Another example of Gauss’s Law

Last time we looked at applying Gauss’s law to work out the electric field around a charged wire.

Now let’s consider a sphere of radius $r_0$ with a uniform density of charge, totalling $Q$, centred on the origin.

In the lecture we will use Gauss’s law to show that:

\[\vec{E} = \begin{cases} \frac{Q}{4 \pi \vert \vec{r} \vert^2 \epsilon_0} \hat{\vec{r}} & \text{if } \vert \vec{r} \vert > r_0 \\ \frac{Q \vert \vec{r} \vert}{4 \pi r_0^3 \epsilon_0} \hat{\vec{r}} & \text{if } \vert \vec{r} \vert < r_0 \end{cases}\]

Note that the electric field is always directed radially, but it increases in strength from zero in the centre up to the boundary of the sphere. Beyond this boundary, the electric field is just what one would expect from Coulomb’s law for a point charge $Q$. The fact that the charges are actually spread out is invisible from the outside (provided they are distributed spherically symmetrically).

Types of material

Could one ever construct a sphere of constant charge density like the example above?

Materials hold onto their charges and respond to electric fields in very different ways. Every material is different, but broadly we can divide them into four categories, three of which we will study in the current course.

Type of material Charges can move? Currents can flow? Effect on $\vec{E}$ field
Conductor Yes Yes Internal field must be zero
Dielectric A little bit No Internal field is reduced
Insulator No No None (unless charged)
Semiconductors (not examinable) Depends on conditions Depends Depends
  • The above table assumes the material is in a steady state, with no magnetic fields present, otherwise things can look more complicated.
  • Conductors and insulators are the topic of the current lecture.
  • Dielectrics are actually a type of insulator because currents cannot flow through them. We will look more at dielectrics in lecture 8.
  • Semiconductors are a fourth case. They change their conductivity dependent on conditions such as temperature, exposure to light, or provision of extra electrons from a neighbouring material. This is a topic for future courses and won’t be covered just yet.
  • All of electronic engineering (and therefore much of the modern world!) is about placing these materials in the right arrangements to do the things we want them to.

Materials classify in different ways under different conditions:

  • Air, for example, is essentially an insulator.
  • But it can behave as a dielectric, albeit a rather weak one.
  • And if electric fields become large enough, air can suddenly and dramatically start to conduct – this is the phenomenon of lightning.

Pretty much any insulating/dielectric material has a ‘breakdown voltage’ beyond which it will conduct, because the electric field is so strong that it can rip electrons off atoms.

Insulators

Imagine one has a perfect insulator (i.e. put to one side any concerns over it reaching a breakdown voltage), that is not dielectric (which we will cover later). From the perspective of electromagnetism, can it do anything interesting at all?

  • If you place it in the path of an electric field it will do nothing – the electrons are tightly held in place by the rigid structure of the material itself.
  • But, if you somehow provide additional electrons to an insulator, it may be able to hang onto them.
  • This is because on microscopic scales the electric field is not exactly zero – the orbiting electrons are not distributed completely symmetrically around the nuclei.
  • There are little pockets of attraction or repulsion on the surface of the material, and if you provide some extra electrons it is possible for them to get stuck in these pockets. The insulator becomes net negatively charged.
  • Similarly, electrons can be ripped out if they are loosely bound and come into contact with another material that is happy to take them. This leads to a net positively charged insulator.
  • Rubbing two different insulators together will often have the effect of ripping out electrons from one and depositing them on the other. The material which has tighter-bound electrons ends up negatively charged, and vice versa. This is known as the ‘triboelectric effect’.
  • It can be a big problem, especially in industrial applications where you don’t want accidental ‘sparks’ from static electricity.1
  • Generally, any charge on an insulator will be found on its surface because, practically speaking, this is where it is easiest to deposit or remove electrons.

Conductors

In metals, unlike in insulators, electrons can flow between neighbouring atoms. While this requires quantum mechanics to understand in detail, an intuitive and reasonably accurate picture is as follows:

  • Individual atoms have electrons orbiting around them, but when they come together in a material there is an interaction between the electrons in neighbouring atoms, meaning the orbits can overlap.
  • Electrons are still bound to the material as a whole, but they are only very loosely bound to any individual atom.
  • In the case of conductors, the electrons orbiting furthest from the atoms are mobile enough to move through the whole material.

The way to think of a conductor is as a lattice of fixed positive charges surrounded by lots of mobile negative charges.

  • The negative charges are individual electrons, so they can move easily. In fact, they must be constantly moving due to thermal energy.
  • There are a huge number of mobile electrons in metals, e.g. $\simeq 8.5 \times 10^{28} \mathrm{m}^{-3}$ in copper.
  • However this is still only a fraction of the total number of electrons, which is $\simeq 2.5 \times 10^{30} \mathrm{m}^{-3}$ for copper. Only the outermost electrons are mobile.
  • As well as the thermal random motions of the electrons, they can also be made to have an average motion in a particular direction, normally by applying an electric field – this is then a current.

Let’s suppose a current of 1A is flowing through a copper wire 1mm in radius. How fast are the electrons drifting on average, and how does this compare to their speed from purely random thermal motion?

