Download Chapter 13

Capacitance and Electric Fields
Chapter 13
 Introduction
 Capacitors and Capacitance
 Alternating Voltages and Currents
 The Effect of a Capacitor’s Dimensions
 Electric Fields
 Capacitors in Series and Parallel
 Voltage and Current
 Sinusoidal Voltages and Currents
 Energy Stored in a Charged Capacitor
 Circuit Symbols
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Introduction
13.1
 We noted earlier that an electric current represents a
flow of charge
 A capacitor can store electric charge and can
therefore store electrical energy
 Capacitors are often used in association with
alternating currents and voltages
 They are a key component in almost all electronic
circuits
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Capacitors and Capacitance
13.2
 Capacitors consist of two conducting surfaces
separated by an insulating layer called a dielectric
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 A simple capacitor
circuit
– when switch is closed
electrons flow from
top plate into battery
and from battery onto
bottom plate
– charge produces an
electric field across
the capacitor and a
voltage across it
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 For a given capacitor the stored charge q is directly
proportional to the voltage across it V
 The constant of proportionality is the capacitance C
and thus
C
Q
V
 If the charge is measured in coulombs and the
voltage in volts, then the capacitance is in farads
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 Example – see Example 13.1 in course text
A 10 F capacitor has 10 V across it. What quantity
of charge is stored in it?
From above
Q
C
V
Q  CV
 10 5  10
 100 μC
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Alternating Voltages and Currents
13.3
 A constant current cannot flow through a capacitor
– however, since the voltage across a capacitor is
proportional to the charge on it, an alternating voltage
must correspond to an alternating charge, and hence
to current flowing into and out of the capacitor
– this can give the
impression that an
alternating current
flows through the
capacitor
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 A mechanical analogy may help to explain this
– consider a window - air cannot pass through it, but
sound (which is a fluctuation in air pressure) can
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The Effect of a Capacitor’s Dimensions
13.4
 The capacitance of a capacitor is directly
proportional to its area A, and inversely proportional
to the distance between its plates d. Hence C  A/d
– the constant of proportionality is the permittivity  of
the dielectric
– the permittivity is normally expressed as the product of
the absolute permittivity 0 and the relative
permittivity r of the dielectric used
C
A
d

 0 r A
d
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Electric Fields
13.5
 The charge on the capacitor produces an electric
field with an electric field strength E given by
E V
d
the units of E are volts/metre (V/m)
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 All insulating materials have a maximum value for the
field strength that they can withstand
– the dielectric strength Em
 To produce maximum capacitance for a given size of
capacitor we want d to be as small as possible
– however, as d is decreased the electric field E is
increased
– if E exceed Em the dielectric will break down
– there is therefore a compromise between physical size
and breakdown voltage
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 The force between positive and negative charges is
described in terms of the electric flux linking them
– measured in coulombs (as for electric charge)
– a charge Q will produce a total flux of Q coulombs
 We also define the electric flux density D as the flux
per unit area
 In a capacitor we can almost always ignore edge
effects, and
DQ
A
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 Combining the earlier equations it is relatively easy to
show that
 D
E
 Thus the permittivity of the dielectric within a
capacitor is equal to the ratio of the electric flux
density to the electric field strength.
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Capacitors in Series and Parallel
13.6
 Capacitors in parallel
– consider a voltage V applied
across two capacitors
– then the charge on each is
Q  VC
1
Q  VC
1
2
2
– if the two capacitors are replaced with a single
capacitor C which has a similar effect as the pair, then
Charge stored on C  Q  Q
1
2
VC  VC  VC
1
C  C C
1
2
2
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 Capacitors in series
– consider a voltage V applied
across two capacitors in series
– the only charge that can be
applied to the lower plate of C1
is that supplied by the upper plate
of C2. Therefore the charge on
each capacitor must be identical.
Let this be Q, and therefore if a
single capacitor C has the same
effect as the pair, then
V  V V
1
2
Q QQ
C C C
1
2
1 1  1
C C C
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Voltage and Current
13.7
 The voltage across a capacitor is directly related to
the charge on the capacitor
1
Q
V    Idt
C C
 Alternatively, since Q = CV we can see that
dQ
dV
C
dt
dt
and since dQ/dt is equal to current, it follows that
dV
I C
dt
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 Consider the circuit shown here
– capacitor is initially discharged
 voltage across it will be zero
– switch is closed at t = 0
– VC is initially zero
 hence VR is initially V
 hence I is initially V/R
– as the capacitor charges:




VC increases
VR decreases
hence I decreases
we have exponential behaviour
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 Time constant
– charging current is determined by R and the voltage
across it
– increasing R will increase the time taken to charge C
– increasing C will also increase time taken to charge C
– time required to charge to a particular voltage is
determined by the product CR
– this product is the time constant  (greek tau)
 See Computer Simulation Exercises 13.1 and 13.2
in the course text
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Sinusoidal Voltages and Currents
13.8
 Consider the application of a
sinusoidal voltage to a capacitor
– from above I = C dV/dt
– current is directly proportional to
the differential of the voltage
– the differential of a sine wave is
a cosine wave
– the current is phase-shifted by
90 with respect to the voltage
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 Since I = C dV/dt the magnitude of the current is
related to the rate of change of the voltage
– in sinusoidal voltages the rate of change is determined
by the frequency
– hence capacitors are frequency dependent in their
characteristics
 We will return to look at frequency dependence in
later lectures.
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Energy Stored in a Charged Capacitor
13.9
 To move a charge Q through a potential difference V
requires an amount of energy QV
 As we charge up a capacitor we repeatedly add
small amounts of charge Q by moving them through
a voltage equal to the voltage on the capacitor
 Since Q = CV, it follows that Q = CV, so the
energy needed E is given by
1
E   CV dV  CV 2
0
2
V
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 Alternatively, since V = Q/C
2
2
1
1
Q
1
Q


E  CV 2  C   
2
2 C 
2C
 Example – see Example 13.7 in the course text
Calculate the energy stored in a 10 F capacitor
when it is charged to 100 V.
From above:
1
1
2
E  CV   10  5  1002  50 mJ
2
2
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Circuit Symbols
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13.10
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Key Points




A capacitor consists of two plates separated by a dielectric
The charge stored on a capacitor is proportional to V
A capacitor blocks DC but appears to pass AC
The capacitance of several capacitors in parallel is equal to
the sum of their individual capacitances
 The capacitance of several capacitors in series is equal to
the reciprocal of the sum of the reciprocals of the individual
capacitances
 In AC circuits current leads voltage by 90 in a capacitor
 The energy stored in a capacitor is ½CV2 or ½Q2/C
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