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A device designed to store charge is called a capacitor.
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A typical capacitor consists of two conducting layers separated by an insulator.
Relationship between charge and p.d.
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 coulomb meter with the switch set to B. In
this way pairs of readings of voltage and
charge are obtained.
Charge is directly proportional to voltage.
Q
0
V
Q
= constant
V
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For any capacitor the ratio Q/V is a constant and is called the capacitance.
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which is usually written as:
Intercept (c) = E (emf)
Gradient (m) = –r (internal
resistance)
p.d. across capacitor (V)
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The unit of capacitance is the Farad (F). 1 Farad = 1 Coulomb per Volt. 1 F = 1 C V-1.
The farad is too large a unit for practical purposes.
In practice the micro farad (µF) = 1 × 10-6 F and the nano farad (nF) = 1 × 10-9 F are used.
Example
A capacitor stores 4 × 10-4 C of charge when the potential difference across it is 100 V.
Calculate the capacitance.
C=
Q
4  10-4

V
100
= 4 µF
Energy stored in a capacitor
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
A charged capacitor stores electrical energy.
Consider the charging of a parallel plate capacitor as shown below:
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When charging a capacitor, the negatively charged plate will tend to repel the electrons
approaching it.
In order to overcome this repulsion work has to be done in charging the capacitor. This energy is
supplied by the battery.
Note that current does not move through the capacitor, electrons build up on one plate and
move 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 p.d. of V and holding a charge Q.
The work done is stored as energy in the capacitor.
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The work done in charging a capacitor is stored as electrical energy, so:
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Since Q = CV the relationship can be re-written in the following forms:
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NOTE:
o E = ½ QV for a capacitor
o EW = QV for the work done on a charge in an electric field.
o 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.
Example
A 40 μF capacitor is fully charged using a 50 V supply. Cal culate the energy stored in the
capacitor.
C = 40 x 10 -6 F
V = 50 V
= ½ × 40 × 10 –6 × 50 2
E=?
= 0.05 J
Charge / Discharge graphs for a capacitor
Charging
E = ½CV 2

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:
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Consider the circuit at three different times.
Consider this circuit when the capacitor is
fully charged, switch to position B
Discharging
If the cell is taken out
switch is set to A, the
discharge
A
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Consider the circuit opposite
The switch is set to B and the capacitor
is fully charged:
Consider this circuit when the capacitor is
fully charged, switch to position B
The cell is shorted
out of the circuit
A
by setting the switch to A
-- -The capacitor discharges:
A
++ ++
-- -++ ++
A
A
B
If the cell is taken out of the circuit and the
switch is set to A, the capacitor will
discharge
A
B
A
B
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While the capacitor is discharging, the current in the circuit and the voltage across the capacitor
behave as shown in the graphs below:
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The current/time graph has the same shape as that during charging but
o the current on discharge is in the opposite direction.
The discharging current decreases because the p.d. across the plates decreases as charge leaves
them.
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Factors affecting the rate of charge and discharge
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The time taken for a capacitor to charge is controlled by the resistance of the resistor R and the
capacitance of the capacitor C.
As an analogy, consider charging a capacitor is 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 product RC is known as the time constant.
Large capacitance and large resistance both increase the charge / discharge time.
The current-time graphs for capacitors of different value during charging are shown below:
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Note that since the area under the current-time graph is equal to charge Q
For a given capacitor the area under the graphs must be equal.
RC Circuits - Example
The switch in the following circuit is closed at time t = 0.
VS = 10 V
1 MΩ
2 μF
(a)
(b)
Immediately after closing the switch what is
(i)
the charge on the capacitor?
(ii)
the p.d. across the capacitor?
(iii)
the p.d. across the resistor?
(iv)
the current through the resistor?
When the capacitor is fully charged what is
(i)
the p.d. across the capacitor?
(ii)
the charge stored on the capacitor?
Solution:
(a)
(b)
(i)
The initial charge on the capacitor is 0 C.
(ii)
The initial pd across the capacitor is 0 V since there is no charge.
(iii)
p.d. across the resistor is 10 V (V R = V S – V C = 10 – 0 = 10 V)
(iv)
I
(i)
Final p.d. across the capacitor equals the supply voltage, 10 V.
(ii)
Q = CV = 2 × 10 –6 × 10 = 2 × 10 –5 C
V 10
 6  1 × 10 –5 A
R 10
Capacitors and Resistors in a.c. Circuits
Frequency response of resistor
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The following circuit is used to investigate the relationship between current and frequency
in a resistive circuit.
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The results show that the current flowing through a resistor is independent of the frequency
of the supply.
Frequency response of capacitor
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The following circuit is used to investigate the relationship between current and frequency
in a capacitive circuit.
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The results show that the current is directly proportional to the frequency of the supply.
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To understand the relationship between the current and frequency consider the two halves
of the a.c. cycle.
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The electrons move back and forth around the circuit passing through the lamp and charging
the capacitor one way and then the other.
o The electrons do not pass through the capacitor).
The higher the frequency the less time there is for charge to build up on the plates of the
capacitor and oppose further charges from flowing in the circuit.
More charge is transferred in one second so the current is larger.
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Some applications of capacitors
Blocking capacitor
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A capacitor will stop the flow of a steady d.c. current .
This is made use of in the a.c./d.c. switch in an oscilloscope.
In the a.c. position a series capacitor is switched in allowing passage of a.c. components of the
signal, but blocking any steady d.c. signals.
Flashing indicators
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A low value capacitor is charged through
a resistor until it acquires sufficient
voltage to fire a neon lamp.
The neon lamp lights when the p.d.
reaches 100 V.
The capacitor is quickly discharged and
the lamp goes out when the p.d. falls
below 80V.
1 - 2 M
120 V
Crossover networks in loudspeakers

In a typical crossover network in low cost
loudspeaker systems, the high
frequencies are routed to LS-2 by the
capacitor.
LS 1
LS 2