3.1d Capacitors
Capacitance
is the ability (or capacity) to store charge. A device that stores charge is
called a capacitor.
Practical
capacitors are conductors separated by an insulator. The simplest type consists
of two metal plates with an air gap between them. The symbol for a capacitor is
based on this:
Relationship between
charge and pd
The capacitor is charged to a
chosen voltage by setting the switch to A. The charge stored can be measured
directly by discharging through the coulometer with the switch set to B. In
this way pairs of readings of voltage and charge are obtained.
It is
found that the charge stored on a capacitor and the pd (voltage) across it are
directly proportional:
so Q = a
constant
V
This
constant is defined as the capacitance, C:
or Q = CV
The formal
definition of capacitance is therefore the charge stored per unit voltage.
Notice that this means that capacitance is the gradient of the above graph. The unit of capacitance is the farad
(F).
From the
above formula: 1 F = 1 coulomb per volt.
The farad
is too large a unit for practical purposes and the following submultiples are
used:
1 mF
(microfarad) = 1 × 10–6 F
1 nF
(nanofarad) = 1 × 10–9 F
Note: When a capacitor is charging,
the current is not constant (more on this later). This means the formula Q = It
cannot be used to work out the charge stored.
Worked example
A
capacitor stores 4 × 10–4 C of
charge when the potential difference across it is 100 V. Calculate the
capacitance.
F (4 mF)
Energy
stored in a capacitor
Why work must be done
to charge a capacitor
Consider
this:
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When
current is switched on electrons flow onto one plate of the capacitor and away
from the other plate.
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This
results in one plate becoming negatively charged and the other plate
positively charged.
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Eventually
the current ceases to flow. This happens when the pd across the plates of the
capacitor is equal to the supply voltage.
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The
negatively charged plate will tend to repel the electrons approaching it. In
order to overcome this repulsion work has to be done and energy supplied. This
energy is supplied by the battery. Note that current does not flow through the
capacitor, electrons flow onto one plate and away from the other
plate.
For a
given capacitor the pd across the plates is directly proportional to the charge
stored. Consider a capacitor being charged to a pd of V and holding a charge Q.
This work
is stored as electrical energy, so:
E = ½QV
(Contrast
this with the work done moving a charge in an electric field where W = QV.
In a capacitor the amount of charge and voltage are constantly changing rather
than fixed and therefore the ½ is needed as an averaging factor.)
Since Q = CV
there are alternative forms of this relationship:
energy = ½CV2
and energy = ½
Worked example
A 40 mF
capacitor is fully charged using a 50 V supply. Calculate the energy stored in
the capacitor.
energy
= ½CV2
=
½ × 40 ×
10–6 × 502
= 0.05 J
Charging
and discharging a capacitor
Charging
Consider the following
circuit:
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When the switch is
closed the current flowing in the circuit and the voltage across the capacitor
behave as shown in the graphs below.
Consider the circuit at
three different times.
Discharging
Consider
the circuit opposite in which
the
capacitor is fully charged:
If the
cell is shorted out of the circuit
the
capacitor will discharge.
While the
capacitor is discharging, the current in the circuit and the voltage across the
capacitor behave as shown in the graphs below:
Although
the current/time graph has the same shape as that during charging, the currents
in each case are flowing in opposite directions. The discharging current
decreases because the pd across the plates decreases as charge leaves them.
A
capacitor stores charge, but unlike a cell it has no capability to supply more
energy. When it discharges, the energy stored will be used in the circuit, eg
in the above circuit it would be dissipated as heat in the resistor.
Factors affecting the
rate of charge and discharge
The time
taken for a capacitor to charge is controlled by the resistance of the resistor
R (because it controls the size of
the current, ie the charge flow rate) and the capacitance of the capacitor
(since a larger capacitor will take longer to fill and empty). As an analogy,
consider charging a capacitor as being like filling a
jug with
water. The size of the jug is like the capacitance and the resistor is like the
tap you use to control the rate of flow.
The values
of R and C can be multiplied together to form what is known as the time
constant. Can you prove that R × C has units of time, seconds? The time
taken for the capacitor to charge or discharge is related to the time constant.
Large
capacitance and large resistance both increase the charge or discharge time.
The I/t
graphs for capacitors of different value during charging are shown below:
The effect
of capacitance on charge The effect
of resistance on charge
current current
Note that
since the area under the I/t graph is equal to charge, for a given
capacitor the area under the graphs must be equal.
Worked example
The switch
in the following circuit is closed at time t
= 0.
(a) Immediately after closing the switch what is
(i) the
charge on C?
(ii) the
pd across C?
(iii) the
pd across R?
(iv) the
current through R?
(b) When the capacitor is fully charged what is
(i) the
pd across the capacitor?
(ii) the
charge stored?
(a) (i) The
initial charge on the capacitor is zero.
(ii) The
initial pd across the capacitor is zero since there is no charge.
(iii) pd
across the resistor is 10 V
(VR = VS
– VC = 10 – 0 = 10 V)
(iv) 
1 × 10–5 A
1 × 10–5 A
(b) (i) Final
pd across the capacitor equals the supply voltage, 10 V.
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