Potential difference

  • The difference in electrical potential between two points is called the potential difference, and is measured in volts (V).
  • Because the potential at any point in a perfect conductor is the same, if we have some way to generate a potential difference, that potential difference can easily be used to generate a current e.g. through a resistor or other component that does something useful, just by connecting the two points with a conductor.
  • Electrons will flow from the point of lower potential to the point of higher potential, and this flow of electrons is what we call an electric current.
  • But, annoyingly, the “direction of the current” is defined to be the exact opposite of the way electrons flow: from higher to lower potential. This is because historically people imagined positive charges to be carrying the current.

Batteries

  • One way to generate a potential difference is to use a battery.
  • The potential difference between the two terminals of a battery is generated by a chemical reaction inside the battery. The reaction is split into two parts – half at one terminal, and the other half at the other terminal1.
  • The electrons move between terminals through the external circuit, and the positive ions move through the battery to the other electrode.2
  • How exactly to make all this work reliably and reversibly (so that you can recharge the battery by forcing electrons in the other direction) isn’t obvious at all and has been an area of research for more than 200 years.
  • In a charged state, a battery is a dense store of energy – around 500 kJ per kg (that’s well over 100 Wh per kg, i.e. 1 kg of battery can power 100 watts for over an hour). That’s why your phone can run all day from a battery that weighs less than 100 grammes.
  • Petrol, however, can store around 100 times as much energy per kg, which is one reason why electric cars end up being very heavy.
  • Batteries also cannot charge or discharge particularly quickly. As you know from charging your phone, getting a meaningful amount of energy into the battery can take an hour or two. This is because the rate of the chemical reaction is limited by how quickly ions can move through the electrolyte.

Capacitors

  • Another way to generate a potential difference is to somehow put some charge on a conductor.
  • For as long as the charge can’t flow off the conductor, there will be a potential difference between the conductor and the rest of the universe.
  • If the conductor then does get connected to a circuit, the charge can flow off the conductor, generating a current, although the potential difference will drop to zero fairly quickly as the charge flows off.
  • Because this doesn’t involve a chemical reaction, currents can flow in and out of capacitors much more quickly than they can in and out of batteries.
  • But capacitors aren’t used much for energy storage, for reasons we’ll see later in this lecture.

To understand capacitors, you need to work out the electric field generated by a sheet with a known net charge per unit area $\sigma$.

  • In the lecture, we will derive that the electric field is $E=\sigma / 2 \epsilon_0$, where $\sigma$ is the charge per unit area.
  • The field is uniform on both sides.
  • The field is perpendicular to the sheet and directed away from it (for positive $\sigma$) or towards it (for negative $\sigma$).
  • If you have two sheets with opposite charges, the electric field between them will be twice as strong as the field from one sheet alone, $E = \sigma/\epsilon_0$.
  • However the electric field outside the sheets will be zero, because the fields from the two sheets cancel each other out.

This is arrangement of two parallel conducting sheets is known as a capacitor3; the overall net charge is zero, but there is a potential difference $\Delta V$ between the sheets.

  • $\Delta V=Ed$ where $d$ is the separation of the plates.
  • Since the total $E=\sigma/\epsilon_0$, the potential difference is $\Delta V=\sigma d/\epsilon_0$.
  • We can also write $\sigma = Q/A$ where $Q$ is the total charge on one plate and $A$ is the area of the plate.
  • So $\Delta V=Qd/A\epsilon_0$.

  • Note that $A\epsilon_0/d$ is a constant for a given capacitor, and is called the capacitance $C$ of the capacitor. Thus,

    \[\Delta V=\frac{Q}{C};\quad C=\frac{A\epsilon_0}{d}.\]
  • SI units of capacitance is the farad (F), where $1\,{\rm F} = 1\,{\rm C/V}$.
  • When we talk about the charge $Q$ in a capacitor, we do always mean the charge on one plate. The charge on the other plate is $-Q$, and the net charge is zero. Capacitors do not typically ‘store charge’ because they have two plates with zero net charge; they do genuinely store energy though.

Charging a capacitor

The amount of energy that is stored in a capacitor can be calculated by the work done in charging it:

  • If the final potential difference is $V$ and the quantity of charge moved is $Q$, a first guess of the amount of energy stored would be $Q V$ — however this is wrong.
  • When the very first electron moves around the circuit from one plate to the other, there is no potential difference between the plates, and thus it takes no net energy to pull it around the circuit!

