Anyone who has ever had the pleasure of understanding Maxwell’s equations knows it quite well that the magnetic fields are not very different from the electric fields. This approach although very elegant, requires a very deep understanding of the fundamentals in a mathematical sense. Here is another approach using special relativity and in the form of a thought experiment to show that in their essence, electric and magnetic fields are just different ways of looking at the same phenomenon. We will begin our thought experiment by imagining a straight copper wire as shown below.
Here, the negative charges are the electrons and the positive charges are “holes”. A hole in this context is the empty space left behind when an electron moves from point A to point B, and it has a positive charge due to the nucleus. Although these holes are imaginary particles, it is very important to understand that these positive charges are not just the protons or the nucleus itself. This distinction is necessary because when an electron moves in one direction, a corresponding hole will move in the opposite direction with the same velocity.
Knowing that, inside our wire the number of electrons equal the number of holes, and thus if we place a positive test charge near the wire, it will not experience any net force due to the electric fields of the electrons and holes inside the wire.
When a current starts flowing through the wire, the net effect is that the electrons start flowing in one direction and as a consequence, the holes start flowing in the opposite direction. Lets say for the sake of simplicity that the electrons move in a straight line towards right, and the holes move towards left. However, even in this scenerio, at any one instant, the number of electrons and holes in the wire are the same, and thus our test charge will experience no net force on it. But what happens if our test charge starts moving too?
To keep it simple we will assume that our test charge is moving in the same direction as the electrons, and with the same velocity. We know from observations that our test charge will start experiencing a repulsive force from the wire. We define this force as the force due to the magnetic fields of the test charge and the wire. This effect is more prevalent when two parallel wires have current flowing inside them in opposite directions and the wires repel each other. It does however raise the question that what has changed? How is the movement of the test charge producing this effect?
We need to look at this situation from our test charge’s reference frame. In this frame of reference, our test charge is stationary along with the electrons, while the holes are moving towards left with double the velocity. Now, we need to consider the effect of this velocity on the holes, according to special relativity there will be some length contraction depending on the speed of the holes.
We can imagine this by saying that since the holes are moving, they will contract in length and thus, there will be extra holes that can fit inside the wire. The net effect will be that we end up with more holes than electrons because of this length contraction, and now there is a net positive charge inside the wire and the test charge will therefore feel a repulsive force. This the same force that we define to be due to the magnetic fields, but from the perspective of the test charge, it is due to the electric field of the holes. Although, this is a very simplified thought experiment, it does give you a kind of an intuitive understanding of how similar the magnetic and electric fields are in their essence.