2.1
Electrons at Rest
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:
The magnitude of the force between two point charges at some distance depends on the product of their charge values .
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 , it will feel an attraction toward each electron in . If these were measured in units of number of elementary charges, then would simply be the number of possible pairs. The rest is just a unit conversion from counts to Coulombs.
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 per unit length
- an area charge has some areal density of charge per unit area
- a volume charge has some volumetric density of charge 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 .
Any other charge, for example , will feel a force proportional to the field strength & direction at its current position, multiplied by the charge value of .
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.)
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.
That potential function is voltage. Because the electric field exists at all of space, so does the potential function.
We’ll talk about this more in the Voltage and Current section.
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:
Instead of imagining a test charge in a magnetic field, which most of us have little intuition about, let’s try imagining a mass (say, a tennis ball in your hand) in the presence of an external gravitational field (like the one pulling that tennis ball towards the floor right now). First, start with your arm pointed toward the floor, and then rotate your arm up toward the ceiling: you will have to use your muscles and do work on the ball to lift it. Now continue, rotating until your arm is again toward the floor: the ball will do work against your arm, even if your biology isn’t optimized to capture that energy. The amount of work done on the way up equals that done on the way down; they sum to zero. If it were any other way, you could either gain or lose energy as you just spun your arm in a circle – and if that were the case, then you might be able to build a perpetual motion machine. But you can’t, because the gravitational field is a conservative field.
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.
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.
Induced Charge
If a conductor is placed in the presence of an external electric field, the charges in a conductor are free to move, and will be accelerated by the field. They will rearrange themselves on the conductor’s surface in a way that creates an opposing electric field. The charges will move until they achieve steady-state, when the surface charges perfectly cancel out any electric field within the conductor, because any remaining non-zero field will cause more charges to move.
Even though charges have moved within the conductor, the voltage drop in the conductor returns to zero, because the electric field from the surface charges perfectly cancels out the external field.
Throughout the whole process, conservation of charge is maintained. Charge that moves along the surface from one area to another leaves an equal and opposite charge on a different part of the surface.
In some early electricity experiments, you may hear this called induced charge. If we take a neutral conductive rod and place a positive static charge closer to one end, negative charges in the conductor will migrate closer to that end, leaving positive charges at the other end. If we then break the rod in half (or less violently, separate two half-rods that clip together in the middle), we will find that each half-rod now is non-neutral: one has positive charge and one negative.
This is said to be an induced electrostatic charge, because the charge was induced without any contact with the external charge. It happened merely because the conductive object was in the presence of an external electric field, and the charges in the conductor acted to cancel out its influence within the rod.
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.
These charges are not free to move to repel an external electric field, so we don’t get the same induced charge effect.
Origins of Capacitance
The tendency of charges in a conductor to rearrange themselves along the surface to perfectly cancel out an external electric field within the conductor leads us directly to the concept of capacitance. The rearrangement zeros out fields within the conductor, but produces surface charge distributions, and nonzero fields outside the conductor.
If we enforce a voltage difference between two conductive surfaces (such as with a battery or other voltage source), the electric potential difference necessarily implies a nonzero electric field, which also implies a non-neutral distribution of charges in space. The potential, field, and charge distribution are inextricably linked as a mathematical fact.
If we apply a fixed potential difference between two initially-neutral conductors, some charge will flow between them until the conductors have the correct potential difference. A charge will be present on the positive-potential conductor and an equal-and-opposite on the other.
The actual quantity of charge displacement (and electric field strength) is a complicated function of the geometry and materials of the situation. The overall effect is captured by the basic equation of a capacitor:
where is the displaced charge in Coulombs (symmetrically and on opposite ends), is the potential difference in Volts, and is the capacitance in Farads. The capacitance captures the overall effect of the geometry and materials of the situation between the two conductors in question.
Parallel Plate Capacitor
When two conductive plates are placed a short distance apart, with an insulating gap of constant width, this is often called a parallel-plate capacitor.
When an external voltage is applied, charges move between the two plates, through the external voltage source, until the electric potential difference across the capacitor’s gap is equal to the external potential difference.
This geometry is simple enough that the capacitance can be solved with only a few assumptions, and it turns out that:
where is the permittivity of free space, for the specific material in the gap, is the overlapping surface area of the plates, and is the distance of the gap between the plates.
This tells us that, for a given external voltage , the amount of charge displacement:
goes up if we fill the gap with a dielectric material
goes up if we make the plates have larger surface area
goes up if we make the plates closer together
In fact, because there are positive and negative charges in close proximity, the plates experience electrostatic forces trying to attract them toward each other!
Parallel Plate Capacitor with Interstitial Conductor
Suppose we insert a third metal plate into the air gap, but not touching either of the two capacitor plates. What happens?
This metal plate is in the electric field of the capacitor, so charges rearrange on its surface to cancel out the internal field as shown.
As the plate is inserted, the effective capacitance increases because the effective gap between plates falls. There is less distance over which the electric field can exist, so the electric field must be correspondingly stronger for a fixed potential difference. A stronger electric field is equivalent to saying that a higher surface charge density is required. This manifests as an increased capacitance.
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 .
In non-vacuum materials, the effective Coulomb force can be reduced because . 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.
Dielectrics in Capacitors
High-permittivity materials are often used as dielectrics within capacitors. We can store energy in the slight dislocation caused by polarizing the materials within.
One way to visualize the effect of the dielectric this is to think about the parallel plate capacitor with the interstitial conductor, but in place of the additional metal plate (where charges can rearrange freely), imagine inserting a collection of molecules where the positive charges are fixed and a negative charge is connected to each positive charge by a tiny “spring”.
