### Further Reading

# Coulomb's law

#### Prerequisites

- Formulation of Newton's Laws as foothold principles
- Electric forces
- Charge and the structure of matter

There are two kinds of charge and there are three qualitative properties of the electrical force between charged objects. ( You can see this for yourself in a simple experiment with Scotch Magic Tape^{TM},: Simply pull two strips of tape off each other - both strips become charged. Then you will see that the dangling ends of the tape strips attract each other. When you bring them close to another pair of charged tape strips, they either repel or attract.)

- Charges of the same kind repel each other, charges of different kinds attract.
- The forces between charges get weaker as the charges get farther apart.
- The direction of the force between charges is along the line from one to the other.

From our work with the Newtonian theoretical framework, we also expect

- The forces two charges exert on each other should be equal and opposite (satisfy Newton's 3rd law).

In this webpage, we will elaborate these four ideas to create a model of electric force that both encompasses these qualitative ideas and allows us to quantify electric forces so that we can use them in Newton's 2nd law. The critical idea seems to be charge.

We first have to answer some questions about charge: What is it? How does it behave? And what mathematical model is appropriate for describing it?

## Quantifying charge

"Charge" is what we are calling that property of the basic particles of matter -- electrons and protons -- that create and feel the electric force. One basic fact that is easily demonstrated is:

*The electrical effects of an electron and a proton on other charged particles are opposite to each other and equal in strength.*

This is why atoms with the same number of electrons and protons don't seem to have obvious electric effects. Since they seem to cancel and be equal, it seems that a natural way to assign a number to charge is to count the number of protons and subtract the number of electrons. The total effect of neutral matter would be zero and observable effects would come from having an excess of one or another. (We are temporarily assuming all these charges are pretty much in the same place so we are ignoring polarization effects for now, since they are much smaller than the electrical effects from unbalanced charges -- though polarization does turn out often to be important!)

One way to assign a number to the charge on an object is just to count.

*A plausible way to quantify charge would be to assign the number: Q = (number of protons) - (number of electrons). *

Unfortunately, we don't do this. Electrical forces were analyzed and the theory worked out in the last half of the 18th century. The understanding of matter as made up of atoms didn't come till almost 100 years later and the understanding of atoms as made up of electrons and protons not till 50 years after that.

As a result, the scale for measuring charge was based on other things. The result is the standard unit for charge is taken to be the Coulomb:

1 Coulomb = the charge on 6.24 x 10^{18} protons.

Actually, it seems rather reasonable to do this, since counting up to 10^{18} protons might take us a very long time! But since we have a new arbitrary measurement scale, we can assign a new dimension to go with M, L, and T (mass, length, and time). We'll call it "Q". It will turn out to be quite convenient for checking and proposing new electrical equations using dimensional analysis.

Since we are going to use "Coulombs" instead of counting protons and electrons, it's useful to introduce the number that is the value of the amount of charge on an electron or proton in our Coulomb unit. This is just the reciprocal of the number of charges in a Coulomb:

*e* = the magnitude of the charge on a proton or electron = 1.6 x 10^{-19} Coulombs.

## Creating the force law

#### How does the force depend on the charge?

First, let's ask ourselves how the force between two charges must depend on the magnitude of the charges. We've drawn a sketch of two charges, labeled *Q* and *q*, separated by a distance *r*. Just for convenience, we'll talk about them as if they are positive. (When we get to talk about how vectors go into the law, we'll worry about whether it works for negative charges too.) For two positive charges, the forces they exert on each other tend to push them apart. We have labeled the force each charge feels from the other. By N3, we expect them to be equal and opposite. (In this diagram, since we haven't put arrows over the symbols, the symbols for the forces only refer to magnitudes of the forces).

In the second diagram, we've now asked what would happen if we put a second charge, q, on top of the first. First of all, we'd expect that charge q would feel the same electric force from Q that the original one did. So if we lumped these two charges together, considering them a single charge of amount 2q, we expect that the force that 2q would feel from any other charge would be twice what a q charge would feel.

This suggests that the force that a charge feels should be proportional to the amount of charge feeling the force.

If we now look at what has happened to Q, it should feel the force from the original charge q, but it should feel an equal amount of force from the second charge, q. So the force that Q would feel from a charge of 2q should be twice the charge that it feels from a charge q alone.

This suggests that the force that a charge causes should be proportional to the amount of charge causing the force.

Our first conclusion is therefore that when two charges exert electric forces on each other, the magnitude of the force each one exerts on the other is proportional to ($\propto$) the product of the two charges.

$$F^E_{q \rightarrow Q} = F^E_{Q \rightarrow q} \propto qQ$$

Note that we have no vectors over the symbols so we are only talking about the magnitude of the forces.

#### How does the force depend on the distance?

Now we have to get the distance dependence. We know it gets weaker, but how? This can be determined simply by experiment.* The result is that the force between the two charges falls off like the square of the distance between them.

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

We've been careful to label the "$r$" to show that we mean the distance between the two charges.

### Do we need a constant?

Now we have to decide if we're done or if we need to put anything else in. By looking at the dimensions it's clear that we have a problem. The left side of the equation has dimensionality of force:

[$F$] = ML/T^{2}

while the right side has something else:

$\big[\frac{qQ}{r^2}\big]$ = Q^{2}/L^{2}

It has our new dimensionality, Q in it and is missing other factors. So let's introduce a "universal constant with units". This is typically what we have to do when we introduce new arbitrary measurement scales (here, charge) without really knowing how they relate to other things we have already defined. These constants are an indication that we didn't know enough physics at the time to define what charge "really" is. (There is a system of measurement that defines charge in terms of the already existing M, L, and T measurements, but it didn't catch on.)

So our final law is going to be (for the magnitudes of the forces)

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

This result is called **Coulomb's law of electric forces** and the constant k_{C} is called *Coulomb's constant.* Sometimes you'll see it written as K, and sometimes (for peculiar historical reasons) as $\frac{1}{4π\epsilon_0}$. (Sorry about that.**) Since you will actually sometime see this in more advanced biology texts, we give the value here, but we'll stick with Coulomb's constant for this class -- and will write it in the easy to remember form *k*_{C} = 9 x 10^{9} N-m^{2}/C^{2 }(good to about 0.1%). Notice that the units of this are just those needed to cancel the charge squared in Coulombs (C) on the top and the meters (m) squared on the bottom to give Newtons (N).

$k_C =$ 8.99 x 10^{9} N-m^{2}/C^{2}$\epsilon_0 =$ 8.85 x 10^{-12} C^{2}/N-m^{2}

We now have to pay attention to the vector character of the law. Go to the follow-ons to get more insight into the structure of the law.

*It's tempting to notice that the area of a sphere surrounding a charge grows like r^{2} as the radius of the sphere grows. If one is trying to create a theory of electric forces, it would be quite natural to say that the charge "emits" something that can be felt as a force by another charge and it "thins out" as you go further away by being spread out over all directions -- over the surface of the sphere surrounding the charge.

** It also makes some of the more advanced equations of electromagnetism slightly simpler.

Joe Redish 10/13/11

#### Follow-ons

Last Modified: May 24, 2019