1.7 Coulomb Law
We have seen that property of matter which is responsible for the electrical interactions is the electric charge. The force associated with the electrical interactions can be either attractive or repulsive in nature, which brought the concept of positive and negative charge.
Now let us try to concentrate on this force. The magnitude of the electrical force between two charged point particles is first introduced by an experimental law at 1785 by Charles Coulomb, which is known as Coulomb’s Law. And it simply states that the magnitude of the force between two point charges is directly proportional to the product of the magnitude of the charges and it is inversely proportional to the square of the distance separating them.
Therefore if you consider two point charges, Q1 and Q2, and let’s assume they are like charges, they are both positive and since they are like charges, therefore they will exert a repulsive force to one another. Namely Q1 will repel Q2 with a force of let’s say F21 and similarly Q2 will repel Q1 with a force of F12. So F21 is the force on Q2 due to Q1 and F12 is the force on Q1 due to Q2. Of course from Newton’s third law, from action-reaction principle, these forces are equal in magnitude. F12 magnitude is equal to F21 magnitude, and let’s say they both are equal to F.
Let’s assume, or let’s denote the distance between these two point charges by R. Then Coulomb’s law simply states that the magnitude of the force that these two charges exert to one another is directly proportional to the magnitude, product of the magnitude of the charges, Q1 times Q2. It also states that magnitude of the force is inversely proportional to the square of the distance separating these two charges. Again where R is the distance between the charges. Okay.
So simply by looking at this relationship we see that this force is a long range force, and in other words F magnitude goes to 0 as the distance between the charges approaches to infinity. And the charges do not need to be in contact to be able to exert this electrical force to one another, they can do this at a distance.
One can find a proportionality constant to be able to express these relationships in the form of an equation. And if you do that it turns out to be that the magnitude of this electrical force, or Coulomb force, becomes equal to a constant which is represented as 1 over 4 pie epsilon 0 times Q1 magnitude and Q2 magnitude divided by the square of the distance separating these two charges.
The proportionality constant over here is known as Coulomb’s constant. And it has the numerical value of 8.99 times 10 to the 9 Newton meter square per Coulomb square in SI unit system. Epsilon 0 which appears in the denominator of this constant is known as permittivity of free space.
Permittivity is a constant related to the electrical properties of the medium of interest. And Epsilon 0 is the permittivity for air and vacuum and this quantity has a numerical value of 8.85 times 10 to the minus 12 Coulombs square per Newton per meter square.
If you recall universal law of gravitation, which is basically associated with the force, gravitational force, between two point masses, and it is this force is equal to, the magnitude of the force is equal to, some constant which is known as gravitational constant times the product of the masses divided by the square of the distance separating them.
So here we have a point mass of M1 and another point mass of M2 which are separated by a distance of R from one another. And the universal law of gravitation states that they attract each other when a force such that it’s magnitude is proportional to the product of the masses, and the magnitude of the force is inversely proportional to the square of the distance separating them.
From the mathematical point of view we observe that this is very similar to the Coulomb’s law we have just introduced. In this case the magnitude of the force between two point charges was equal to Coulomb constant 1 over 4 pie Epsilon 0 times product of the magnitude of the charges divided by the square of the distance separating them. Again if you consider our positive point charges, separated by a distance of R from one another, we ended up with pair of repulsive forces that they exert to one another.
Now from the mathematical point of view, if you just replace the Coulomb constant with the universal gravitational constant and the charges with the masses we basically end up with the force expression for the gravitational interactions. Of course the major difference over here is that in the case of gravitational interactions, force can only be in attractive nature, whereas in the case of electrical interactions, Coulomb force, can be either attractive or repulsive depending on the nature of the charges, whether they are like charges or unlike charges.
The mathematical form of Coulomb’s law is given as the magnitude of the force that two point charges exert to one another is equal to Coulomb constant, 1 over 4 pie Epsilon 0, times the product of the magnitude of the charges, Q1 times Q2 magnitude, divided by the square of the distance, R square, separating these two charges.
We can apply Coulomb’s law only to point charges. Coulomb’s law can be applied only to point charges. Where we will see later on what type of manipulations that we can make to be able to apply this law to the charge distributions.
Second Coulomb’s law applies only to electrostatics. Coulomb’s law applies only to electrostatic where charges are either at rest or moving at very low velocities relative to one another.
Furthermore, since Coulomb force is a force that for like all the other forces they can be superimposed. In other words, if our system consists of more than two point charges, and if we’re interested to figure out the net force acting on a specific charge due to the rest of the charges, then we will calculate the force on that specific charge due to each one of those charges and then we vectorialy add all those forces to be able to get the net force. So F net is going to be equal to sum of all the forces acting on the charge of interest due to all the other charges.
In other words, if I have bunch of charges in the system of interest, like Q1, Q2, and Q3, and Q sub n, and if I’m interested with the net force on a specific charge QJ due to all the other ones, I calculate the force on QJ due to Q1, Q2, Q3, up to Q sub n and then add those forces vectorialy to be able to get the total force. We will do an example related to this feature in the next section.