In the lecture, we will see how to estimate this and find:

  • the speed of the thermal motion [which is not examinable for the current course since it relies on a result from statistical mechanics] is around $1.2 \times 10^{5} \mathrm{m}/\mathrm{s}$ – extremely fast, but in random directions;
  • the speed of the average drift is around $2.3 \times 10^{-5} \mathrm{m}/\mathrm{s}$ – exceptionally slow, but in a single direction.
  • In other words, it is worth bearing in mind when we talk about currents they are not orderly flows of electrons, they are a absolutely tiny drift on top of a chaotic sea of electrons.
  • This is sometimes known as the ‘electron sea’ model, or the Drude model (after its creator Paul Drude). It is not our best understanding of a conductor (which requires quantum mechanics) but it is pretty good.
  • While the random motions will create small-scale electric fields due to small differences in the electron density from one region to another, any large-scale charge imbalance will self-correct because there will be a net force moving electrons back out of that region.

Electric fields in conductors

  • If any electric field is applied to a conductor from the outside, electrons in the sea will start being moved by it, much like liquid in a container will start moving in a particular direction if tilted.
  • Electrons will move backwards along the fieldlines, as we know from earlier lectures. This starts to build up negative charge on one side of the conductor, and the absence of electrons (sometimes referred to as ‘holes’) on the other side of the conductor makes a net positive charge there.
  • The ‘holes’ left where there is a deficit of electrons can be thought of as a positive charge moving in the opposite direction to the electrons. This point of view seems a little odd at first but is actually quite sensible and becomes essential in semiconductor physics.
  • The rearrangement of electrons and holes within a conductor will continue until the electric field from the movement of these internal charges exactly counterbalances the externally-applied field.
  • As a result, the macroscopic electric field inside a conductor is always zero, unless the external field is changing (or magnetic fields can also complicate matters, but that is for later lectures).
  • This also means the electric potential gradient inside a conductor is zero, and therefore the whole of a conductor must be at the same electric potential.
  • As a consequence, the bulk of the conductor is neutral, i.e. there are no net charges anywhere within the interior;
  • But to cancel out an external field the conductor must have net charges on its surfaces, i.e. negative electrons on one side and effectively positive holes (really just ‘lack of electrons’) on the other.2

Nothing is really a perfect conductor, but metals do approximate the above behaviours quite well.

In the example below, there is an external, uniform electric field acting in the x direction. Try moving the negative charges to the left hand end and positive charges to the right hand end in such a way as to cancel out the electric field in the interior. (Remember, the ‘positive charges’ are actually just holes where there are missing electrons, but these are just as moveable as the electrons.)

It isn’t possible to cancel out the field absolutely perfectly. In addition to separating the charges to the far ends, what else do you need to do to get the field as close to zero as possible?

Charged conductors

The above is all considering a conductor that has no net charge. We can also consider a charged conductor. Just as with an insulator:

  • If electrons are added, a conductor becomes negatively charged overall;
  • If electrons are removed, holes are created and a conductor becomes positively charged overall.

The difference compared to an insulator is that the extra electrons can move away from where they are initially placed (in the first case), or electrons can flow in from other parts of the conductor (in the second case). So, unlike with an insulator, the excess charge doesn’t stay where it is generated.

Where does it typically go? One way to answer this is to consider energy; the system of charges will try to minimise its overall energy.

If the individual charges are $q$ and there are $N$ of them, the total potential energy is obtained by summing over all pairs:

\[U = \frac{1}{4 \pi \epsilon_0} \sum_{i=1}^{N} \sum_{\substack{j=1 \\ j \ne i}}^{N} \frac{q^2}{\vert \vec{r}_{ij} \vert}\]

To make $U$ as small as possible, we want all the \(\vert \vec{r}_{ij} \vert\) to be as large as possible, i.e. the charges try to get as far away from each other as possible. This leads to the expectation that the excess charges must be on the surface of the conductor.

Another way to understand where the charges in a conductor go is to consider the electric field. As discussed above, if there is any net large-scale electric field, the free charges in the conductor will try to arrange themselves to compensate for it. So $\vec{E}=0$ everywhere throughout the conductor.

In the lecture we will use Gauss’s law to show that the charge therefore must reside on the surface, in agreement with the energy argument above.

By symmetry on a sphere, the charges will also be evenly distributed (there is no preferred place on the sphere for them to end up). As a result, we can write the electric field for a charged spherical conductor of radius $r_0$ as:

\[\vec{E} = \begin{cases} \frac{Q}{4 \pi r^2 \epsilon_0} \hat{\vec{r}} & \text{if } r > r_0 \\ 0 & \text{if } r < r_0 \end{cases}\]

Contrast this with the electric field for the uniform charge distribution from the start of the lecture. The external field is identical (a direct consequence of Gauss’s law), but the internal field is completely different.

Discussion point: On a more complex, arbitrary shaped conductor, is it still true to say that the charges are on the surface? If so, are they evenly distributed (i.e. is there a constant charge density per unit area of the surface)?

Footnotes

  1. There are lots of documented examples of this, including my personal favourite, which is a risk of creating explosions by using a fire extinguisher! See Chapter 10 of the weirdly interesting book ‘electrostatic hazards’

  2. Talking so much about holes might seem weird. In one sense, they are not a real thing: they are just a lack of electrons in a particular place. Yet, practically speaking, understanding the behaviour of holes is vital for understanding things like semiconductors. And, from a theoretical standpoint, the maths of holes is essentially identical to the maths of antiparticles – thinking about holes led directly to the prediction of positrons.