# 2.1 Electrons at Rest

Electric forces, fields, potentials, and the behavior of electric charge within materials. 9 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:

## Matter

There are two signs of charge: positive charges (protons), and negative charges (electrons). These two, plus neutrally-charged neutrons, are the elementary particles that make up atoms.

(There are other elementary charged particles like muons and positrons. Unless you’re explicitly studying them, you won’t see them in practical electronics.)

Charge is quantized. This means that charges are discrete particles. You can have 1 electron, or 2 electrons, but you can’t have 1.5 electrons.

Nonetheless, in most practical electronics, we’re concerned with so many electrons that we treat charge and its flows as a continuum, a statistically averaged continuous quantity rather than a discrete quantity. (However, exceptions do exist.)

## Electric Forces

Electric charges produce forces on other charges. The magnitude of that force depends on the inverse square of the distance between them.

Charges of the same sign repel each other. Charges of opposite sign attract.

Coulomb’s Law is:

$$F = \frac {Q_1 Q_2} {(4 \pi \varepsilon) r^2}$$

The magnitude of the force between two point charges $Q_1 \ \text{and} \ Q_2$ at some distance $r$ depends on the product of their charge values $Q_1 Q_2$ .

Note that there is nothing magical about this multiplication. In effect, it is simply counting up all the pairwise interactions between discrete quantized charges. For each proton in $Q_1$ , it will feel an attraction toward each electron in $Q_2$ . If these were measured in units of number of elementary charges, then $Q_1 Q_2$ would simply be the number of possible pairs. The rest is just a unit conversion from counts to Coulombs.

## Dielectrics

Dielectrics and permittivity have to do with whether materials can be electrically polarized.

In a vacuum, the “electric permittivity of free space” is a particular constant $\varepsilon = \varepsilon_0$ .

In non-vacuum materials, the effective Coulomb force can be reduced because $\varepsilon_{\text{material}} > \varepsilon_0$ . While this is considered to be a macroscopic, averaged, linear effect, in truth it is just the linearized, steady state, lumped element model approximation of an atomic-scale behavior. Here’s why:

When an electric field is applied to a real material, the material may be polarizable, such that the internal electrons and protons shift relative to each other by a small fraction of an atomic radius. This shift acts to partially cancel out the applied field within the material.

High-permittivity materials are often used as dielectrics within capacitors. We can store energy in the slight dislocation caused by polarizing the materials within.

## Charge Distribution and Geometry

Coulomb’s Law above is for point charges. Protons and electrons can be considered to be point charges.

However, many practical situations involve charge distributed over space:

• a line charge has some linear density of charge $\frac {q} {l}$ per unit length
• an area charge has some areal density of charge $\frac {q} {A}$ per unit area
• a volume charge has some volumetric density of charge $\frac {q} {V}$ per unit volume

These distributions are related to Coulomb’s Law via multivariable calculus.

## Electric Fields

Instead of talking about forces between two charges, we sometimes talk about the electric field generated by one charge $Q_1$ .

Any other charge, for example $Q_2$ , will feel a force proportional to the field strength & direction at its current position, multiplied by the charge value of $Q_2$ .

$$F_{\text{on} \ Q_2} = Q_2 E_{\text{from} \ Q_1}$$

We often use the phrase test charge to indicate that we’re talking about a hypothetical point charge inserted at a particular position. This allows us to use the electric field to determine what forces would result on that particle.

We often talk about fields more than about forces between charges because the geometry of a situation yields useful field descriptions. For example, with a parallel-plate capacitor, the charges are “smeared out” across a plane, and to determine the force on a test charge between the plates, we’d have to sum over all the individual charges. However, this same situation yields a very simple field configuration.

The electric field description is an equivalent description of the same phenomenon as electric forces. Pick either the force or field approach: avoid double counting.

## Electric Potential

An electric field is a vector field: it points in the direction that a test charge would be pushed by the electric force.

