4.9 FORCE BETWEEN TWO PARALLEL CURRENTS, THE AMPERE
We have learnt that there exists a magnetic field due to a conductor carrying a current which obeys the Biot-Savart law. Further, we have learnt that an external magnetic field will exert a force on a current-carrying conductor. This follows from the Lorentz force formula. Thus, it is logical to expect that two current-carrying conductors placed near each other will exert (magnetic) forces on each other. In the period 1820-25, Ampere studied the nature of this magnetic force and its dependence on the magnitude of the current, on the shape and size of the conductors, as well as, the distances between the conductors. In this section, we shall take the simple example of two parallel currentcarrying conductors, which will perhaps help us to appreciate Ampere’s painstaking work.
Two long straight parallel conductors carrying steady currents
Figure 4.20 shows two long parallel conductors
The conductor
It is of course possible to compute the force on
Note that this is consistent with Newton’s third Law. Thus, at least for parallel conductors and steady currents, we have shown that the Biot-Savart law and the Lorentz force yield results in accordance with Newton’s third Law*.
*It turns out that when we have time-dependent currents and/or charges in motion, Newton’s third law may not hold for forces between charges and/or conductors. An essential consequence of the Newton’s third law in mechanics is conservation of momentum of an isolated system. This, however, holds even for the case of time-dependent situations with electromagnetic fields, provided the momentum carried by fields is also taken into account.
We have seen from above that currents flowing in the same direction attract each other. One can show that oppositely directed currents repel each other. Thus,
Parallel currents attract, and antiparallel currents repel.
This rule is the opposite of what we find in electrostatics. Like (same sign) charges repel each other, but like (parallel) currents attract each other.
Let
The above expression is used to define the ampere (A), which is one of the seven SI base units.
The ampere is the value of that steady current which, when maintained in each of the two very long, straight, parallel conductors of negligible cross-section, and placed one metre apart in vacuum, would produce on each of these conductors a force equal to
This definition of the ampere was adopted in 1946. It is a theoretical definition. In practice, one must eliminate the effect of the earth’s magnetic field and substitute very long wires by multiturn coils of appropriate geometries. An instrument called the current balance is used to measure this mechanical force.
The SI unit of charge, namely, the coulomb, can now be defined in terms of the ampere.
When a steady current of
Roget's Spiral for Attraction between Parallel Currents
Magnetic effects are generally smaller than electric effects. As a consequence, the force between currents is rather small, because of the smallness of the factor
With the use of a soft spring, we can increase the effective length of the parallel current and by using mercury, we can make the displacement of even a few mm observable very dramatically. You will also need a constant-current supply giving a constant current of about
Take a soft spring whose natural period of oscillations is about
Let the DC source be put off to begin with. Let the tip be adjusted so that it just touches the mercury surface. Switch on the constant current supply, and watch the fascinating outcome. The spring shrinks with a jerk, the tip comes out of mercury (just by a mm or so), the circuit is broken, the current stops, the spring relaxes and tries to come back to its original position, the tip again touches mercury establishing a current in the circuit, and the cycle continues with tick, tick, tick,... In the beginning, you may require some small adjustments to get a good effect.
Keep your face away from mercury vapour as it is poisonous. Do not inhale mercury vapour for long.
Example 4.10
The horizontal component of the earth’s magnetic field at a certain place is
Hence,
The direction of the force is downwards. This direction may be obtained by the directional property of cross product of vectors.
(b)When the current is flowing from south to north,
Hence there is no force on the conductor.