# How do varying static point charges exert the same force on each other?

If you have two point charges one being 1 Coulomb and the other being 1 Trillion Coulomb, it is said that the electric force from the 1 Coulomb point charge exerted on the 1 trillion Coulomb point charge is equivalent to the electric force from the 1 trillion coulomb point charge exerted on the 1 Coulomb point charge.

How can a 1 coulomb point charge exert the same force as a 1 trillion coulomb point charge?

Perhaps this analogy will help:

Imagine I have two fans - one with a huge diameter, the other with a tiny diameter. When I put them facing each other, with the huge fan running, I will be able to extract a small amount of power from the tiny fan (because only a tiny fraction of the wind generated by the big fan will intersect with it). Conversely, when the tiny fan is running, almost all its air will be "felt" by the huge fan. But the tiny fan only generates a little bit of air movement...

This is how it is with two different charges (or if you like with two different masses). The same thing (charge, mass) that makes them able to generate a field, makes them susceptible to the field of another (charge, mass). This is indeed a necessary consequence of Newton's third law - the attractive force must be reciprocal (force of A on B must equal force of B on A), so there must be symmetry in the equation describing the force.

If you are OK with the force of Moon on Earth being the same as the force of Earth on Moon, then you should be OK with this. And if those forces were not the same, their would either be crashing into each other, or flying apart...

• Don't the charges fly apart though? When you have two like charges, they repel May 8 '17 at 22:24
• Change the sign of one of them, and they attract. The law is symmetrical - that's the point I was trying to make. May 8 '17 at 22:30
• I guess what I'm trying to ask is that in your last sentence you typed, you say that they shouldn't move. What I was trying to say is that they are moving (either repelling or crashing into eachother, depending on how extreme the force) May 8 '17 at 22:39
• Ah, I see - I mean that the moon and earth are orbiting each other and they need the same force on both of them to continue doing so... (they are rotating about their common center of mass - the force balance means $M_1\omega^2 r_1 = M_2\omega^2 r_2$ - so $M_1 r_1 = M_2 r_2$ which is saying their rotate about their common center of mass, which follows from conservation of momentum. My point is "all these things are self-consistent, and if you don't believe me then nothing in the universe should work the way it does". May 8 '17 at 22:44

This makes no sense to me, how can a 1 coulomb point charge exert the same force as a 1 trillion coulomb point charge. I know Newton's third law, but I still can't comprehend this.

I think the issue here is partly your intuition of force.

The force exerted, as the formula shows, is the result of the product of the two charges involved. It is an interaction of two things, not a force produced by one.

The deep reason for this "product rule" is explained by Quantum Electro Dynamics (QED). That's a rather involved theory and probably well beyond the scope of this question. For the sake of a brief and sensible answer, let's settle for "because that's how the hideous mathematics works out".

You may also be confused by the notion of acceleration and also with how gravity works, which may seem different intuitively.

In gravity it is again the product of the two "charges" (which are the masses in this case) that result in the force (at least in Newton's gravitational theory).

But with gravity the acceleration is the force divided by the mass, and that means the acceleration of one body is proportional to the other body's mass.

We don't get that with charges because the mass and the charge can be unrelated and so the acceleration is not conveniently proportional to one charge.

My sense is that you're possibly intuitively mixing up forces and accelerations in this sense (probably not when you think it out logically, but intuition is not logic).

The force exerted on a charge is given by the Coulomb force: $$F=k\frac{q_1 q_2 }{r^2}.$$ Let's look in particular at the product of the two charges on top. We can see this as $q_1$ being our source. Then if $q_1$ is larger, it will lead to a larger electrostatic potential. Now we consider $q_2$ as a charge which is "feeling" the force. We see that the more electric charge you have on $q_2$, the more you "feel" that force.

To address your example, the 1 coulomb charge puts out a small electrostatic potential, but then the 1 trillion coulomb charge will feel this very strongly, due to its high charge.

The other way around, the 1 trillion coulomb charge will put out a very large electrostatic potential, however the 1 coulomb charge will not feel this as strongly, because it has a low charge.

These two effects can be seen to balance out perfectly, as indicated by the fact the Coulomb force law is unchanged by swapping $q_1$ for $q_2$, and so the same force is felt by each.

• Is this equation for all charges or just static charges? May 8 '17 at 22:42
• Electrostatics describes systems of stationary charges. In the above case, if only one of the two charges were allowed to move, we could still use the same equation (although the charge separation r would change in time). The free charge would move according to Newton's law; its acceleration limited by its mass. If the 1 coulomb charge and the 1 trillion coulomb charge had the same mass, either would accelerate in the same way. (Note if both are allowed to move, then you have currents = no longer electrostatics)
– CDCM
May 8 '17 at 22:51