# Is Coulomb's law wrong?

"Magnitude of the electrostatic force F between two point charges $$q_1$$ and $$q_2$$ is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them".

Let us assume a case in which two oppositely charged particles A and B having magnitude of charge as 1 coulomb are separated at the distance of 1 light year at $$t=0$$. The attraction force between them is given by:

$$|F| = \frac{kq_1q_2}{d^2}$$ ($$d$$ is the distance between the charges i.e. 1 light year).

At $$t=1$$, charge A moves 10 metres towards charge B. Now the force of attraction between them increases. According to coulombs law just after the charge A travels 10 m the force between them ($$F_2$$) is given by:

$$F_2=\frac{kq_1q_2}{(d+10)^2}.$$

However since nothing can travel faster than the speed of light it would take some time for the Coulomb force to travel from particle A to particle B implying that particle B will experience $$F_2$$ not immediately after particle A moves 10 m. This is a contradiction since Coulomb's law states that particle B will experience $$F_2$$ just after A travels 10 m which is not possible since the electrostatic force will take some time to reach particle B.

Isn't this a limitation of coulombs law and how can we avoid this incongruence? Help is appreciated.

• A good rule of thumb is that any theory developed to describe non-relativistic physics, especially theories developed before the 20th century, do not necessarily describe things accurately at relativistic velocities. This does not mean that they are "wrong", just that they are only accurate within a certain domain. Commented May 29 at 9:16
• Your formula for the new force value is wrong. Commented May 29 at 10:03
• @my2cts It is not so simple. It is retarded position plus retarded velocity multipled by time elasped. Feynman lectures covered that this little difference is how some no-go theorems were circumvented. Commented May 29 at 10:12
• The force depends on the inverse of the square of distance Commented May 29 at 10:54
• Coulomb's law of electrostatics is no more wrong than, say, Newton's law of motion. Commented May 29 at 11:51

Coulomb's law only holds for static situations. It describes the force between particles that are always at rest with each other. The situation described in the question is not a static, as the particles are in motion relative to each other. Thus, Coulomb's law is not applicable.

To accurately describe this situation, you would need to use Maxwell's equations.

• So what if instead of decreasing their distances i just increase the magnitude of their charges.Now it is a static situation.What can we do in this case? Commented May 29 at 12:04
• @Ishaan: That doesn't count as a static situation. "Static" means that nothing is moving, not even the flow of charges. Commented May 29 at 12:33
• Ok i got it now Commented May 29 at 12:43

You are making a beginner's misunderstanding, even though it confused all the great minds a few centuries back.

Coulomb's law is correct, but it is not what you think it is doing, and you are using it in ways that it is not meant to be. Nobody said that Coulomb's law is the only force that charges can exert on each other. When charges are moving relative to each other, there are magnetic contribution, and other Special Relativisitic corrections to the interaction. These are not captured by Coulomb's law, nor is it supposed to be Coulomb's law's job to get these effects either.

One way to see what is really happening, is that one charge had already set up the fields around it, all over the universe, and the other charge just sees the weaker or stronger forces available in the already-setup fields. Just like in gravity, the spacetime is already curved beforehand, so that when something flies by, it can just read the local information to figure out what to do next.

• So the effect of the moving charge on the stationary charge would be instantaneous? Commented May 29 at 12:05
• No, it is not instantaneous if you consider it that way. Commented May 29 at 12:32
• Thanks for the clarification Commented May 29 at 12:43

Electrostatic as well as electromagnetic force is transmitted with photons, so there is no contradiction. Any "immediate" change is immediate in terms of information travel speed. It cannot be understood more "immediately" than this.

Similar concept is when a locomotive pulls on carriages. The last carriage is pulled last because force needs to travel along ALL atoms on the way. In this case the speed of sound (or rather Young modulus) is relevant factor for information travel speed.