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My confusion arose when going through the additive property of charges where charges with opposite signs subtract each other. My knowledge of electrostatics is limited, but I am sure that signs are associated with quantities to show their direction, and such quantities are called vector quantities. We associate signs with electric charges ($+q$ and $-q$); however, I have come across texts that state that charges have no direction associated with them since they are scalar quantities. This is where I fell into the pit of doubts and confusion and I request for the clarification of my concepts.

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    $\begingroup$ "Scalars" are just numbers, like you learned about in elementary school, and can be positive or negative. The magnitude of a vector is a scalar that can't be negative, but other scalars can be negative. $\endgroup$
    – knzhou
    Apr 1 at 19:52
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    $\begingroup$ Temperature (as on a weather map) is a scalar field. Certainly, temperatures can be less than 0 Celsius. Height above sea level is a scalar field that can taken on negative values. $\endgroup$
    – robphy
    Apr 1 at 20:11
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    $\begingroup$ I wonder if my bank will accept the idea that since money doesn't have a direction, there can be no such thing as an overdraft! $\endgroup$
    – Simon B
    Apr 2 at 21:04
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    $\begingroup$ More than anything, it's a convention that allows us to model the tendency of like charges to repel and unlike charges to attract with a simple multiplication operation. Sure, we could keep track of this with counts of electrons and counts of protons and do some if-then logic to determine which way the resulting forces are directed. But it's easier to just call one of them negative. $\endgroup$
    – Devsman
    Apr 3 at 13:39
  • $\begingroup$ See the first paragraph (and specifically the very first sentence) here: en.wikipedia.org/wiki/Scalar_(physics) $\endgroup$ Apr 4 at 9:44

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As a high school student, my knowledge of electrostatics is limited but what I am sure is that Signs are associated with quantities to show their direction (such quantities are called vector quantities).

Signs are not limited to vector quantities. This is the root of your confusion.

Scalar quantities are numbers and numbers have signs. Case in point is the scalar quantity work, which is defined as the dot product of two vector quantities, force and displacement, as follows:

$$W=\vec F\cdot d\vec s$$

$$W=Fs\cos \theta$$

Where $\theta$ is the angle between the direction of the force vector and the direction of the displacement vector. When the force and displacement vectors are in in opposite directions, $\theta =180^{0}$, $\cos \theta = -1$ and $W=-Fs$ a negative scalar quantity.

Similarly charge $Q$ is a number whose magnitude is the quantity of charge in Coulombs (many electrons or protons). By convention the charge of an electron is negative, $-e$, while that of a proton is positive, +e. Charged particles of opposite signs attract one another while particles of the same charge repel one another per Coulomb's force law.

$$\vec F_{2}=\frac{Q_{1}Q_{2}}{4\pi\epsilon r^{2}}\hat a_{r12}$$

where

$\vec F_{2}$ =the force on charge 2 due to charge 1

$r$ =the distance between charges 1 and 2

$\hat a_{r12}$= a unit vector directed from 1 to 2

$\epsilon$ = the electrical permittivity of the medium between the charges

In order to determine the direction of $\vec F_2$ one needs to include the signs of the two charges. The product of $Q_1$ and $Q_2$ will be positive if they are both positive or negative resulting in a repulsive force between the two like charges. If the signs of the two charges are opposite the product will be negative resulting in an attractive force between the two unlike charges.

Hope this helps.

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but what I am sure is that Signs are associated with quantities to show their direction (such quantities are called vector quantities)

You shouldn't be too sure of this - unless you think of direction in a more abstract sense. A much more fruitful way to think about signs is simply as a way to distinguish in binary choices. Whenever there is a binary choice, a positive and negative sign can be assigned to the two options to make it easier to refer to them. And there are many binary concepts to think of:

  • moving backwards vs forwards, which is the physical/geometric directionality you are referring to.

  • absorbing or supplying e.g. heat.

  • entering or leaving e.g. mass in a system.

  • inwards or outwards directionality, such as flux of light flowing one way or the other through a window.

There are many such binary scenarios. And yet another example is electric charge. It turns out that all electric charges we have ever found either attract or repel other electric charges. Again a binary option. So again here we can choose to define one of them as positive and then the other type as negative, simply to make it easier to refer to and talk about the. Now, whenever you know that you are dealing with one type of charge, you know that any other charge of the same type will repel from yours.

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I think that you can blame Benjamin Franklin who had the idea that electricity was something to do with a fluid moving from one body to another. Excess fluid making some objects positive and a deficit of fluid making others negative.

So nothing to do with vectors just a matter of keeping account.

Fluid theory of electricity.

