Previous:  Twin Paradox

In the preceding sections, we have looked at the consequences of relativity on the behaviour of moving objects.  In particular, we have seen that a moving object exhibits time dilation, length contraction, and an increase in mass.  In this section, we will consider another aspect of relativity, namely how it impacts on electrical and magnetic phenomena.  Indeed, it was the relationship between electricity and magnetism that was one of the motivations that set Einstein going in developing the theory of special relativity.

Understanding how relativity impacts on electrical and magnetic phenomena is a complex and involved study.  However, we can get some idea of the consequences by considering a very simple and well known phenomenon from the study of electricity and magnetism..  We will consider what happens when two parallel wires, next to each other, each have a current flowing through them.

In this situation, it is found that if the current flows through the wires in the same direction, then the wires experience an attractive force towards each other.  Conversely, if the currents flow in opposite directions, then the wires experience a mutual repulsive force.  The question is, how does this force of attraction or repulsion arise between the wires?

Consider the following schematic representation of a wire, with no current flowing and with a current flowing.  Note that, of course, this figure provides a very simplified representation of the wire.

The upper half of this figure shows the situation when no current flows in the wire.  if the wire is made of a good conductor such as copper, then the wire itself consists of negative charges (electrons) and positive charges (copper ions).  If the overall wire is electrically neutral, then the wire contains the same number of positive and negative charges, with the average separation between negative charges being about the same as the separation between positive charges (marked as L in the upper half of the above figure).

When a potential difference is applied across the ends of a conductor such as copper, a current will flow due to the movement of the charges, as shown in the lower half of the figure.  (In fact, in reality only the negative charges move.  However, it is convenient to think of both the positive and negative charges moving in opposite directions as shown.  In any case, whichever approach is adopted, the results are the same).

Now consider the same current flowing in the same direction in two conductors, A and B.  As viewed from the laboratory or room where the conductors are located, the movement of charges in the conductors is as in the diagram below:

In the laboratory frame, the positive and negative charges in both conductors move with the same speed, which we will call v.  To an observer in the laboratory, the movement of the charges leads to a length contraction of the separation between the charges, in just the same way that a moving rod is seen to suffer a length contraction.  The new effective separation is

            .

But to the observer in the laboratory, the positive and negative charges in both conductors are subject to the same length contraction, and as such remain electrically neutral.  However, simple experimentation shows that a force of attraction exists between the conductors, as we mentioned above.

The conventional explanation for this is that, if we consider one of the conductors, then the current flowing in that conductor results in the generation of a magnetic field around the conductor.  When a current flows in the other conductor, the result is a force on the other conductor if it lies in the magnetic field generated by the current flow in the first conductor.  This is a very well-known result – in fact it is the basis for the operation of all electric motors.

So how does this force come about?  To explain this, we need to look at the motion of the charges in one conductor, as viewed from the frame of reference of the charges in the other conductor.  Consider the following diagram.

In this figure, the current is assumed to be flowing in the same direction in both conductors.  The motion of the charges is shown as it would be seen from the frame of reference of one of the negative charges in conductor A.  In this case, since the negative charges are moving at the same velocity, all the negative charges in conductor A appear to be stationary.  For the same reason, the negative charges in conductor B also appear stationary.

However, the positive charges in conductor B appear to be moving as shown, with velocity 2v relative to the negative charges in conductor A.  The separation between the positive charges in conductor B therefore appear length contracted with respect to the negative charges in A.  As such, to a negative charge in A, there appears to be a net positive charge in Conductor B.  The result is a net attractive electrostatic force between the negative charges in conductor A and the positive charges in conductor B.

A similar argument can be used to show that, to the positive charges in conductor A, the negative charges in conductor B appear length contracted.  Therefore, an attractive electrostatic force exists between the positive charges in conductor A and the negative charges in conductor B.

The result of the considerations in the previous two paragraphs is that there should be an attractive force between the two conductors, as expected.

Now consider the situation shown in the following diagram, where the currents in the two conductors are flowing in opposite directions.  Again, the view is from the frame of reference of a negative charge in conductor A.

In this case, it is the negative charges in conductor B that appear to be length-contracted, whereas the positive charges in conductor B are stationary.  As such, to a negative charge in A, there appears to be a net negative charge in Conductor B.  In this case, therefore, there is a repulsive electrostatic force between the negative charges in conductor A and the negative charges in conductor B.

Similarly, it is easy to argue that in this situation, there should be a repulsive electrostatic force between the positive charges in conductor A and the positive charges in conductor B.  Overall, it is concluded that there should be a repulsive force between the two conductors, as would be expected.

The significance of the preceding arguments is as follows.  We mentioned earlier that the conventional explanation of the attraction or repulsion between current-carrying wires is in terms of the force experienced by a current-carrying wire in the magnetic field of another current-carrying wire.  But, when we look at things from the frames of reference of the moving charges, we see that the origin of the force is electrostatic in nature.  This is just one example of a more general result that electric and magnetic forces are different manifestations of a single (electromagnetic) interaction that occurs between charged particles.  The mediating agent is special relativity.

THE END