Why is electric potential scalar?

Question

I can't conceptually visualize why it would be so. Say you have two point charges of equal charge and a point right in the middle of them. The potential of that charge, mathematically, is proportional to the sum of their charges over distance from the point ($q/r$). But intuitively, my thought process keeps going back to the concept of direction and how the electric field at that point would be zero. So why would the electric fields cancel while the electric potentials just add up algebraically?

Answer 1

Let me first comment that the statement

electric fields cancel while the electric potentials just add up algebraically

is not actually correct. Electric fields add due to the principle of superposition (see the section on superposition in the wikipedia article).

However, when two electric field vectors are of the same magnitude but point in opposite directions, then their sum is zero; this is what is happening at the midpoint between two equally charged particles.

Given an electric field $\mathbf E$, the electric potential $\Phi$ is defined through the relation $$ \mathbf E = -\nabla \Phi $$ so it is a scalar by definition. The electric potential also obeys the superposition principle. Provided we set the zero of potential at infinity, the potential due to a point charge $q$ is given by $q/(4\pi\epsilon_0 r)$, and $r>0$, so the potential of a point charge is either everywhere positive or everywhere negative depending on the sign of the charge. Therefore, given two point charges of the same sign, the sum of their potentials will cancel nowhere.

Answer 2

An explanation based on the definition of scalar quantities in physics.

To see why electric potential energy is a scalar quantity you need to understand the following:

A physical quantity is a scalar property of a system, when its value and its effects do not depend on the orientation of the system.

Let us assume we have a system of three electrically charged particles carrying electric charge $+Q_A$, $-Q_B$ and $+Q_C$. Let us also assume the three particles are at positions $A$, $B$ and $C$ with position vectors $\bf {r}_A$, $ \bf {r}_B $ and $\bf {r}_C$ with respect to some arbitrary origin O. We can write the total potential energy of the system of three charged particles as

$E=\frac{1}{4\pi\epsilon_0}(\frac{ Q_AQ_C}{| \bf{r}_A-\bf {r}_C |}-\frac{ Q_AQ_B}{|\bf {r}_A-\bf {r}_B |}- \frac{Q_BQ_C}{| \bf {r}_C-\bf {r}_B|})$.

We can observe that as long as $|\bf{r}_{\mu}-\bf{r}_{\nu}|$ remain fixed, with $\nu\ne\mu$, the three particles can be placed in an infinitely large number of positions in various orientations, and yet $E$ will have the same value. I.e.

The orientation of the system does not bear any measurable effects on the value $E$.

This is why electrical potential/energy is a scalar quantity.