One common “trick” of multivariable calculus is that instead of considering a vector field, where every position maps to a vector, we can sometimes write that field as being the gradient of some potential function. (Usually, a negative sign is added too.)

$$\vec{E} = - \nabla V$$
$$V = - \int \vec{E} \cdot \vec{dl}$$

That potential function is a scalar: every position in 3D space maps to a number. The electric field at a particular point is the (negative) gradient of the scalar field at that point.

$$\vec{E}(x,y,z) = - \left( \frac {d V} {d x} \hat{x} + \frac {d V} {d y} \hat{y} + \frac {d V} {d z} \hat{z} \right)$$

That potential function $V$ is voltage. Because the electric field exists at all of space, so does the potential function.

Again, just as with electric fields, talking about electric potential is just an equivalent description of the same phenomenon: charges attracting and repelling each other.

It might be more convenient to think about forces, fields, or potentials in any particular problem solving application, but all three describe the same underlying physics, so be sure to avoid double counting.

## One Big Assumption

In turning an electric field (vector) into an electric potential (scalar), we relied on a mathematical assumption that only applies to conservative vector fields.

It turns out that because of Maxwell’s equations, electric fields are conservative if and only if there are no time-varying magnetic fields.

In reality, we have lots and lots of time-varying magnetic fields, intentional and unintentional. This means that the electric potential does not actually exist!

However, we’re able to pretend that it does by encapsulating the time-varying magnetic field behavior into discrete components, like inductors and transformers. We’ll revisit this assumption when we discuss Kirchhoff’s Voltage Law and Kirchhoff’s Current Law in a future section.

In order to get a little bit of physical intuition about this, let’s try a physical thought experiment:

An electric field (without any time-varying magnetic fields) is also a conservative field, and energy can’t be gained from or lost to it over a closed loop.

However, an electric field with a time-varying magnetic field is not a conservative field. And we use this every day to great practical effect: we intentionally put currents in a loop within time-varying magnetic fields, and use those to extract electrical energy from the time-varying magnetic field (i.e. in generators), or use it to turn electrical energy into magnetic fields (i.e. in motors).

Regardless, we usually encapsulate these electromagnetic effects into our Lumped Element Model, and go on assuming that the electric field actually is conservative.

Understandably, this can be confusing and disorienting to beginners. Ninty-nine percent of the time, it’s safe to just assume the electric field is conservative, but if you’re doing anything with changing or moving magnetic fields, you should make a little mental note to remember that it’s really not.

## Conservation of Charge

Total charge is conserved in the universe. We know of no processes that create a positive charge without also creating a negative charge at the same time.

Total charge is also conserved in a circuit. It can’t “leak out” and go somewhere unknown. It is possible to build a static charge generator which will put net negative charge (extra electrons) on an object, however some other object will be left with a corresponding net positive charge (electron deficit).

## Coulombs

The unit of charge is the Coulomb.

Charge is defined as positive for protons and negative for electrons, which we’ll discuss more in the next section, Electrons in Motion.

\begin{align} 1 \ \text{proton} & = 1.602\dots \times 10^{-19} \ \text{C} \\ 1 \ \text{electron} & = -1.602\dots \times 10^{-19} \ \text{C} \\ -1 \ \text{C} & = 6.242\dots \times 10^{18} \ \text{electrons} \end{align}

## Materials and “Free” Electrons

In matter made of protons and electrons, most electrons are not free to move. Instead, they are tightly bound to their nuculeus.

However, in some materials and at some temperatures, some electrons are indeed free to move to neighboring atoms. These materials are called conductors.

In general, only a tiny fraction of total charge within the material is able to move. If you’ve studied chemistry, recall the concept of valence electrons, which are the least-tightly bound to to the nucleus.

## Static Charges Within a Conductor

Static charges within a conductor are free to move through the conductor.

Because these charges can move, and charges of the same sign repel each other, they will naturally seek to move as far away from each other as possible. They will, therefore, spread out to the surface or edges of the conductor, rather than staying within the bulk of the material.

## Static Charges Within an Insulator

In contrast, charges in an insulator can’t move, so any distribution of charge is possible depending on how the charges were placed there. They could be within the insulator material, or could be on the surface.

## What’s Next

In the next section, Electrons in Motion, we’ll discuss what happens when we allow charges to move.