- State that the charge Q on two parallel conducting plates is directly proportional to the p.d. V between the plates.
- Describe the principles of a method to show that the charge on a capacitor is directly proportional to the p.d. across the capacitor.
- State that capacitance is the ratio of charge to p.d.
- State that the unit of capacitance is the farad and that one farad is one coulomb per volt.
- Carry out calculations involving the relationship between capacitance, charge and potential difference.
- Explain why work must be done to charge a capacitor.
- State that the work done to charge a capacitor is given by the area under the graph of charge against p.d.
- Draw qualitative graphs of current against time and of p.d. against time for the charge and discharge of a capacitor in a d.c. circuit containing a resistor and capacitor in series.
- Carry out calculations involving p.d. and current in CR circuits.
- State the relationship between current and frequency in a capacitive circuit.
- Describe and explain the possible functions of a capacitor: storing energy, storing charge, blocking d.c. while passing a.c.

Aids to Understanding

The Meaning of Capacitance

Whenever we are asked to explain what a capacitance of 2000 m F means we first look at the units of F. Thus 2000 m F means 2000 mCV^{-1}. This means little on its own so we need to elaborate a little further. If we recall the activity that first led us to the concept of capacitance it should be clear that 2000 mCV^{-1} means that for every 2000 mC placed on the capacitor, its pd rises by 1 V.

The Coulombmeter

The Coulombmeter consists of a capacitor. Whenever we use it to measure the charge on a capacitor's plate we connect the Coulombmeter in parallel with the capacitor and the charge is transferred from the capacitor to the Coulombmeter. However unless the Coulombmeter's capacitor (2000 mF) is much larger than the capacitance of the the capacitor whose charge it is we wish to measure then the capacitors will share the charge and an inaccurate reading will occur. In the first activity we used a capacitor of 1.0 m F

Capacitors in parallel

For resistors in parallel we use the fact that the pd across components in parallel are equal. Use this fact to find an expression for capacitors inparallel. Compare this with the expression for resistors in parallel. If we have a number of resistances in parallel then the largest resistance will have the smallest current flowing in it. This is a fact. Write an equivalent expression for capacitances in parallel. Remember C = Q/V.

Energy and Capacitors

In unit 2.1 you learned that the work done in pushing a charge, Q, through a pd of V, is equal to QV and we say W = QV. The work done in charging a capacitor, by an external source, is exactly 1/2 QV, however. Thus although the source is placing charges on the capacitor against the field the field it is working against is only at a maximum at the end of the charging process. Given the field is zero inititally we can think of the source working against an average field and thus the energy stored in a capacitor with a charge Q on one of its plates and a pd of V between the plates, is equal to E = 1/2QV. In this argument it was assumed that the E field between the plates is proportional to the pd between the plates. This is entirely justified and it can be shown that E = V/d where d is the distance between the plates.

Capacitors in dc circuits

The characterist curves described in the main text should be memorised but also understood. To help us here we need to recall our circuit rules for dc circuits, ie the current in a dc circuit is the same at all points. This means there is only one characteristic graph for current building up in a CR circuit. The fact that the current falls to zero can be explained by thinking of the capacitor gradually becoming charged up and therefore opposing more charge from joining it. When the pd across the capacitor equals the pd of the cell the current is zero.

The voltage across components in series always sums to the voltage of the cell therefore as the pd rises across the capacitor it must fall across the resistor. whenever a capacitor discharges through a resistor the voltage across each must be the same across both components so the characteristic voltage graphs are identical.

Read this paragraph in conjuction with the graphs in the main text on the right.

Problem Solving

The following points must be borne in mind.

1. The charge stored on a capacitor is the area under its I-t graph.

2. The initial charging current is independent of the value of the capacitor and depends only on Voltage of cell and resistance of resistor, ie I = v/R.

3. The energy stored on a capacitor is the area under its Q-V graph.

4. For a CR circuit the voltage across each component always sum to the voltage of the cell at ALL times during charging or discharging.
This list is not comprehensive but represents some points that students occasionally miss or forget.

ac supplies and Capacitors

The result that current increases with frequency in ac circuits contaning capacitors might surprise you but it ties in with our previous work.
In a dc circuit the capacitor gradually charges up and prevents further charge from joining it. with an ac supply, the charge never quites get time to charge up the capacitor before the signal changes. As the frequency increases this effect is further enhanced therefore the opposition to charge decreases with increasing frequency and the current increases. We call the opposition to ac current reactance and clearly the **reactance** of a capacitor decreases with increasing currenty. This result should be compared with resistors in ac circuits. The current in an ac circuit with a resistance **does not change** with changing frequency.

A circuit capacitor has a capacitance of 2000 mF and a maximum working voltage of 15 V. This means if the voltage applied to the capacitor exceeds 15 V then it is likely the capacitor will be damaged. The following questions could now be asked

i) Calulate the maximum charge this capacitor can hold if it is not to be overloaded. Answer: Q = C V = 2000 mF x 15 V = 30 000 mC.

ii) If the capacitor is connected into an ac supply, then what is the maximum V

Answer: The peak voltage of the ac supply must not exceed 15 V, therfore V

A capacitor consists of two conducting plates that are insulated from each other. The insulation can either be air or some other material. Since no charges can flow through a capacitor we might expect the current in circuits containing capacitors might also be zero. We find this is not the case as the investigations show.

The image above is colour coded. Red represents the circuit, data and graph for matters involving charging the capacitor. Blue represents the circuit, data and graph for matters involving discharging the capacitor. Note the initial current in both cases is 3 mA, which is V/R nd the area under each graph is equal to the charge transferred to the capacitor during charging and the charge that leaves the capacitor during discharging.

Whenever a current flows in an RC circuit, charges build up on the capacitor thus raising the pd across its plates V

V

Note that the right hand columns of each table add up to 3.0 V. This is to be expected since V

If we throw the switch to position 2, the capacitor will discharge through the resistor and the pd between its plates will fall. Note that during discharging the capacitor and resistor are in parallel. This means the pd across them will be equal and thus their pd discharging graphs are identical.

The pc terminals above monitor the pd across the capacitor and resistor when the switch is closed. If the capacitor takes 100.0s to completely charge...

i) Draw what will be seen on the pc monitors. Numerical values on axes are required for a full answer.

ii) Draw the current-time graph for the circuit. Numerical values on axes are required for a full answer.

iii) How much charge is stored on the capacitor when fully charged.

iv) How much energy is stored on the capacitor when it is fully charged.

The fact that current increases with frequency is a useful fact to engineers when they are designing electronic circuits.

1. Why are capacitors are needed in flash lights for cameras.

2. What property of capacitors allow it to block dc signals but pass ac signals

3. Modern electronic circuits work with dc signals, but they are connected in the ac mains supply. Find out what rectification means and then find out what smoothing means. Find out how capacitors smooth a rectified signal.