# What is meant by difference in electric potential?

I know this question like this has been asked many times but I am still so confused about the concept of electric potential and voltage etc… Like I know what voltage and electric potential energy is when example involves single plus charge and single negative charge. But when example involves bunch of positive and negative charges in two different position (in all combinations, like only plus charge, only negative charges together, or some mixed) and you are told to see if there is electric difference between two location than I feel like I don’t understand these concepts properly. Like there is an example online that shows two bunch of negative charges only that were lovated at two different location. One bunch has more negative charges than other, and the teacher says that there is actually voltage difference between them. I couldn’t understand this, because wouldn’t these bunch of charges repel each other? Why wouldn’t these negative charges in one bunch would want to travel to location of negative bunch with less negative charges? I don’t know if I am rambling. But Please help!!!!

• "Like there is an example online that shows two bunch of negative charges only that were lovated at two different location." - Please provide the link. – Farcher Jan 18 '17 at 5:00
• 4:23, he starts talking about it. youtube.com/watch?v=g287MugJC9E&t=633s – TLo Jan 18 '17 at 5:46

The fact that a grouping of several charges will exert forces on each other is irrelevant to what the potential difference between two locations in space is. Assuming that you know the instantaneous positions of all the charges what you do is calculate the sum of the individual potentials due to each charge for the first location, then for the second location: $$\Phi_1=\Sigma \dfrac{kq_j}{r_{j1}}\text{ and }\Phi_2=\Sigma \dfrac{kq_j}{r_{j2}}$$ where $r_{j1}$ and $r_{j2}$ are the respective distances from the individual charges to location 1 and location 2, and $\Phi_1$ and $\Phi_2$ are the potentials at those locations due to the charges.

Notice that we don't care how the charges interact with each other while calculating these potentials. Those interactions may cause the charges to change position which will, consequently, change the values of the potentials, but they don't change the method we use to calculate the instantaneous potentials at the locations.

If someone tells us that the charges are fixed in position, that means the potentials won't change, but, again, the charges interacting with each other is irrelevant.

Because potential is related to the work done by a unit positive charge I have changed the charges on the conducting spheres to be positive to make the explanation a bit easier and I have also assumed that the conducting spheres are sufficiently far apart that the charges on one sphere do not influence the distribution of charges on the other sphere.

Note that as the spheres are conductors the charges reside on the surface of the spheres.

The surface of a conductor is all at the same potential and potential at a point is defined as the work done in taking unit positive charge from infinity (the arbitrary zero of potential) to the point.

Now imagine bringing a unit positive charge from infinity up to the each of the sphere.
Less work would be done in bringing the unit positive charge up to the sphere with the larger radius because the charge on that sphere is more spread out than the charge on the smaller sphere.
So the potential of the larger sphere is less than the potential of the smaller sphere.
So there is a potential difference between the two spheres with the smaller sphere being at the higher potential.

If the two spheres were connected together the because there is a potential difference between the sphere charges would move along the conductor (a current would flow) until the potentials of each sphere was the same as shown in the diagram below.
I will explain later where the numbers come from.

In fact a positively charge conductor means that it has a deficit of electron so in this case it is not really positive charges moving from the smaller sphere to the larger sphere but electrons moving from the larger sphere to the smaller sphere.

With the assumptions that I have made above one can be a bit more formal in the analysis.

The potential $V$ of a conduction sphere of radius $R$ carrying a charge $q$ is given by $V = \dfrac{1}{4 \pi \epsilon_o} \dfrac q R$ so initially the potentials of the two spheres are $\dfrac{1}{4 \pi \epsilon_o} \dfrac {Q} {2r}$ and $\dfrac{1}{4 \pi \epsilon_o} \dfrac Q r$

Joining them with a conductor means that charge moves from one sphere to the other with the charges on the spheres now being $Q_2$ and $Q_1$ as in the diagram below.

The final state must be that the potentials of the spheres must be the same so $\dfrac{1}{4 \pi \epsilon_o} \dfrac {Q_2} {2r} =\dfrac{1}{4 \pi \epsilon_o} \dfrac {Q_1} {r} \Rightarrow Q_2 = 2Q_1$

Charge must be conserved so $Q_2+Q_1 = 2Q$ which gives $Q_2 = \dfrac {4Q}{3}$ and $Q_1 = \dfrac {2Q}{3}$

• Hello. Thanks for the detailed answer. It was informative. But can I ask you, how does electron traveling from larger sphere to smaller sphere equal out the potential difference? I think this may deal with electric field canceling out of one proton, one electrons making it neutral. But what is happening at a fundamental level that one positive and one negative charge coming together in the sphere equalizes the potential difference between them one by one? Thanks in advance. – TLo Jan 18 '17 at 17:36
• At the beginning there is a deficit of electrons on both spheres. On connecting the conductor between two he two spheres electrons move from the larger sphere, making it have more positive charge, to the smaller sphere, making it have less positive charge. – Farcher Jan 18 '17 at 17:54