an electric force, no matter what, is a vector quanitity

I want to ask a question about the Coulomb force or the electrostatic force.

As you can see, there are these statements:

1) The electric force is a "vector quantity".

2) The electric force between two stationary charged objects is called the Electrostatic force.

3) The electrostatic force between two charges can be repulsive or attractive.

My problems are for each question:

1) If an electric force is a vector quantity, then why do we call the force between two charges "force"? Should it be called "forces" as there are two vectors in the case of "Coulomb's law" (meaning F12 and F21)?

2)+3) I only know what is called "the magnitude of the electrostatic force is $F=k q_1 q_2/ r^2$. But what is the electrostatic force?

Does the term "electrostatic force" mean (see the example in the picture):

  1. The vector F12 and the vector F21 combined, or

  2. either the vector F12 or the vector F21?

  • 1
    $\begingroup$ You could say that for any force. It's Newton's Third Law. It seems like you're confused about what vectors and forces are, but I'm not sure how to help. $\endgroup$
    – Wood
    Commented Oct 17, 2016 at 8:03
  • $\begingroup$ When we say electrostatic force, we of course mean electrostatic force acting on a charged particle. Hence the 'electrostatic force' acting on q1 is $F_{21}$ and that acting on q2 is $F_{12}$. $\endgroup$ Commented Oct 17, 2016 at 8:09
  • $\begingroup$ Oh I get it so you mean that F12 is an electrostatic force caused by q1 and the F21 is another one but caused by q2. But the problem is in the statement of Coulomb's law: "The electrostatic force between q1 and q2 is repulsive or attractive" $\endgroup$ Commented Oct 17, 2016 at 8:18
  • $\begingroup$ According to that law or statement, the electrostatic force cannot be as we thought. $\endgroup$ Commented Oct 17, 2016 at 8:20

4 Answers 4


Think of it this way, the Earth exerts a force on a pebble which is directed towards the centre of the Earth and this force is equal to its weight $ mg$.
Also, the pebble exerts an equal force just opposite in direction on the Earth.

Similarly, $\ q_1$ exerts a force on $q_2$, so does $q_2$ on $q_1$. Only a single force acts between the two charged particles. The only difference is the direction of the force is opposite for both the particles.

When we say the electrostatic force, it is the magnitude $ F = kq_1q_2/r^{2}$ , and not the sum of $F_{12}$ and $F_{21}$.

Just think, if you combine the two vectors what you get is $0$ as the forces exactly cancel each other.

So, the Electrostatic force is equal to the magnitude of $F_{12}$ (and of course $F_{21}$) and acts in opposite directions on the two particles.

  • $\begingroup$ Hmmph ! So I get what you meant. The reason why they can't use "forces" because if they use forces (as F is a vector quantity), it would become "vector F12 + vector F21", right !? $\endgroup$ Commented Oct 17, 2016 at 8:39
  • $\begingroup$ When you said "single", did you mean "one type of force" ? $\endgroup$ Commented Oct 17, 2016 at 8:39
  • $\begingroup$ In any case you won't say forces,although you say it this way the Force on q2 is F and acts towards the right(when both are negatively charged). And by single I mean that the particles interact through a single force. Imagine the two particles(with opposite charges) are tied with each other by a rope.The first particle will pull the second one towards itself,and it will feel the same pull from the other particle. $\endgroup$
    – P_RS
    Commented Oct 17, 2016 at 8:49
  • $\begingroup$ Oh I get it, it's like the case of temperature equilibrium right. $\endgroup$ Commented Oct 17, 2016 at 8:50
  • $\begingroup$ Object A 20 degree Celsius, Object B 100 degree Celsius. If they touch, the degree of either of two objects is 50 degree Celsius (not talking about the precise number), so it means there is only one value of Celsius degree but two objects !? $\endgroup$ Commented Oct 17, 2016 at 8:51

(1) When we call it the electrostatic force, it is a matter of semantics. As others have noted:

  1. The magnitude of the electrostatic forces is equal, i.e. $|F_{12}|=|F_{21}|$. This follows from Newton's third law, so when we say electrostatic force, we actually mean the pair.
  2. In most cases, we have a source charge (which we will assume to be stationary for some reason), influencing another charge that will move due to the source charge, to simplify the problem. So when we refer to electrostatic force, we refer to the force acting on charge 2 caused by charge 1 (charge 1 is stationary)

(2) The force is $\textbf{F}=\frac{q_{1}q_{2}}{4 \pi \epsilon_{0} r^{2}}\hat{r}$, where $\hat{r}$ is the unit vector pointing outwards from the origin. See: Spherical Coordinate System. To convert $\hat{r}$ into cartesian xyz, it can be done using as $\hat{r}=\sin{\theta}\cos{\phi}\,\hat{x}+\sin{\theta}\sin{\phi}\,\hat{y}+\cos{\theta}\,\hat{z}$.

