# 2.2 Electrons in Motion

Current, current density, Amps, magnetic fields, and why electron flow and charge flow are in opposite directions. 8 min read

As described in the Introduction to this book, familiarity with the physics of electricy & magnetism is a prerequisite. See that section for further references.

Here is a brief conceptual review:

## Flow of Electrons

The motion, or flow, of charges is measured in units of Amperes, usually just called an “Amp”, which is a unit of charge per time. From the earlier Electrons at Rest section, the unit of charge is the Coulomb, so the flow rate is measured in units of Coulombs per second:

\begin{align} 1 \ \text{Amp} & = 1 \ \frac {\text{Coulomb}} {\text{second}} \\ 1 \ \text{A} & = 1 \ \frac {\text{C}} {\text{s}} \\ I & = \frac {Q} {t} \end{align}

A reasonable way to think about measuring any flow, including the flow of charge, is to create an imaginary surface (a thin 2-D shape in 3-D, like a piece of paper, with top and bottom sides) and consider inserting that surface so it cuts through our circuit somewhere (but imagine that the circuit continues working as before). For a short time $dt$ , count the particles that flow through in each direction, keep track of their sign, and add up the total charge $dQ$ that has flowed through. The instantaneous current through that surface is then $I = \frac {dQ} {dt}$ .

This surface approach generalizes to the idea of “current density,” which is current per unit area with units $\frac {A} {m^2}$ :

$$J = \frac {I} {A}$$

Note that the flow of electrons doesn’t necessarily imply a change of the net quantity present at the point of interest. For example, imagine the flow of water in a small artifical “lazy river” closed loop. Water is in motion and is flowing at every point, so there is a nonzero current, but there is no change in the amount of water present in any particular volume of interest. The inflow and outflow at each section is balanced.

The motion or flow of electric charge creates a magnetic field around that flow. This is how electromagnets are made, as well as solenoids, motors, and other applications.

Electrons are commonly found in motion in conductive objects, such as in solid metals, or as completely unbound free electrons, such as in a cathode ray tube.

In liquids and gases, protons can move too. Sometimes these are free protons, like a bare hydrogen nucleus carrying zero electrons. Other times, they are ions of more substantial atoms or molecules. A number of important processes in molecular biology are based on controlling the motion of charged molecules within a liquid in an electric field, known generally as electrophoresis.

It’s possible to calculate the average speed of motion of an electron within a current. This is known as the drift velocity, and it is generally very small compared to the random thermal velocities just due to the material being at a particular temperature. This fact is actually critical to making Ohm’s Law be a linear relationship, which we’ll discuss further in the Resistance and Ohm’s Law section.

Electrons have mass, though quite lightweight individually (roughly 1/2000th as massive as a proton), and they follow the same basic laws of motion of other massive particles. They must be acted on by some net force in order to be accelerated. This is, in general, the electric force due to the external electric field $\vec{E}$ . Without much detail here, the magnetic field also contributes an acceleration term to a charge in motion. The Lorentz force law combines these two:

$$\vec{F} = q \big( \vec{E} + \vec{v} \times \vec{B} \big)$$

The magnetic field component of the Lorentz force law plays a direct effect in applications like cathode ray tubes, where a magnetic field is used to bend the path of an electron, or in Hall effect sensors, where it’s used to sense a magnetic field. This term is also responsible for force generation in motors and generators.

Within a conductor, random thermal collisions play a dominant role in the motion of electrons.

The direction of current matters deeply, and is frequently a confusing point for those new to studying electronics.

## Charge Flow and Electron Flow

For the rest of this book, and almost certainly the rest of your electronics career, you’ll talk about current as a flow of positive charge. But, in most cases, the particles actually in motion are electrons, which each have negative charge. This causes lots of confusion for beginners.

Imagine we hold a piece of wire horizontally. If we could look inside and see a bunch of electrons moving from left to right, we’d say there’s an electron flow to the right. However, because electrons have negative charge, there’s a current flow to the left.

Unless you’re doing specialized physics, in the electronics world we always talk in terms of current flow – never electron flow.

These views are completely identical with regards to electronics.

If there are really electrons flowing from left to right, how can this situation be equivalent to something (charge) flowing from right to left? Let’s try a thought experiment:

Imagine a row of three cups. These represent atomic positions along our wire. The left two cups start with one blue marble each. Each blue marble represents an electron. The rightmost cup begins empty. Call this “Step 1.” Imagine that each cup can either hold one marble (electron) or zero at any time.

Now, let’s move the marble from the middle cup to the right. The marble (electron) has moved to the right, but now the middle cup is empty. Call this “Step 2.” (Notice that we could say that the empty position has moved to the left.)

Next, repeat: move the marble from the left cup to the middle. Another electron has moved to the right, but now the 1st cup is empty. Call this “Step 3.”