When an external field is applied, the negative charges in the material are attracted to the positive capacitor plate, and move slightly toward it. This offset between the positive and negative charge partially cancels out the electric field within the material. (In contrast, the metal plate completely cancels out its internal electric field.)
As the capacitor is being charged up, work must be done on the spring to stretch it, requiring more charge on the capacitor plates compared to a vacuum gap. This energy is stored within the dielectric material. When the capacitor is discharged, the dielectric gives this energy back to the circuit as the “spring” relaxes and releases its attractive force on the extra plate surface charges.
Parasitic Capacitance
Capacitances are everywhere, and not only in places where we want them. An undesired, unintentional capacitance is called a parasitic capacitance.
Every surface has some small, but nonzero capacitance to every other surface in the vicinity. A voltage difference (such as that imposed by a battery or other voltage source) between two surfaces results in electric field lines between them. The presence of a voltage difference, or equivalently of electric field lines, is equivalent to a spatial separation of positive and negative charges. These electric field lines are free to interact with other materials in the environment, and they do, causing charge separation in other materials too (while remaining compatible with conservation of charge).
Through this process of electric fields inducing charge separation in conductors, basically everything is capacitively coupled to everything else!
Capacitive coupling is especially problematic within electronic devices, where adjacent pins of a component are in close proximity to each other and adjacent wires on a printed circuit board (PCB) are often run as little as apart or less. Close proximity causes higher capacitance.
As a rough ballpark, it’s not uncommon to see parasitics of between nearby pins and traces on a PCB. Careful PCB layout can minimize but not eliminate these.
A common design approach is to place a low-impedance ground plane layer of the PCB immediately above or below signal tracks so that electric field lines preferentially couple to this ground plane (like our conductive dining table). Adding a ground plane actually increases parasitic capacitance. However, by proximity, it can shape the electric field lines and steer the capacitance toward a low-impedance ground node (where it’s less likely to be harmful), and away from adjacent signal traces (where it’s likely to do the most harm).
Shielding
The exception to everything being capacitively coupled to everything else is if we enclose a system in a conductive box. When enclosed, there can be no capacitive effects between objects inside and outside the box. This shielding effect works because static charges on the inner surface of the conductive shield rearrange precisely to cancel any electric field lines from escaping. Similarly, static charges on the shield’s outer surface rearrange to prevent external electric fields from entering.
(This applies strictly for static charges. The shielding discussion gets substantially more complicated when we allow time-varying electromagnetic fields, or equivalently, when we allow charges to move.)
Electrostatics Thought Experiments
It’s important to remember that capacitance isn’t something that occurs only in explicitly designed and purchased capacitors. To illustrate this, let’s walk through a thought experiment of a do-it-yourself capacitor you could easily build in your dining room.
Imagine a insulating dining table, with an insulating tablecloth, on which we place a battery, plus two conductive metal dining plates at adjacent placesettings. Initially, the plates are neutrally charged. But when we take some alligator clips and wire up each of the battery terminals to one of the metal plates resting on the table, we impose a voltage difference between the plates. A small amount of charge flows from the negative plate, into the negative battery terminal, through the battery, out the positive terminal, and into the positive plate. This leads the positive plate to hold charge and the negative plate to hold charge . The capacitance of this arrangement is .
Because there is a voltage difference between the two plates, there are electric field lines starting from the positive plate and ending at the negative plate. The field strength is low because the field lines are long in distance, flowing through the air and the table.
The actual amount of charge displaced (and therefore the capacitance) is highly dependent on the geometry of this situation. More charge is displaced if the plates are closer, for example. The charge will be maximized if we put a thin insulating sheet of paper between the two plates, and stack them as they’d sit in the cabinet, basically like a parallel plate capacitor with a paper dielectric! But even if the plates are spread far apart on the table, there is some small amount of charge displacement and therefore capacitance.
Next, suppose that the table itself is steel (a conductor), but is still covered with an insulating tablecloth. We now connect the battery terminals to the two dining plates at the adjacent placesettings. The conductive table now has a bit of a problem: the electric field lines originating from the two dining plates pass through the metal table.
As discussed earlier, static charges are free to move within a conductor, and due to the electric field lines, there are forces on the charges within the conductive table. Therefore, we get charge separation within the steel table. A sheet of negative charge forms on the surface of the table just underneath the positive plate. An equal and opposite group of positive charge forms on the surface of the table just underneath the negative plate.
The table is still net-neutrally charged, but we have induced a charge separation. These charges continue to move until they perfectly cancel out any electric field within the conductive table, because any nonzero field will produce a force that causes the charges to move.
After this situation settles out, is there a voltage difference within the table? No: the charges rearrange precisely to cancel out the externally-applied electric field, leading to zero net field and therefore zero voltage drop within the conductive table.
But the consequence for the capacitance between the two battery terminals is dramatic. The voltage drop now is carried by very short electric field lines from the positive plate to the table (through only the insulating tablecloth), and again from the table to the negative plate. The electric field strength is much higher now than it was without the conducting table, because the length of distance over which it acts is much smaller. This requires a much higher charge density to produce the more concentrated electric field, so compared to the insulating table, the capacitance of the scenario with the conducting table (with insulating tablecloth) is much higher than the insulating table.
We can walk through other modifications of this scenario to explore other concepts of electrostatics. One example would be charging the plates, disconnecting the wires, and then moving the charged plates around. We’ll leave these as beyond the scope of this section. More generally, however, this kind of thought experiment can be quite useful to illustrate different concepts such as conservation of charge and electric field lines, especially since these effects are hard to observe directly.
What’s Next
In the next section, Electrons in Motion, we’ll discuss what happens when we allow charges to flow.