(3) Again, it boils down to semantics, when it doubt just understand it as the magnitude of the electrostatic force $|F|$, and exists as a pair of force.

  • $\begingroup$ This is the explanation I was looking for! Typically, the unit vector portion gets omitted from the formula, and it just becomes a magnitude. $\endgroup$
    – LamGyro
    Commented Aug 10, 2021 at 18:12

1) If an electric force is a vector quantity. Then why do we call the force between two charges "force" because it should be "forces" as there are two vectors in the case of "Coulomb's law". (meaning F12 and F21)

F12 is an electrostatic force. F21 is another electrostatic force.

Remember that when talking about forces, we always talk about a particular object that the forces work on. So, pick one particle, and you can talk about one electrostatic force exerted on it by something else.

2)+3) I only know what is called "the magnitude of the electrostatic force is F=k q1 q2/ r^2. But what is the electrostatic force.

Yes, the magnitude of an electrostatic force is:


This is the magnitude of F12. It also happens to be the magnetude of F21. (This is because of Newton's 3rd law.)

But they have different directions and are different forces, who just happen to be equal in magnitude. Each of the two shown particles only feel one of these forces on itself.

The electrostatic force with the magnitude above is as any other vector defined by both magnitude and direction. And this is fairly easy to write, we just have to multiply the magnitude onto the direction of the force:

$$\vec F=k\frac{q_1q_2}{r^2}\vec{\hat r}$$

The $\vec{\hat r}$ is the direction as a unit vector (one unit long). Naturally this direction can be anything depending on how exactly the charges are placed compared to each other and depending on signs.

Does the term "electrostatic force means (for example in the picture)" ? The vector F12 and the vector F21 combined. Either the vector F12 or the vector F21.

Either. Both are two different electrostatic (that just happen to have equal magnitude because of Newton's 3rd law).

  • $\begingroup$ Thank you ! Oh ! But about this I am pretty confused $\endgroup$ Commented Oct 17, 2016 at 8:32
  • $\begingroup$ The statement "the electrostatic force between...", what about this one !? $\endgroup$ Commented Oct 17, 2016 at 8:32
  • $\begingroup$ @ProtonUpUpDown When someone says "the electrostatic force between..." then this is just a short way to talk about one of these forces. Each charged particle will feel one force, and Newton's 3rd law says that they will always feel the same force magnitude (the forces are just opposite). Because of this symmetry, that statement is often said. But only one force on one particle is meant. $\endgroup$
    – Steeven
    Commented Oct 17, 2016 at 8:35
  • $\begingroup$ Oh I get it. So this matter depends on situation, is it right !? $\endgroup$ Commented Oct 17, 2016 at 8:41
  • $\begingroup$ Oh I get it. I feel like this is just like "temperature equilibrium", am I right. Like if one object is 100 Celsius degree and the other is 20. When they touch (wait I am not stress the precision here". For example the temperature of equilibrium will be 50 celsius degree. It means the temperature is the same, only "one magnitude" but "two temperature on two objects") $\endgroup$ Commented Oct 17, 2016 at 8:47

It boils down to the use of English.
Change "Coulomb force" into "Coulomb interaction" and what do you get?

1 (Electric) force is a vector
2 The interaction between two charged particles is called the Coulomb internation. The magnitude of this Coulomb interaction $F$ is given by $F = \frac{kq_1q_2}{r^2}$. And it is often the case that the words "magnitude of" are omitted when the word force is used.
3 The Coulomb interaction can either be attractive or repulsive.

So in common usage "Coulomb force" and "Coulomb interaction" are synonyms.
So although the word "force" in "Coulomb force" is used in the singular it actually refers to the two action and reaction forces.

  • $\begingroup$ Thank you. But if the word force means "two forces" then doesn't it mean the magnitude will change into 2F ? $\endgroup$ Commented Oct 17, 2016 at 8:35
  • $\begingroup$ @ProtonUpUpDown I have changed the word "two" into "action and reaction forces" or it could be "both". $\endgroup$
    – Farcher
    Commented Oct 17, 2016 at 8:40

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