasoning as above. It has the same magnitude as Fba and is aimed at ‘b.’ As a result we derive

Fba = –Fab.

This is in complete accordance with Newton’s Third Law, which states that every action has its equal and opposite reaction. Thus, we have demonstrated that, at least for parallel conductors and steady currents, the Biot-Savart law and the Lorentz force produce results that are consistent with Newton’s Third Law.

From the above equations, we can see that currents flowing in the same direction attract each other. It is possible to demonstrate that oppositely directed currents repel each other, so separate calculations aren’t required. As a result, we now know that parallel currents attract while antiparallel currents repel. This rule is diametrically opposed to what we find in electrostatics. Charges with the same sign repel each other, but currents with the same sign attract each other.

Definition of an Ampere 

The ampere is the value of the steady current that would produce a force of 2 x 10–7 newtons per metre of length on two very long, straight, parallel conductors with negligible cross-sections placed one metre apart in a vacuum.

This ampere definition was adopted in 1946. It’s a purely academic definition. In practice, the earth’s magnetic field must be removed and extremely long wires must be replaced with multi-turn coils with appropriate geometries. This mechanical force is measured using a device known as a current balance. If we let Fba denote the magnitude of the Fba per unit length. Then, using the above equation we get, 

Fba = 0IaIb2𝜋d

The expression above is used to define the ampere (A), one of the seven SI base units and even the coulomb. The SI unit of charge can now be defined in terms of the ampere.

When a constant current of 1A is applied to a conductor, the amount of charge that flows through its cross-section in one second is one coulomb (1C).

Conclusion

Current carrying conductors exhibit magnetic forces and are hence attracted and repelled towards each other due to the presence of the magnetic field. To reduce the effect of the earth’s magnetic field, multi-turn coils should be used in the place of long wires ensuring a reduced magnetic force on the conductor. It’s critical to remember that the force on one wire is caused by the magnetic field of the other. Most students make mistakes when calculating force on a specific wire by taking its magnetic field into account. When the currents are moving in the same direction, the force attracts; when they are moving in opposite directions, the force repels should be the only concept while calculating force.