# Calculate Your Way to Special Relativity Just Like Einstein

A central idea of physics is that reference frames don’t change fundamental laws. It shouldn’t matter where you observe an event or how fast you are moving, the laws of physics should remain the same independent of your motion.

If we find a situation in which two different reference frames would disagree, we must abandon the idea that physics is the same in different frames or fix the physics. Let’s examine one of these odd situations.

### Two Moving Electric Charges—A Stationary View

Suppose I have two positive charges (both with a charge of +q) moving in parallel directions with a velocity v and separated by a distance r.

If I remain stationary while watching these charges move, there is a force on the bottom charge (well, actually on the top charge too) from both the electric and magnetic field. There is an electric field from the top charge, but because both charges are moving there also is a magnetic field and a magnetic force.

First let’s look at the electric force on the bottom charge. The top charge makes an electric field that is pointing in the downward direction at the location of the bottom charge. If I call “up” the positive y-direction then:

Second, I will calculate the magnetic force. The first step is to find the magnetic field due to the top charge. I will skip some of the details but using the Law of Biot-Savart the magnetic field at the location of the bottom charge should be in the negative z direction (into the screen).

The magnetic force on the bottom charge will be:

Just to be clear: The magnetic field due to the moving top charge is into the screen. Since the magnetic force uses the cross product, the resultant force is up (positive y-direction). Now there are two opposing forces on the bottom charge. There is the electric force pushing the two particles away from each other and there is a magnetic force pulling them together. Both of these forces decrease with distance (they are both 1 over r^{2}) so that if I take the ratio of magnitude of the magnetic to electric force the r term will drop out.

Since both μ_{0} and ε_{0} are both constants, I will replace them with the following expression:

Yes, here *c* is the speed of light with a value of approximately 3 x 10^{8} m/s. This means that the ratio of magnetic to electric force will be:

If the speed of the charged particles is small compared to the speed of light, the ratio of magnetic to electric force is also small, very small. This interaction should be dominated by the electric force and push the two particles away from each other.

### Two Moving Electric Charges—A Moving View

What if I am viewing these moving charges from a reference frame moving alongside the two charges at the same speed? In this case, it appears the two charges are initially stationary with respect to the frame. If the charges are not moving there is no longer a magnetic field produced by the charges and no magnetic force pulling them together. There is only the electric force.

Of course at low speeds, you wouldn’t notice the difference in net forces on the charges since the magnetic force is so tiny. But what about at higher speeds? In that case there is clearly a problem. The stationary observer would see the two charges repelling each other and moving away at a slower speed than an observer moving along with the charges.

### Fixing the Problem

So there is a problem. The two observers don’t agree about what happens—but something does indeed happen. The two charges will repel and move apart, but with what net force? Which observer is correct? At this point, we have some options:

- The nature of interactions and reality depend on your reference. This would mean that the laws of physics are not constant. It would also suck because you can’t really study a star that is moving fast and far away if the fundamental physics interactions are different.
- The electric and magnetic forces in the above calculation are wrong. In fact, they actually are wrong in that they do not take into account the fields of a fast moving particle. However, the relativistic corrections to both the electric and magnetic field have the same factor such that the ratio of magnetic to electric force still depends on the velocity.
- There must be some other physics thing that hasn’t been accounted for.

For the case of two moving charges, there is one thing we can change so that everyone agrees. Remember that the two reference frames would calculate different net forces (but still repelling net forces). The two charges would move away from each other at different times in the two frames. But what if time isn’t the same in the two frames? What if the stationary frame sees the two moving charges with a slower time rate? In that case the moving frame and the stationary frame would agree on the motion of the two charges. Problem fixed.

This is exactly the kind of situation that Einstein used to develop his Theory of Special Relativity. In fact, one of his early papers was titled “On the Electrodynamics of Moving Bodies”.

If you like this connection between electromagnetic forces and Special Relativity, you should probably check out the awesome textbook—*Matter and Interactions*. There is a similar derivation in there and other great examples that relate fundamental physics principles to cool stuff.

See the article here –

## Leave a Reply