  • In fact, the potential difference is always $\Delta V=q/C$, where $q$ is the charge moved so far. So the energy needed to move a small charge ${\rm d}q$ is

    \[{\rm d} U = \Delta V{\rm d}q = \frac{q}{C} {\rm d}q.\]
  • Therefore the total energy for charging the whole capacitor is:
\[U = \int_0^Q \frac{q}{C} {\rm d}q = \frac{1}{2} \frac{Q^2}{C}\]

  • Writing the final potential difference as $V$, applying $V=Q/C$ again, we can also write this as

    \[U = \frac{1}{2} CV^2\]

    and as

    \[U = \frac{1}{2} QV.\]

Discussion point: But wait, you might object: aren’t all points connected up in a circuit supposed to have the same potential? How can there be no p.d. between the capacitor plates at the start, if there is a p.d. between the battery terminals? (As you might guess, this line of reasoning is wrong, but it’s worth thinking about why.)

Where is the energy?

Assuming the energy is stored in the electric field, YF 24.3 shows that the energy per unit volume ($u$) is given by:

\[u = \frac{1}{2} \epsilon_0 E^2\,.\]

We will go over this derivation. Though we can’t prove it in this first year course, the really exciting point is it actually turns out to be a general rule, valid not just for capacitors but for any electric field!

Why capacitors aren’t used for long-term energy storage

To increase the energy stored in a capacitor, using $U=\frac{1}{2} CV^2$, you can increase the potential difference $V$ or the capacitance $C$.

There is an upper limit to how high $V$ can be because the electric field strength between the plates will eventually break down the insulating gap, short-circuiting the plates.

Since $C=A\epsilon_0/d$, to increase $C$ you can:

  • Increase the area of the plates $A$;
  • Decrease the separation of the plates $d$;
  • Use a dielectric material in between the plates, which introduces an extra factor multiplying $\epsilon_0$ (more on this later).

However, there are limits to how much you can increase $C$:

  • The area of the plates $A$ can’t be increased indefinitely in a finite volume.
  • The separation of the plates $d$ can’t be decreased indefinitely, for the same reason that $V$ can’t be increased indefinitely.
  • There are limits on how good dielectrics can be (more on this later).

So capacitors are not very good for storing large amounts of energy.

So what are capacitors good for?

  • Unlike a battery, capacitors can deliver a lot of power (energy per unit time).
  • They are not limited in how quickly they can charge, whereas batteries are limited by the flow of the positive ions through the electrolyte.
  • Applications which need a lot of power for a short time, like a professional camera flash or a defribrillator, do make use of capacitors.
  • They are also widely used in electronics to smooth out the power supply, because they can deliver a lot of power if the external voltage supply momentarily drops.
  • Finally, the device you’re reading this on almost certainly has loads of tiny capacitors in it, used to store not energy but information: the 1s and 0s of the digital world are often stored as charges on capacitors.

Here’s an interesting (non-examinable) plot of the amount of energy you can store per kilogram and the amount of power you can generate per kilogram for various energy storage devices.

Plot of specific energy vs specific power for various energy storage mechanisms

Footnotes

The first half-reaction happens at the zinc electrode, and the second half-reaction happens at the copper electrode. Electrons can’t flow through the lemon juice, and nor can $\text{H}_2$ be generated at the zinc electrode (which requires copper as a catalyst), so instead the electrons flow through the external circuit.

  1. For example, in a ‘lemon battery’, the overall exothermic chemical reaction is $\text{Zn} + 2 \text{H}^+ \rightarrow \text{Zn}^{2+} + \text{H}_2$. This is split up into two half-reactions: $\text{Zn} \rightarrow \text{Zn}^{2+} + 2e^-$ and $2 \text{H}^+ + 2e^- \rightarrow \text{H}_2$. The hydrogen ions come from the lemon juice, and the zinc comes from the zinc electrode. ↩

  2. The material inside the battery is an electrolyte which permits the movement of charged ions, but does not allow electrons to flow. Therefore, electrons are forced to go through the external circuit. ↩

  3. Note that in reality a capacitor has edges to the sheets, and the electric field at these edges isn’t totally uniform. But the field is still very close to uniform in the middle, and this is where the majority of the energy is stored, so the above approximation is good enough for our needs. Also, capacitors typically wind the plates up into a coil to pack a big area into a small volume, and this in theory also needs to be taken into account when calculating the capacitance. ↩