A teacher explains the potential at a point due to a point charge to his students.

He says, "Let's assume we have a point charge $q$ and we want to find out potential at point $P$ which is at $r$ distance from the charge."

He further explains that the work done in holding a unit positive charge at any point in the electric field is the electric potential of the field at that point.

He tells them that we have a familiar expression to calculate the potential in such cases.

Then, he asks them what if there is more than one charge at different distances from point $P$.

So, let's further elaborate on this situation and try to understand it better.

Imagine we have a point P surrounded by “n” number of charges $q_{1},q_{2},q_{3}$ and so on.

Imagine we have a point P surrounded by “n” number of charges $q_{1},q_{2},q_{3}$ and so on.

Let’s assume the distances of charges from point P are $r_{1},r_{2},r_{3}$……. $r_{n}$ respectively.

Now, we need to find the potential at P due to these surrounding charges.

We know, the electric potential at a point P due to a point charge ”q” separated by distance r is as given in the above message.

Similarly, for $q_{1}$, the potential at point “P” is given in the above image.

Likewise, for $q_{2}$, the potential at point “P” is given in the above image.

And similarly, for $q_{n}$, the potential at point “P” is given in the above image.

Now, the total potential acting at a point “P” due to all these charges is added due to the superposition principle.

That is as shown above.

Now, let's learn the same concept in vector notation.

Suppose the whole system is located in the Cartesian coordinate system as shown in the figure.

The positions of vectors are supposed to be $r,r1,a$as shown above.

We know that,

Also

Similarly,

Therefore, the total potential at “P” due to these charges is as shown above.

We will learn how to find the electric potential at a point P for more than one charge.

Revision

The electric potential on P due to a point charge is denoted as shown in the above image.

The electric potential on P due to “n” point charges can be written as shown in the above image.

The electric potential on P due to “n” point charges in vector form is as shown in the above image.