I know that electric potential energy is defined for a system of charges, not for a single charge. But what about electric potential?

Question

My professor explained me in the best manner possible about the electric potential energy and why it is NOT defined for a single charge, BUT my question is as Electric potential is just electric potential energy with one charge being a unit positive charge, therefore is electric potential also defined for a system of charges?

Answer 1

The relationship between electric potential and electric potential energy is like the relationship between electric field and electric force; in the sense that every electric charge produce an electric field (that has a particular value in any point in space), and it does that regardless of whether thare are any other charges around. But in order to talk about (electric) force, you need at least another charge in the picture, because what happens is that the electric field created by charge $q_1$ acts on the charge $q_{2}$ and accelerates it.

$$E\ \overbrace{\rightarrow}^{\text{is for}} \ V$$ what $$F\ \overbrace{\rightarrow}^{\text{is for}} \ U$$

It's the same with electric potential and electric potential energy. A charge (or system of charges) produce an electric potential at any point in space, but we can only talk about electric potential energy between two charges (or between a system of charges and a test charge, for example).

So $E$ and $V$ refer to what a charge (or a system of charges) create in a particular point in space, while $F$ and $U$ are quantities (or terms) that refer to an interaction between two or more electrically charged entities.

The answer to your question "therefore is electric potential also defined for a system of charges?" is yes. The electric potential that a system of charges creates in a given point $P$ is:

$$V(P)=\frac{1}{4\pi\varepsilon_{0}}\sum_{j=1}^{N}\frac{q_{j}}{r_{jP}}$$ $N$ is the number of charges, $r_{jP}$ is the distance between charge $q_{j}$ and point $P$.

Answer 2

Well Electric field and electric potential are related by the relation: $$ \vec{E}=-\nabla \phi $$ There is actually a even more general relationship between them. It says: $$ \phi=-\int{\vec{E}\cdot \vec{d\ell}} $$ now let's multiply both sides by Charge: $$ Q\phi=-\int{(Q\vec{E})\cdot\vec{d\ell}} $$ and we know that $\vec{F}=Q\vec{E}$.
$$ Q\phi=-\int{\vec{F}\cdot\vec{d\ell}} $$ but the term on right side could be identified as Work done. $$ Q\phi=-W $$ but dW=-dU(U for potential energy) $$ \phi=\frac{U}{Q} $$ for more general purposes: $$ \phi=\frac{dU}{dq} $$ Now you might be wondering that what Does this mean?
Well here is the definition of potential:Potential is the amount of work that is needed to be done to bring a unit positive charge from infinity to a point P.
TO respond to your question:"therefore is electric potential also defined for a system of charges?" My answer is Yes. Because electric potential is nothing but a quantity related to a system(See the definition).
In the end just to clarify it all for once: Think of electric potential as a useful tool to find the electric field of a distribution of charges cuz it is hard to deal with electric field as a vector but potential is just a useful "Tool". The relation is just a reward that it related potential energy to potential.The "actual" purpose of potential field was to find the fields(electric) of charge distributions.(You will learn how after some time.)

Answer 3

Yes.

Electric potential is the answer to the question, "how much energy would it take to move a small charge into this system from far away?". (where "small" is defined as small enough not to disturb the fields around the existing charges in the system significantly)

But it is only a question about a hypothetical new charge added to the system. The new charge doesn't have to actually exist for the potential exist.