Capacitors
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
1
Notes
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February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
2
Capacitors
! Capacitors are devices that store energy in an electric field
! Capacitors are used in many every-day applications
• Heart defibrillators
• Camera flash units
• Touch screens
! Capacitors are an essential part of electronics
! Capacitors can be micro-sized on
computer chips or super-sized
for high power circuits such as
FM radio transmitters and exist
in a variety of shapes
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
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Capacitance
! A capacitor consists of two separated conductors, usually
called plates, even if these conductors are not simple planes
! If we take apart a typical capacitor, we might find two sheets
of metal foil separated by an insulating layer of Mylar
! The sandwiched layers can be rolled
up with another insulating layer into
a compact form that does not look
like parallel sheets of metal
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
4
Capacitance
! Assume a convenient geometry and then generalize the
results
! The geometry we choose is a parallel plate capacitor, which
consists of two parallel conducting plates, each with area A,
separated by a distance d, in a vacuum
! The capacitor is charged by
placing a charge of +q on
one plate and –q on the
other plate
! The electric potential, ΔV, between the plates is proportional
to the amount of charge on the plates
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
5
Capacitor potential
! Because the plates are conductors, the
charge will distribute itself uniformly
over the plates
! We can use the techniques in
Chapter 23 to calculate the
potential numerically using
a computer
! The potential has a steep drop
between the plates and a gradual drop
outside the plates
! Thus the electric field will be strong between the plates and
weak outside the plates
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
7
Capacitor potential
! We can take a
slice through the
x-y plane
! The
equipotential
lines are close
together between
the plates and far
apart outside the
plates
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
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Capacitor field
 
 
E ( r ) = −∇V ( r )
! Here we show the
electric field vectors at
regularly spaced grid
points in the x-y plane
! The field between the
plates is perpendicular
to the plates and has a
much larger
magnitude than the
field outside the plates
! The field outside the
plates is the fringe field
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
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Capacitance
! The potential difference between the two plates is
proportional to the amount of charge on the plates
! The proportionality constant is the the capacitance of the
device
q
C=
V
q
C=
ΔV
! The capacitance of a device depends on the area of the plates
and the distance between the plates but not on the charge or
potential difference
! The capacitance tells how much charge is required to
produce a given potential difference between the plates
! We can rewrite this equation as
q = CΔV
February 2, 2014
q = VC
Physics for Scientists & Engineers 2, Chapter 24
10
Capacitance
! The units of capacitance are coulombs per volt
! A new unit was assigned to capacitance, named after British
physicist Michael Faraday (1791 – 1867) called the farad (F)
1C
1F=
1V
! One farad represents a very large capacitance
! Typical capacitors have capacitances ranging from 1 pF to
1 μF
! With this definition, we can write the electric permittivity of
free space as
ε 0 = 8.85⋅10−12 F/m
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
12
Circuits
! An electric circuit consists of wires that connect circuit
elements
! These elements can be capacitors
! Other important elements include resistors, inductors,
ammeters, voltmeters, diodes, and transistors
! Circuits usually need a power source
! A battery can provide a fixed potential difference commonly
called voltage
! An AC power source provides a time-varying potential
difference
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
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Circuit Symbols
! Circuit elements are represented by commonly used
symbols
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
15
Charging and Discharging a Capacitor
! A capacitor is charged by connecting it to a battery to create
a circuit
! Charge flows from the battery to the capacitor until the
potential difference across the capacitor is the same as the
potential difference across the battery
! If the capacitor is then disconnected, it will hold its charge
and potential difference
! We can use a circuit diagram to illustrate
the charging/discharging process
• Switch position a charges the capacitor
• Connects the battery across the plates
• Switch position b discharges the capacitor
• Shorts the plates together
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
16
Parallel Plate Capacitor
! We will consider an ideal parallel plate capacitor
! Two parallel conducting plates in a vacuum with charge +q
on one plate and –q on the other plate
! The field is constant between the plates and zero outside
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
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Parallel Plate Capacitor
! We can calculate the field using Gauss’s Law
  q

∫∫ E • dA = ε 0
! We use the Gaussian surface shown by the dotted red line
! We add the contributions to integral from the top, the
bottom, and the sides
! The sides are outside the capacitor, so the field is zero
! The top is inside the conductor, so the field is zero
! The bottom part is in the constant field between the plates
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
18
Parallel Plate Capacitor
! The electric field is constant and points downward
! The vector normal to the surface also points downward
! So the integral over the Gaussian surface becomes
 
q

∫∫ E • dA = ∫∫ E dA = E ∫∫ dA = EA = ε 0
! The electric potential difference across the two plates is
ΔV = − ∫
f
i
 
E • ds
! The path of integration is chosen to be from the negatively
charged plate to the positively charged plate, which gives us
qd
ΔV = − ∫ ( −Eds ) =Ed =
0
ε0 A
d
! Combining these equations gives us the capacitance of a
parallel plate capacitor
ε A
q
εA
C=
= 0
ΔV
d
February 2, 2014
C=
Physics for Scientists & Engineers 2, Chapter 24
0
d
19
Demo: Large Capacitance
! Energy stored in this particular capacitor: 90 J
! This is equivalent to the kinetic energy of a
mass of 1 kg moving at a velocity of 13.4 m/s!
E = 12 mv 2
2E
2⋅90 J
v=
=
= 13.4 m/s
m
1 kg
February 2, 2014
Physics for Scientists & Engineers 2, Chapter 24
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Example - Capacitance of a Parallel Plate Capacitor
! We have a parallel plate capacitor
constructed of two parallel plates,
each with area 625 cm2 separated
by a distance of 1.00 mm.
! What is the capacitance of this
parallel plate capacitor?
ε0 A
d
A = 625 cm 2 = 0.0625 m 2
C=
d = 1.00 mm = 1.00 ⋅10 −3 m
( 8.85 ⋅10
C=
−12
)(
F/m 0.0625 m 2
1.00 ⋅10
C = 0.553 nF
−3
m
) = 5.53 ⋅10
-10
F
A parallel plate capacitor constructed out of square conducting
plates 25 cm x 25 cm (=625 cm2) separated by 1 mm produces a
capacitor with a capacitance of about 0.55 nF
February 2, 2014
Physics for Scientists&Engineers 2
22
Example 2 - Capacitance of a Parallel Plate Capacitor
! We have a parallel plate
capacitor constructed of two
parallel plates separated by a
distance of 1.00 mm.
! What area is required to
produce a capacitance of 0.55
F?
ε0 A
d
d = 1.00 mm = 1.00 ⋅10 −3 m
C=
(
)
1.00 ⋅10 −3 m ( 0.55 F )
dC
8
2
A=
=
=
0.62
⋅10
m
ε0
8.85 ⋅10 −12 F/m
(
)
A parallel plate capacitor constructed out of square conducting
plates 7.9 km x 7.9 km (4.9 miles x 4.9 miles) separated by 1 mm
produces a capacitor with a capacitance of 0.55 F
February 2, 2014
Physics for Scientists&Engineers 2
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