The Coulomb Force
In the beginning of our studies of mechanics we discussed four fundamental interactions found in nature. Among these was the electromagnetic force, which for static charges without intrinsic magnetic moments, becomes the Coulomb force that we discussed.
Coulomb's law applies to interactions among spherically symmetric charges. With macroscopic, charged, spherical objects, there is another complicating effect that takes place - polarization. We will discuss polarization in a later chapter of the text, but it just means that a charged object will cause another nearby spherically symmetric, charged object to shuffle its charge so that while the object may be spherical, the charge is no longer spherically symmetric when in proximity to the first charged object. This occurs because charges will get moved a bit (or a lot in conductors) due to the presence of other nearby charges.
Because of these complicating (and real-life) issues, Coulomb's law can be very limiting, but we will see later in the course that Gauss's law, one of the five fundamental laws of electromagnetism that is very powerful and applicable to real-life scenarios, reduces to Coulomb's law for spherically symmetric charge distributions. So Coulomb's law is useful, but isn't fundamental to the subject.
Why then do we discuss Coulomb's law if it doesn't belong in that fundamental set of laws of electromagnetism? The biggest reason I can think of is that Gauss's law is limited, in practice, by the symmetry of a charge distribution. For all but the simplest symmetry classes, Gauss's law cannot be worked out by hand because the calculus gets so complicated. In those cases, one needs to revert back to Coulomb's law, but written as an integrand, to do such calculations. In that sense Coulomb's law will be an easier route to doing the mathematics.
So a quick recap on Coulomb's law: 1) It only works for spherically symmetric charge distributions. 2) It is useful later on for doing the mathematics to find fields near asymmetric distributions, but will be written as an integrand under an integral for such cases.
Coulomb's law looks mathematically the same as Newton's law of universal gravitation:
Recall that based on our subscript conventions, the first force should be read "the Coulomb force on A due to B equals...". The on/by notation is an arbitrary choice adopted in this textbook (as well as many others), but thinking of it the other way (by/on) will give you exactly the opposite answer (opposite sign) to problems in this book, every time.
The terms q (quantity of charge) are the net charges residing on each object. The letter 'c' is reserved for the speed of light, and the upper case 'C' will be used for capacitance in this course, so unfortunately charge gets a 'q'.
The constant So 'k' is not a fundamental constant. The fundamental constant among these is (spoken as 'epsilon zero') and not 'k', but it is common to write 'k' since it is easier and neater to write.
Written as a constant the value is If it helps to memorize it, just realize that it's essentially where the number and exponent are the same.
Please note that there is no minus sign in Coulomb's law. This implies that charges repel one another when they have the same sign unlike gravity which attracts when both masses are positive (the only type of mass in nature).
Coulomb's Law is an Inverse Square Law
There are many inverse square laws in nature due to the three spatial dimensions that we seem to live in. I say "seem to" since it is suspected that there may be something more to reality than that. It's possible that there are either more or fewer spatial dimensions than the three we perceive, but that's a discussion for a day after you've digested all that we presently know about the three assumed dimensions. If you can't wait for that day, try searching terms like 'Holographic universe' and 'string theory' on the internet.
In any case, Coulomb's law written as above, looks like an inverse cube law unless you recall that any vector may be written as a product of a magnitude and a direction.
Here are the steps in case it isn't obvious to you: First substitute and then rewrite Coulomb's law as
Next, expand the numerator as and get Using our shorthand of , we get leaving an inverse square law.
