# Why charge is not a measure of inertia?

We know that mass is the measure of inertia of rest. Then why charge is not a measure of inertia? What causes the difference between charge and mass?

I think the key point here is that gravitational mass and inertial mass are not the same concept.

The $m$ in Newton's second law, $F=ma$, is called inertial mass. One way to describe this might be the resistance and object has to be accelerated by a force. It takes more force to give a more massive object the same acceleration as that given to a less massive object by a lesser force.

The $m$ is Newton's law of universal gravitation is called gravitational mass. It is completely unrelated to inertial mass except for the puzzling fact that the two are identical. The fact that inertial mass is equal to gravitational mass hasn't been satisfactorily explained, although one part of Einstein's weak equivalence principle states that the two are the same.

My guess is that the reason you've considered charge to be a factor in a version of Newton's second law is that Coulomb's law looks similar to Newton's law of universal gravitation, and that therefore charge, too, should play a role akin to mass in classical mechanics. However, because inertial mass is the main important quantity, not gravitational mass, this logic doesn't work.

There is no reason whatsoever for charge to show up in Newton's laws. Charge plays no role whatsoever in inertia.

Neutrons and neutrinos have zero charge, but they have mass. Protons and positrons have the same charge, but very different masses. Positrons and electrons have the same mass as one another, but their charges are of opposite sign. Charge has a sign, but mass does not. As far as we know, there is no such thing as a particle with negative mass. Charge and mass are very distinct attributes.

David Hammen has given a fine example. Here is another example:

If charge were measure of inertia:

1. Then an electron would have some inertia. Let it be $I$

2. A proton would have same inertia as an electron, since the magnitude of charge is same. Since inertia would be a positive value, proton's inertia = $I$

Now Consider a Hydrogen atom, a system of 1 electron and 1 proton.
By superposition principle, net charge $= 0$. Does that that the system has $0$ inertia ?

Separately, both constituent particles had some inertia!

• Electron offered resistance to motion.
• Proton offered resistance to motion.
• But together, they offered no resistance to motion !?

We may logically deduce that measure of inertia must be a scalar, as well must be strictly positive.

Therefore, if charge were taken to be measure of inertia, it would cause a havoc, since charge can be both negative and positive.