By Don Lincoln, Fermilab
Relativistic length contraction, according to relativity dictates that, if we measure the length of an object when it is moving, it is shorter than if we measure its length when it is stationary. Furthermore, the faster the object is moving, the shorter it is. This effect just sounds unreasonable and demonstrating length contraction is a lot harder than the changes in time.
Magnetism and Length Contraction
There is a simple demonstration from introductory physics class that conclusively shows that length contraction is real.
The simplest way is to use magnetism. Suppose we have somehow generated a magnetic gravitational field. If we put a magnet in the magnetic field, it will be either attracted or repelled or rotate. The details depend on the specific configuration of that particular magnetic field.
However, a completely different thing happens when we put an electric charge in a magnetic field. If we do that, the electric charge completely ignores the magnetic field. It’s as if it’s not there. Things change if we cause the electric charge to move. If it moves, it is pushed by the magnetic field in some direction. And this is a key point, which is that stationary electric charges are not affected by magnets, while moving electric charges experience forces.
Moving and Stationary Charges
There is another physical phenomenon and that is that moving charges make magnetic fields; stationary charges don’t. The easiest way to make charges move is to move them through a wire, connected to a battery. If we do that, it’s easy to see that the current carrying wire will create a magnetic field. We can even do the experiment with a compass, a wire, and a battery. When we connect the wire to a battery, we can deflect the wire of a compass.
So, let’s imagine a thought experiment that both shows that we have to be careful about magnetism and how relativity saves the day. Suppose we have a long, straight, and stationary wire which is carrying a current, say to the right. This current will make a magnetic field near the wire. For purposes of this example, we will completely ignore gravity. Now imagine that we have a stationary and positively charged particle immediately above the wire. What will that cause to happen? Nothing. That’s because the particle is stationary and stationary particles experience no magnetic forces.
Now, let’s change the stationary charged particle with a moving one. What happens then? In this case, the charged particle feels an attractive force and, so, while the charged particle still moves generally to the right, it also moves somewhat downward due to the magnetic force.
Relativity
Keeping in mind how relativity works, in the case of the charged particle, the particle is stationary. That’s the essence of relativity. So, according to the particle, the wire is moving to the left. It still has a current, but the current is different, which means that the magnetic field is different, too.
On the other hand, the particle is stationary, which, of course, means zero velocity experiencing zero magnetic forces. Relativity also dictates that a person who is stationary with respect to the wire sees that the particle has an attractive magnetic force, while the person who is stationary with respect to the particle sees no magnetic force at all. The two observers see very different things. One sees a force and the other doesn’t.
This article comes directly from content in the video series The Evidence for Modern Physics: How We Know What We Know. Watch it now, on Wondrium.
Relativity and Length Contraction
Keeping their principles of relativity and length contraction in mind, we start in the frame in which the wire is stationary. The wire has a current in it that is to the right. This means that the wire has in it positive particles moving to the right. They have some density, meaning many charged particles per unit length.
We also know that the wire is electrically neutral. That means that it has to have an equal number of negative particles and, furthermore, those negative particles have to have the same density; again, so many charged particles per unit length.
In the frame in which the wire is stationary, the moving positively charged particles set up a current and a magnetic field and the moving charged particle feels an attractive force.
In the reference frame in which the positive particle is stationary, relativistic effects come into play. From the point of view of the charged particle, the negative charges in the wire are moving to the left. The positively charged particles in the wire are either moving to the right, or, at the very least, are moving to the left at a slower speed than the negatively charged particles.
Wire Having a Net Negative Charge
According to the single charged particle, the negatively charged particles in the wire are moving faster than the positively charged particles in the wire and therefore, the distance between them is shorter than that between the positive particles. Also, since the distance is shorter, the charge is more concentrated and the net effect is that there are more negatively charged particles per unit length. Thus, according to the stationary particle above, the wire has a net negative charge.
Because the single particle is stationary, it does not experience any magnetic force. But the wire has a net negative charge. Since the single particle has a positive electrical charge and opposites attract, the single particle feels an attractive force to the wire and moves downward.
Magnetic and Electric Force
This is a staggering observation and a consequences of relativity. In the reference frame where the wire is stationary, the single particle experiences a magnetic force and is pulled downward. In the reference frame where the single particle is stationary, there is no magnetic force, but there is an electric force and that force is identical to the one experienced in the other frame.
Hence, in one frame a magnetic force is experienced, but no electric force. In the other frame, an electric force is experienced, but no magnetic force. And the two forces are the same, demonstrating relativistic length contraction.
Common Questions about Demonstrating Relativistic Length Contraction
The simplest way is to use magnetism. Suppose we have somehow generated a magnetic gravitational field. If we put a magnet in the magnetic field, it will be either attracted or repelled or rotate.
Moving charges make magnetic fields; stationary charges don’t. The easiest way to make charges move is to move them through a wire, connected to a battery.
According to the single charged particle, the negatively charged particles in the wire are moving faster than the positively charged particles in the wire and therefore, the distance between them is shorter than that between the positive particles.
