storyline chapter 25: capacitance and dielectrics...definition of capacitance. the capacitance cof a...
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Storyline
Chapter 25:Capacitance and Dielectrics
Physics for Scientists and Engineers, 10e
Raymond A. Serway
John W. Jewett, Jr.
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Capacitor
Capacitor: combination of two conductors, called plates
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Definition of Capacitance
The capacitance C of a capacitor is
defined as the ratio of the magnitude of
the charge on either conductor to the
magnitude of the potential difference
between the conductors:
QC
V
1 F 1 C/V=
Q C V=
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Definition of Capacitance
Consider capacitor formed from pair of parallel plates:
• Each plate connected to one terminal of battery
• Acts as source of potential difference
• If capacitor initially uncharged:
• Battery establishes electric field in connecting wires
For plate connected to negative terminal of battery:
• Electric field in wire applies force on electrons in wire immediately
outside this plate:
• Force causes electrons to move onto plate
• Movement continues until plate, wire, and terminal all at same
electric potential
• Once equilibrium attained potential difference no longer exists
between terminal and plate:
• Result: no electric field present in wire and electrons stop
moving
• Plate now carries negative charge
• Similar process occurs at other capacitor plate:
• Electrons move from plate to wire, leaving plate positively
charged
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Calculating Capacitance
0 0
QE
A
= =
0
QdV Ed
A = =
0
0
/
Q QC
V Qd A
AC
d
= =
=
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Capacitance
0 AC
d
=
The parallel-plate capacitor:
• Proportional to area of plates
• Inversely proportional to plate separation
( )04π
2ln /
LC
b a
=The cylindrical capacitor:
• depends on the radii a and b
• is proportional to the length of the cylinders
An example of this type of geometric arrangement is a coaxial cable, which consists of two concentric
cylindrical conductors separated by an insulator. You probably have a coaxial cable attached to your
television set if you are a subscriber to cable television. The coaxial cable is especially useful for shielding
electrical signals from any possible external influences.
The spherical capacitor:
• depends on the radii a and b ( )04π ab
Cb a
=
−
4πbC a→ =
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Combinations of Capacitors
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Parallel Combination
1 2V V V = =
tot 1 2
1 1 2 2
Q Q Q
QC
V
C V C V
= +
=
= +
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Parallel Combination
tot eqQ C V=
( )
eq 1 2
parallel combination
C C C= +
( )eq 1 2 3+ + parallel combinationC C C C= +
tot 1 1 2 2Q C V C V= +
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Series Combination
1 2Q Q Q= =
tot 1 2
1 2
1 2
V V V
Q Q
C C
= +
= +
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Series Combination
1 2
eq 1 2
Q QQ
C C C= +
( )eq 1 2
1 1 1 series combination
C C C= +
tot
eq
QV
C =
( )eq 1 2 3
1 1 1 1+ + series combination
C C C C= +
1 2tot
1 2
Q QV
C C = +
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Energy Stored in a Charged Capacitor
• Because positive and negative charges separated in
system of two conductors in charged capacitor:
Electric potential energy stored in system
• If plates of charged capacitor connected by conductor
such as wire → Charge moves between each plate
and connecting wire until capacitor uncharged.
