4.5 Potential Energy of system of a point charges
Let’s consider the electric potential energy of system of charges. Electric potential energy.
Let’s assume that we have two point charge system, with charge of q1 and q2 sitting over here. And they are separated from one another by a distance of r. How do we determine the electric potential energy of this system?
In order to do this, we follow a procedure such that in the first step, we calculate the potential of one of these charges, let’s say q1 at the location of the other charge, and that is q2. So at this point we calculate the potential of this point charge q1. And that’s going to be equal to v1, which is equal to q1 over 4 Pi Epsilon 0 r.
And then as a second step, we bring charge q2 from infinity to this point of interest. Therefore we bring the charge q2 to this location from infinity and we look at how much work is done during this process. And that work will be equal to the potential energy of the system.
So u is going to be equal to work done in bringing charge q2 from infinity to this point. And that work then is going to be equal potential generated by q1 times the charge q2. Therefore it’s going to be equal to v1 times q2.
In [inaudible 02:33] form, since v1 is q1 over 4 Pi Epsilon 0 r, we multiply this by q2. And this expression will give us the potential energy of this two point charge system. U is going to be equal to q1 q2 over 4 Pi Epsilon 0 r.
We can generalize this result to systems which involve more than two point charges. Let’s assume that we have three point charges. That they’re located at the corners of an equilateral triangle. So here we have plus charge q1 and here we have plus charge q2. And then here we have minus negative charge q3. And we’d like to express the electrical potential energy of this system.
Let’s assume that these distances are equal to one another, and it is equal to d. Therefore u is going to be equal to 1 over 4 Pi Epsilon 0 is going to be common for each term. So the potential energy of q1 q2 system is q1 q2 divided by the distance between them, which is d. And then plus potential energy q1 q3 pair will be q1 times minus q3, divided by d, the separation distance between them.
And finally, q2 q3 pair, we’re going to have q2 times minus q3 divided by d. So we look at every possible pair and express their potential energy. And the total potential energy of the system will be sum of the potential energies of every possible pair in our system.