# Physical significance of product of two charges

In Coulomb's Law we see the product of two charges in numerator, and I was wondering what it really signifies, because charge multiplied by charge is not a charge. So what exactly is a charge multiplied by a charge? This question could also be asked about the product of two masses in Newton's law of gravitation.

• Let me ask a counter-question: Do you see this product anywhere else, without Coulomb's constant to take away its units? – probably_someone Dec 30 '16 at 12:26
• sorry i couldn't understand the purpose of your question perhaps i should say i didn't understand your question as whole .please if u can elaborate . – Syed Ilyas Dec 30 '16 at 12:30
• In some rough sense charge multiplied by charge at a point distant from the system is proportional to the the electrostatic potential energy of the system at that point. – UKH Dec 30 '16 at 13:02

Charge multiplied by charge is... charge multiplied by charge. Unfortunately, we don't have a name for this quantity, nor do we have any sort of physical intuition with it, because it simply never appears in varied enough locations and situations (i.e. somewhere other than Coulomb's Law) that a name would be useful. The same can be said of squared mass. We just never have a reason to think of that quantity separately from its law.

• why its appearance only in Coulomb's Law can be a reason that we don't need to have any physical intuition and how is appearance in varied enough locations a condition to have the need for physical intuition . please if you can make it more clear – Syed Ilyas Dec 30 '16 at 12:49
• Let's try for a moment to get some physical intuition on squared charge. By rearranging Coulomb's Law we get that squared charge is proportional to force times area, or mass times volume per squared time, if you will. So we get another quantity that we have no intuition for! Unless you can find a way to link squared charge directly to another thing that we already have an intuition for, it seems impossible to gain any intuition from it. – probably_someone Dec 30 '16 at 12:54
• Any deeper than this, and we stray into the realm of philosophy: "Why do we measure the things we do? Why do we consider basic units basic? Where does intuition on these basic things come from?" Unfortunately, not being a philosopher or a metrologist, I am not qualified in any sense to answer these questions. – probably_someone Dec 30 '16 at 12:56

Ultimately it is correct because it matches the experiments, but we can give some a posteriori justification to this.

Physicists describe electromagnetism by means of electric and magnetic field, rather than Lorentz force; the same holds for gravitational field vs. Newton's force.

The way the interaction between (two, for simplicity) particles is modelled is the following: every particle per se radiates a field that permeates spacetime. Then, if some other particle "can feel" that field, it experiences a force.

So let's for a moment focus on the fact that is some region of spacetime. The ability of a particle to feel some field is described by saying that the particle is charged with respect to that field. For EM field the charge is electric charge; for (newtonian) gravitational field the charge is mass. Therefore we have that the force experienced by the particle is proportional to electric charge or mass.Therefore we have $$\text{force} \propto (\text{charge}) \times (\text{field})$$

Now let's go back to the creation of the field. For electromagnetism the property that allows a particle to create the field is electric charge; for gravitational field it is mass. Therefore the field will be dependent on the electric charge or the mass; in the simplest case it will be proportional to the electric charge and the mass, ie $$\text{field} \propto (\text{charge})$$

Therefore we have $$\text{force} \propto (\text{charge})^2.$$

Note that the first relation ($\text{force} \propto (\text{charge}) \times (\text{field})$) is really the definition of field. The second relation, on the opposite side, is a very special feature of these forces. We could as well have forces for which "the generator" is different from "the probe", or for which the field is not proportional to charge. But for electromagnetism and (newtonian) gravity nature seems to be simple.