Discharge often observed as visible spark
• If you accidentally touch opposite plates of charged capacitor → your fingers act as
pathway for discharge. Result: electric shock – Could be dangerous if high voltages
present (i.e., power supply of home theater system)
• Because charges can be stored in capacitor even when system turned off →
Unplugging system does not make it safe to open case and touch components
inside
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Energy Stored in a Charged Capacitor
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Energy Stored in a Charged Capacitor
d d dq
W V q qC
= =
2
0 0
1d d
2
Q Qq QW q q q
C C C= = =
( )2
21 1
2 2 2E
QU Q V C V
C= = =
Work necessary to transfer increment of charge dq from
plate carrying charge –q to plate carrying charge q (at
higher electric potential) is:
Total energy:
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Energy Stored in a Charged Capacitor
( )21
2EU C V=
( ) ( )2 20
0
1 1
2 2E
AU Ed Ad E
d
= =
For parallel-plate capacitor:
• potential difference related to electric field through V = Ed
• capacitance: C = 0A/d
Energy per unit volume (energy density) is: 2
0
1
2
EE
Uu E
Ad= =
Expression generally valid regardless of source of electric field:
• Energy density in any electric field square of magnitude of
electric field at given point
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Portable Defibrillator
• Stored energy released through te heart by conducting electrodes (paddles) which are placed
on both sides of victim’s chest
• Defibrillator can deliver energy to patient in 2 ms
• Paramedics must wait between applications of energy because of time interval necessary for
capacitors to become fully charged
• Capacitors serve as energy reservoirs that can be slowly charged and then quickly discharged
to provide large amounts of energy in a short pulse
• When cardiac fibrillation (random contractions) occurs →
heart produces rapid, irregular pattern of beats
• Fast discharge of energy through heart can return the
organ to normal beat pattern
• Emergency medical teams use portable defibrillators that
contain batteries capable of charging capacitor to high
voltage
• Circuitry actually permits capacitor to be charged to
much higher voltage than that of battery
• Up to 360 J stored in electric field of large
capacitor in defibrillator when fully charged
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Capacitors with Dielectrics
0VV
=
0 0
0
0
0
0
/
Q QC
V V
Q
V
C C
= =
=
=
0 AC
d
=
Dielectric: nonconducting material (i.e., rubber, glass, or waxed paper)
𝜀 = dielectric constant of the material
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Capacitors with Dielectrics
0 AC
d
=
1) Dielectric provides:
• Increase in capacitance
• Increase in maximum operating voltage
2) If magnitude of electric field 𝐸 = 𝑈/𝑑 in dielectric exceeds dielectric
strength → Insulating properties break down and dielectric begins to
conduct, causing discharge.
3) When selecting capacitor for given application, must consider capacitance
as well as expected voltage across capacitor in circuit.
( )2
21 1
2 2 2E
QU Q V C V
C= = =
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Electric Dipole in an Electric Field
2p aq
Electric dipole: two charges of equal magnitude and opposite sign separated by distance a
Electric dipole moment: vector Ԧ𝑝 directed from −q toward +q along line joining
the charges, with magnitude
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Electric Dipole in an Electric Field
sinFa =
sin sinaqE pE = =
= τ p E
sinFa + −= =
sinr F rF = =
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Energy of Dipole
cosEU pE = −
dW d =
sinpE =
( )
sin sin
cos cos cos
f f f
i i i
f
i
f i
i f
U U d pE d pE d
pE pE
− = = =
= − = −
EU = − p E
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Polar Molecules
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Microwave cooking
Many molecules (such as those of water) are electric dipoles, meaning that they have
a partial positive charge at one end and a partial negative charge at the other, and
therefore rotate as they try to align themselves with the alternating electric field of
the microwaves. Rotating molecules hit other molecules and put them into motion,
thus dispersing energy. This energy, dispersed as molecular rotations, vibrations
and/or translations in solids and liquids raises the temperature of the food, in a
process similar to heat transfer by contact with a hotter body
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Soap and the Dipole Structure of Water
• Grease and oil made up of nonpolar molecules
• Generally not attracted to water
• Plain water not very useful for removing this type of grime
• Soap contains long molecules called surfactants
• In long molecule →
• Polarity characteristics of one end of molecule can be different from those at
other end
• In surfactant molecule:
• One end acts like nonpolar molecule and other acts like polar molecule
• Nonpolar end can attach to grease or oil molecule
• Polar end can attach to water molecule
• Soap serves as chain → linking dirt and water molecules together
• When water rinsed away:
• Grease and oil go with it
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Induced Polarization
• Symmetric molecule (figure): no permanent polarization
• Polarization can be induced by placing molecule in electric field.