Here’s what this would look like, step by step, in terms of the blue marbles (electrons):

$$\begin{array}{c|ccc} \text{Step} & \text{Left Cup} & \text{Middle Cup} & \text{Right Cup} \\ \hline 1 & \color{blue}{\text{ELECTRON}} & \color{blue}{\text{ELECTRON}} & \color{gray}{\text{empty}} \\ 2 & \color{blue}{\text{ELECTRON}} & \color{gray}{\text{empty}} & \color{blue}{\text{ELECTRON}} \\ 3 & \color{gray}{\text{empty}} & \color{blue}{\text{ELECTRON}} & \color{blue}{\text{ELECTRON}} \\ \end{array}$$

It’s clear that the blue electrons have each moved one position to the right: i.e. they’re flowing from left to right, just as they were in our wire.

If we looked at any intermediate position (i.e. inserting our “piece of paper” surface from above between the left and middle cups, or between middle and right) and counted how many electrons passed by in a certain amount of time, we could measure the flow of electrons.

We could extend this analogy by making our wire longer with a longer series of cups, as long as we keep putting in “new” electrons on the left and removing them on the right.

How much charge is in each cup at each step? Each electron is $-1$ quantum units of charge, so let’s just put $-1$ in each corresponding cell:

$$\begin{array}{c|ccc} \text{Step} & \text{Left Cup} & \text{Middle Cup} & \text{Right Cup} \\ \hline 1 & \color{blue}{-1} & \color{blue}{-1} & \color{gray}{0} \\ 2 & \color{blue}{-1} & \color{gray}{0} & \color{blue}{-1} \\ 3 & \color{gray}{0} & \color{blue}{-1} & \color{blue}{-1} \\ \end{array}$$

However, we can make a simple redefinition: instead of tracking the blue marble $\color{blue}{\text{ELECTRON}}$ , instead let’s keep track of the empty spot that the electrons can move into; let’s call this a $\color{red}{\text{HOLE}}$ .

As each $\color{blue}{\text{ELECTRON}}$ moves to the right, it swaps position and displaces the empty spot toward the left. Let’s call this red marble a $\color{red}{\text{HOLE}}$ and track its motion over the same sequence of steps:

$$\begin{array}{c|ccc} \text{Step} & \text{Left Cup} & \text{Middle Cup} & \text{Right Cup} \\ \hline 1 & \color{gray}{\text{empty}} & \color{gray}{\text{empty}} & \color{red}{\text{HOLE}} \\ 2 & \color{gray}{\text{empty}} & \color{red}{\text{HOLE}} & \color{gray}{\text{empty}} \\ 3 & \color{red}{\text{HOLE}} & \color{gray}{\text{empty}} & \color{gray}{\text{empty}} \\ \end{array}$$

This is just an alternative view of the same situation described above. While the electrons move from left to right, the hole appears to move from right to left.

This hole actually behaves a lot like the electron. Each cup can only hold a $\color{red}{\text{HOLE}}$ or nothing at any time.

How much charge does each cup hold at each step? a $\color{red}{\text{HOLE}}$ is the lack of (i.e. a deficit of) one $\color{blue}{\text{ELECTRON}}$ , so it must be $+1$ charge as charge is conserved:

$$\begin{array}{c|ccc} \text{Step} & \text{Left Cup} & \text{Middle Cup} & \text{Right Cup} \\ \hline 1 & \color{gray}{0} & \color{gray}{0} & \color{red}{+1} \\ 2 & \color{gray}{0} & \color{red}{+1} & \color{gray}{0} \\ 3 & \color{red}{+1} & \color{gray}{0} & \color{gray}{0} \\ \end{array}$$

This charge table is equivalent to taking the above electron table and simply adding +1 to each cell. That’s a valid thing to do because we only care about the flow, rather than the absolute quantity of charge at any position.

In a real conductive material there are electrons able to move to neighboring positions in a lattice. While not as simple as our cups-and-marbles example, there are in fact positions able to hold an electron which can be occupied or not at any instant. And, if we examine the motion of those available positions (holes) rather than the electrons themselves, they are de-facto equivalent to positive charge carriers.

Keeping track of the flow of negatively-charged electrons, or keeping track of the flow of positively-charged holes, is really identical (in opposite directions). It’s just an odd bookkeeping convention that we’ve all elected to follow to track charge and current in terms of positive charge, even though the predominant practical charge carrier is negative.

Note that there are some cases in which true positive charge carriers can move: for example, positively charged ions in a battery or electrolyte solution can physically move. If you’re working on electrochemical cells or particle accelerators, take note. But these are the exception: usually it’s electrons in electronics.

## What’s Next

In the next section, Voltage and Current, we’ll discuss the two primary variables of electronics.