Physics A Level
Chapter 23: Capacitance 23.6 Capacitor networks
Physics A Level
Chapter 23: Capacitance 23.6 Capacitor networks
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There are four ways in which three capacitors may be connected together. These are shown in Figure 23.14. The combined capacitance of the first two arrangements (three capacitors in series, three in parallel) can be calculated using the formulae. The other combinations must be dealt with in a different way:
- Figure 23.14a – All in series. Calculate Ctotal as in Table 23.3.
- Figure 23.14b – All in parallel. Calculate Ctotal as in Table 23.3.
- Figure 23.14c – Calculate Ctotal for the two capacitors of capacitances ${C_1}$ and ${C_2}$, which are connected in parallel, and then take account of the third capacitor of capacitance ${C_3}$, which is connected in series.
- Figure 23.14d – Calculate Ctotal for the two capacitors of capacitances ${C_1}$ and ${C_2}$, which are connected in series, and then take account of the third capacitor of capacitance ${C_3}$, which is connected in parallel.
These are the same approaches as would be used for networks of resistors.
Questions
15) For each of the four circuits shown in Figure 23.14, calculate the total capacitance in $\mu F$ if each capacitor has capacitance $100\,\mu F$.
16) Given a number of $100\,\mu F$ capacitors, how might you connect networks to give the following values of capacitance:
a: $400\,\mu F$?
b: $25\,\mu F$?
c: $250\,\mu F$?
(Note that, in each case, there is more than one correct answer; try to find the answer that requires the minimum number of capacitors.)
17) You have three capacitors of capacitances $100 pF, 200 pF$ and $600 pF$. Determine the maximum and minimum values of capacitance that you can make by connecting them together to form a network.
State how they should be connected in each case.
18) Calculate the capacitance in $\mu F$ of the network of capacitors shown in Figure 23.15.
Sharing charge, sharing energy
If a capacitor is charged and then connected to a second capacitor (Figure 23.16), what happens to the charge and the energy that it stores? Note that, when the capacitors are connected together, they are in parallel, because they have the same p.d. across them. Their combined capacitance Ctotal is equal to the sum of their individual capacitances.
Now we can think about the charge stored, Q. This is shared between the two capacitors; the total amount of charge stored must remain the same, since charge is conserved. The charge is shared between the two capacitors in proportion to their capacitances. Now the p.d. can be calculated from $V = \frac{Q}{C}$ and the energy from $W = \frac{1}{2}C{V^2}$.
If we look at a numerical example, we find an interesting result (Worked example 3).
Figure 23.17 shows an analogy to the situation described in Worked example 3.
Capacitors are represented by containers of water. A wide (high capacitance) container is filled to a certain level (p.d.). It is then connected to a container with a smaller capacitance, and the levels equalise.
(The p.d. is the same for each.) Notice that the potential energy of the water has decreased, because the height of its centre of gravity above the base level has decreased. Energy is dissipated as heat, as there is friction both within the moving water and between the water and the container.
Questions
19) Three capacitors, each of capacitance $120\,\mu F$, are connected together in series. This network is then connected to a $10 kV$ supply. Calculate:
a: their combined capacitance in $\mu F$
b: the charge stored
c: the total energy stored.
20) A $20\,\mu F$ capacitor is charged up to $200 V$ and then disconnected from the supply. It is then connected across a $5.0\,\mu F$ capacitor. Calculate:
a: the combined capacitance of the two capacitors in $\mu F$
b: the charge they store
c: the p.d. across the combination
d: the energy dissipated when they are connected together.
Capacitance of isolated bodies
It is not just capacitors that have capacitance – all bodies have capacitance. Yes, even you have capacitance! You may have noticed that, particularly in dry conditions, you may become charged up, perhaps by rubbing against a synthetic fabric. You are at a high voltage and store a significant amount of charge. Discharging yourself by touching an earthed metal object would produce a spark.
If we consider a conducting sphere of radius r insulated from its surroundings and carrying a charge Q it will have a potential at its surface of V, where $V = \frac{1}{{4\pi {\varepsilon _0}}}\frac{Q}{r}$
Since $C = \frac{Q}{V}$, it follows that the capacitance of a sphere is ${C = 4\pi {\varepsilon _0}r}$.
Question
21) Estimate the capacitance of the Earth given that it has a radius of $6.4 \times {10^6}\,m$. State any assumptions you make.