We now have ditched the fluid idea but kept the words positive and negative to differentiate between two types of charge. Franklin also introduced the idea that the fluid flowed from the matter with excess (positive) to the matter with the deficit (negative) which we now call the conventional current direction.

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    $\begingroup$ I don't think this is what OP is asking. They are asking why charges can be considered negative at all; not why negative is somewhat arbitrarily assigned to the charge that electrons carry, rather than the charge that protons carry. $\endgroup$ Apr 2 at 6:03
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In physics, scalar, vectors and tensors in general, are defined with respect to transformations of space-time. For simplicity, let us use classical physics and say that the only symmetries of space that we care for are rotations, and reflections.

If you have a one dimensional space, the position of particle is given by a single variable $x$ that can be positive or negative. Similarly velocity is given by $v=\dot{x}$. The particle charge is $q$, can also be positive or negative. So how do we know charge is a scalar and not a vector in this space?

Well, under reflections in one dimension, vectors transform as $A\to-A$. We know that $x\to-x$, $v\to-v$ but charges keep their sign $q\to q$. Thus charges are scalars.

You can do the same thing with higher dimensions. Position is usually given by a vector $\mathbf r=(x,y,z)$, and a rotation of $\theta$ radians around the $z$ axis brings it to $(x,y,z)\to (x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta,z)$ but the charge is still given by a single number $q\to q$ that does not transform under rotations (that can be positive or negative). Thus again charge is a scalar.

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The simple answer to this question is that "It is just a convention".

Sir Benjamin Franklin conducted many experiments (like the famous kite experiment), to explain the results of these experiments he coined the term charge, which gets transferred from one body to other. So one body is losing something and other is gaining it, and therefore he had to distinguish these two opposite, but complementary, observations which were evident by the virtue of attraction and repulsion. Now he has to name these entities:

Franklin's mind: opposite properties, gain and loss. What should I name this? Positive and negative.

It is as simple as it is.

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    $\begingroup$ "Sir" in what sense? $\endgroup$ Apr 2 at 16:14
  • $\begingroup$ Just a salutation $\endgroup$ Apr 9 at 6:57
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Here are three ways of thinking about your dilemma.

  1. Charge is a scalar, and it is a real number: real numbers can be positive or negative.
  2. Charge is a measure of the number of electrons in an atom, and integers can be positive or negative. A proton can be seen as hydrogen atom minus one electron. Add an electron (+1) to a proton (-1), and, voilà, you have a charge of zero, a neutral atom. You multiply this charge by 160.2176634 zeptocoulombs/electron to get the charge in more familiar units (This ignores quarks having a fractional charge: perhaps count an electron as 3 instead of 1?).
  3. Charge is a vector that lives in a 1-dimensional vector space.
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In physics, a lot of scalars are important only down to changes of their value and the exact value is of less or no importance. E.g. gravitational potential energy - in orbital mechanics it is easier to assume that the potential energy goes to zero at infinity (and is always negative), but in civil engineering the useful zero is the ground level so we have both positive and negative values. The same goes for the electrostatic potential and we assume zero wherever we feel like - in electric engineering and electronics, we pick a conductor and call it zero and it is as good zero as any other so we usually use the one that simplifies our calculations.

The electric charges are different - they do have a natural zero point (no charge, no electrostatic interaction) and their interactions clearly depend on the sign of the charge so we need both positive and negative ones.

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To simplify let us just consider two mathematical objects: scalars and vectors.

In order to start the discussion properly we need to define a reference frame. And for reference frames you need to things: a $0$ and $1$. You can set up different axis on this reference frame and the $0$ will serve as the origin and with the $1$ you are able to define all other numbers.

On these reference frames you can "insert" different objects that are related with the numbers that you just created with the $0$ and $1$ entities.

Notice that I saying reference frames and not reference frame. This plural indicates that there several reference frames to consider. It is then very natural for us to pose the following question: "how we can describe objects in one reference frame, given that we know how to describe them in another reference frame?"

Well some objects are always described the same way on all reference frames, while for other objects you need to write a transformation rule (which is only dependent on the two reference frames and not on the objects themselves). The first objects are we call scalars while the second objects are what we call vectors.

The sign is just a matter of convention. after you defined the $0$ and the $1$ you have also defined a preferred direction on your axis. Going opposite that direction is then $-$ sign.

Notice that in my answer I am trying to give a somewhat rigorous explanation but I am letting a lot of things out. For a more rigorous explanation any book (be it more mathematical or more physics orientated) will give the right idea. One suggestion is a really old book by Butkov on mathematical methods. A more recent suggestion would be "Classical Mechanics" by Tai L. Chow

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