Reading the content in Coulomb's law


The vector form of Coulomb's law is compact — just one term, but there are lots of symbols and the symbols have lots of "diacritical markers" — hangers on, like subscripts and different vector marks — arrows, hats, and subscripts:

$$\overrightarrow{F}^E_{Q \rightarrow q} =  \frac{k_CqQ}{r^2_{qQ}} \hat{r}_{Q \rightarrow q}$$

This is because, as with many of our equations, we pack a lot of conceptual ideas  into an equation and must be carefully interpreted to make sense of what the equation is telling us. Each symbol in the equation reminds us of some important conceptual idea. Let's make sense of Coulomb's law — "see the dog in the equation" — by unpacking what you have to know in order to understand this relationship.

1. What does it represent? The complicated symbol on the left tells us that what we are doing here is calculating something. It's not like Newton's 2nd law — a relationship between quantities that are defined elsewhere by other means. The left hand side here tells us what physical concept we are calculating: the electric force that one charge exerts on another and the expression tells us what properties of the objects and the physical situation it depends on. Both the main symbol and all its "hanger ons" tell us how this quantity fits into the Newtonian framework.  It's a force (F) of a particular kind — electrical (E). The object causing the force has the property that its charge is $Q$ and the object feeling the force has the property that its charge is $q$.

2. What does this force depend on? Remember that a (physical) force is an interaction between two objects that have a tendency to make both objects change their velocities (through being part of a net force in Newton's second law). This law tells us that the force depends on the magnitude and direction of the distance between the two objects ($r_{Qq}$ and $\hat{r}_{Q \rightarrow q}$ respectively); and also on the magnitude of both of the charges.

3. $\pmb{qQ}$ — The fact that the law has the product of the charges of the two interacting objects on top tells us that the force is proportional to both the charge that is creating the force and the charge that is feeling the force. That's sensible since if we look at the charge creating the force, if it has multiple pieces, each piece of the charge will exert a force on the charge feeling the force and the total will be that all added together. (If there are N electrons creating the force, the force felt will be N times as big as the force created by 1 electron.) This is also sensible for the charge feeling the force. If the charge feeling the force is made up of a number of charges, each of the charges feeling the force created by the other charge will feel the same force, so if we add all those charges together as one object, the total force felt will add those forces together. (If there are M protons feeling the force, the force felt will be M times as big as the force felt by 1 proton.)

4. $\pmb{r^2_{Qq}}$  — This symbol, and the fact that it is in the denominator, tells us that the force falls of like the square of the distance between the two charges.

5. $\pmb{\hat{r}_{Q \rightarrow q}}$ -- This symbol tells us the direction of the force that an object with charge Q exerts on an object with charge q.  Recall that this symbol is defined as

$$\hat{r}_{Q \rightarrow q} = \frac{\overrightarrow{r}_{Q \rightarrow q}}{r_{Q \rightarrow q}}$$

and means: the displacement vector from the position of charge $Q$ to the position of charge $q$ divided by the magnitude of that distance. It leaves us with a vector pointing in the direction from charge $Q$ to charge $q$ but just a unit vector that gives direction — no units and it has unit length. This is the natural direction that we would choose for two positive charges that we know repel each other.

This form of the law also treat the charges differently than when we write just the magnitude of the force. Since changing the sign of one of the charges changes the forces from attraction to repulsion, and since the negative of a vector is just the flip of the vector (the same length vector but in the opposite direction), the signs from the charges work correctly to give the directions.

This is a complicated looking equation, but if you remember the physical ideas it is representing —

  • charges exert electric forces on each other, like charges repelling, opposite charges attracting;
  • the forces each charge feels are proportional to the strengths of the charges creating the force and the charges feeling the force;
  • the force gets weaker with the square of the distance between the charges;

then you ought to be able to make sense of what information the symbology is carrying.

When does this equation work? This equation explicitly gives the force between two "point charges". What this means is that the charges are very small compared to the distance between them. For that case, this equation always works. If you have a distributed charge you can't use this equation. You have to break up the distributed charge into tiny charges. You can then use this equation and add up the results (perhaps do an integral). But because the distances will change for each of the tiny pieces, the result won't look exactly like Coulomb's law anymore.

Joe Redish 10/15/11


Article 389
Last Modified: May 14, 2019