Unit 1 - Coulomb's Law

Coulomb's Law gives the force exerted between two electrical charges at rest. Things are more complicated when charges are in motion; we will consider that type of case later.

In words, as we stated, the force is: proportional to each charge; inversely proportional to the square of the distance between them; and directed along the line between them, like charges attracting, unlike charges repelling.

In a formula, F = 1/(4pe_o) Q_onQ_by (r_on - r_by)/|r_on - r_by|³

Several things can be noted, in hopes of reducing the problems you can give yourself. First, although the formula is symmetric between the charges (Newton's Third Law at work), I have written the two Q's with subscripts. This is to start you out from the beginning with a distinction that enters into a lot of problems: any force situation always involves both an object acted on and something acted by. From Newton's Third Law, there is always a related entity that has them vice versa, but almost never do both of them enter into a particular problem, and if you use the wrong one you will have trouble. Also, I wrote both Q's as capitals, while our text shows qQ. I think the text's notation has already led to confusion among students; we are used to thinking of a 'test charge' acted on, and of a 'test charge' being small; and we also think of q meaning smaller than Q. But our author has introduced (big) Q for the (small) (test) charge acted on. He has a reason, I think; in the next section, he will (as it were) factor off the Q, and he doesn't want the different confusion of changing the remaining q from capital to the lower-case form which he will usually be using.

Second, there are the same labels on the r's. Here it is more important to keep a distinction; the force formula is antisymmetric in the position vectors. Our textbook's author introduces his 'script r' vector with a standard meaning which involves this difference, and that's one way to handle it. But definitely not all books will use that notation, and if a student isn't careful in his note-taking, he'll lose it. I would favor accepting from the start that many letters in physics (and other fields) get overused, and also from the start practicing the good habit of writing meaningful labels on those repeatedly-used letters.

Third, there is the exponent in the denominator. Coulomb's Law is inverse-square, why does the formula say cube? Well, any force is a vector, with a direction, and in the electrical force takes its direction from the positions. You can write the formula in terms of its magnitude multiplying a unit vector for the direction. This is simpler in concept, and actually is simpler when the direction is that of a named unit-vector. But if the direction is not a recognized unit vector, how do we express it? We find a vector with the right direction and make a unit vector out of that, by dividing by its size. Now, in this case, the right direction is given by the difference of position vectors, and the magnitude of the difference vector is the distance between the charges. That's the same factor as we already had, so we collect the like factors: squared from inverse-square, a third factor from making a unit vector.

Fourth, there is a proportionality constant. We write it in a fancy way instead of as a simple constant; why? Here, the reason is part historical (like a good many things that have odd labels or odd structures), but also part practical. The practical part comes from the fact that, while Coulomb's Law tells you everything if you know exactly where the charges are, unfortunately most of the time you don't know exactly where the charges are. So instead of Coulomb's Law, most work on electrical problems starts with a different law. That law has e_o, but not the 4p. By writing Coulomb's Law in a less-simple way, that more-used law is simpler.

E-M Home Page/Unit 1/

Format for this page adapted with permission from pages
constructed by Dr. Lionel D. Hewett for his course
Modern Physics 1.