# Charge signs in current

I've had recently an argument with my friend about different charge carriers in an electric current. Suppose that electrons and holes are moving in the same direction. It effectively means we have half of the electrons moving in one direction and half in another direction. Thus, he says, the current is zero. So in general:

1. If charge $Q$ moving, say, to the right and charge $Q$ of the same sign is moving to the left the current is zero.
2. If charge $Q$ of one sign and charge $Q$ of the opposite sign are moving in the same direction the current is also zero.

But why? Why then when we have a positive charge $Q$ moving to the right and negative charge $Q$ moving to the left the current is $|2Q|/t$ and not zero? I don't see why the directions are so important and why in some cases the charge "cancels" while in others it doesn't. Moreover, it seems more intuitive that the charge will "cancel out" in the last situation that I have described (because then we have an algebraic sum $+Q-Q=0$).

## 3 Answers

The general definition for a current (electrical, matter, energy, probability) is as follows: consider a small area $\def\dS{\mathrm d\boldsymbol S}\dS$ through which a quantity $Q$ flows. We call $\def\j{\boldsymbol j}\j_Q$ the $Q$-current density, i.e the current density of $Q$. During a small time $\delta t$ the total of the quantity $Q$ flowing through $\dS$ is $$\mathrm \delta Q=\j_Q\cdot\dS\,\delta t.\tag{1}$$

Now we consider a wire of section $\dS=\mathrm d S\,\mathbf e_x$. For charges $+q$ moving with velocity $\def\v{\boldsymbol v}\v=v\mathbf e_x$ the current density multiplied by the wire's section is the current $$I=\frac{\delta Q}{\delta t}=\j_Q\cdot\dS=qv.$$

Let us examine the three cases you mention.

• Two charges of opposite sign $$I=\j_Q\cdot\dS=\bigl(q\v+(-q)\v\bigr)\dS=0.$$ The total charge is zero: no current. This means that opposite charges in same directions create opposite currents.
• Same charges with opposite velocity $$I=\j_Q\cdot\dS=\bigl(q\v+q(-\v)\bigr)\cdot\dS=0.$$ The charges moving in opposite direction create opposite currents as well.
• Two opposite charges in opposite directions $$I=\j_Q\cdot\dS=\bigl(q\v+(-q)(-\v)\bigr)\cdot\dS=2qv.$$ Opposite charges in opposite directions create the same current, because of $(-1)\times(-1)=1$.
• @ V.Rossetto : OP does mention these 2 cases but has asked about the case when opposite charges move in opposite direction ! – Rijul Gupta Jan 16 '14 at 9:45
• Thank you very much, sir! Well, the third case is then simply $q \mathbf{v} + (-q) (-\mathbf{v})=2qv$ – user37433 Jan 16 '14 at 9:49
• @rijulgupta. Thanks for your help. I have been distrubed and lost track. – Tom-Tom Jan 16 '14 at 9:52
• Velocity is the vector quantity, charge is the scalar quantity. Vector velocity multiplied by scalar charge must give the vector which is in same direction of velocity, but current is not a vector quantity. – Immortal Player Jan 16 '14 at 10:03
• @VINAY. This may be a point a language. $\j$ as defined by (1) is a current density. Electric current is a vector quantity for one dimensional systems (wires) so it is often seen as an algebraic quantity. Observe that one needs to define an orientation for each wire in an electric circuit, which is not necessary for a scalar quantity like charge or temperature. So electric current is indeed a vector quantity, but in a one dimensional system one can treat it a bit differently. – Tom-Tom Jan 16 '14 at 10:11

In current flow in normal wires, we do not have flow of both positively and negatively charged particles. In solid conductors and semi-conductors only electrons flow hence we have current as $Q/t$. What you call as flow of holes is actually just the absense of electrons flowing from one point to next because electron is jumping forward in the opposite direction.

We do have $2Q/t$ as current in liquids where ions of both charges are present, for example if you dissolve $NaCl$ in water it gets dissociated as $Na^+$ and $Cl^-$ now if you apply a potential difference both ions flow and current is measured due to both ions and is therefore $2Q/t$ Now these charges do not simply add up and reform$NaCl$ and give a zero current because they are surrounded by water molecules, in several other cases wherever oppositely charges flow in opposite direction they do not simply combine/annihiliate because there has to be collision between then, usually when a multitude of such charges flow, indeed some of them collide and combine but majority of them just flow and do not add up.

Also, suppose through a given area in given time $n$ positively charged go towards right and $n$ negatively charged go towards left, it can be taken as $2n$ particles going either way and giving $2nQ/t$ current towards right.

• Please, read my question again. You didn't answer my question at all. – user37433 Jan 15 '14 at 17:17
• I have edited it, I misinterpreted something last time, comment if you still have some query. – Rijul Gupta Jan 15 '14 at 17:19

# What is electric charge?

Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter.

Remember, if the particle has electric charge, there is an electric field around it.

# What is a Nuetral particle?

In physics, a neutral particle is a particle with no electric charge.

So, a particle will be neutral, if it has no electric charge. In other words, a particle will be neutral, if there is no electric field around it.

Suppose, there is a system of two oppositely charged particles, would you like to say it is neutral? or would you like to say there is no electric field around the system? It may or may not be.

Let me explain you. You can consider a system of oppositely charged particles as a dipole. Magnitude of electric field ($E$) due to an electric dipole at a distance $r$ from its centre in a direction making an angle $\theta$ with the dipole is given by the equation,$$E=\frac{1}{4\pi\epsilon}.\frac{p\sqrt{3\cos^2\theta+1}}{r^3}$$
where, $p=2aq$ ($2a$ is the distance of separation of the charges $q$).

From the above equation it follows that, electric field around the dipole will be zero if and only if the distance between the two oppositely charged particles is zero. So, a system of two opposite charges will have no electric field around it, if the the distance between opposite charges is zero. In other words a system of two opposite charges will be neutral if and only if, the distance between the opposite charges is zero.

So, even if the system consists of two oppositely charged particles it may not be neutral, if the distance between them is not zero.

# What is electric current?

An electric current is a flow of electric charge. Specifically it is defined as the rate of flow electric charge through the section of a wire.

So, if you say that, through a section of the wire, an electron passes in one direction and a proton (not in reality, if you assume hypothetically) in the opposite direction, current will be zero, if charge passing through that section is zero. As explained above, charge will be zero if distance between electron and proton is zero. So, it depends on the situation you consider. If you say distance is non zero, current will be non zero or other wise it will be zero.

• Nice try, but it doesn't answer my question. In fact, if you have a conductor with some charge flowing, it will still be neutral. Electric field has nothing to do with the definition of current (and of course, charges are not zero distance apart), and even if it had, it contradicts the 1. and 2. statements in my question. – user37433 Jan 16 '14 at 9:19