Physics 272
February 3
Spring 2015
www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm
go.hawaii.edu/KO
Prof. Philip von Doetinchem
philipvd@hawaii.edu
PHYS272 - Spring 15 - von Doetinchem - 186
Gauß's law
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Electric flux:
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Gauß's law:
PHYS272 - Spring 15 - von Doetinchem - 188
Electric potential energy and electric potential
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Charged particle moving in a field: field exerts work on particle
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Conservative force → required work independent of exact path
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Moving with the direction of the electric field means moving in the
direction of decreasing potential
Moving a charge slowly against an electric field requires an external
force, equal and opposite to the electric force.
PHYS272 - Spring 15 - von Doetinchem - 189
Capacitance and dielectrics
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Capacitor: just insulate
two conductors (with
same amount of
negative and positive
charge)
Work must be done to
move charges through
the resulting potential
→ Electric field can be
seen as a store-house
of electric potential energy
Source: http://de.wikipedia.org/wiki/Kondensator_%28Elektrotechnik%29
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Capacitor has a certain capacitance depending on its
properties: size, shape, material
Many applications (flashs, filters, microphone, etc.)
PHYS272 - Spring 15 - von Doetinchem - 190
Capacitors and capacitance
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Charging capacitor:
conductors initially
uncharged
Transfer electrons from
one side to the other
Net charge on capacitor
is zero
Common way of charging:
connect sides to different
terminals of a battery
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Potential difference is proportional to the stored charge
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Capacitance stays constant:
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Capacitance is a measure of the ability of a capacitor to store energy.
PHYS272 - Spring 15 - von Doetinchem - 191
Capacitance of a parallel plate capacitor
Gauss' law:
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One farad is a very large amount:
typical values:
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flash unit in a camera: microfarad (F, 10-6)
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radio tuning unit: 10-100 picofarad (pF, 10 -12)
PHYS272 - Spring 15 - von Doetinchem - 192
Spherical capacitor
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Outer sphere makes no contribution to the field
between the sphere
PHYS272 - Spring 15 - von Doetinchem - 193
Capacitors in series
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Combining capacities
helps you to get the
capacitance you need
for your application
Series connection:
capacitors are connected
one after the other
Charges on all plates have
the same magnitude
Equivalent capacitance of a series combination of
capacitors is always less than any individual capacitance.
Charges on plates are the same, but if the dimensions are
different
→ potential for each capacitor different
PHYS272 - Spring 15 - von Doetinchem - 196
Capacitors in parallel
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Charges can reach capacitors independently from the source
Imagine one big capacitor that you split into multiple smaller
capacitors
The parallel combination of capacitors always has a higher
capacitance than the individual capacitances
Charges are generally not the same on each capacitor
PHYS272 - Spring 15 - von Doetinchem - 197
Capacitor network
PHYS272 - Spring 15 - von Doetinchem - 198
Energy storage in capacitors and electric-field energy
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Many important applications of capacitors rely on storing
energy
Electric potential energy stored in a charged capacitor is equal
to the amount of work to separate opposite charges
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Discharging of a capacitor: electric field does work
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Work required for charging a capacitor:
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Potential energy of uncharged capacitor set to zero
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Less work is required to transfer charge if capacitance is
higher
PHYS272 - Spring 15 - von Doetinchem - 203
Energy considerations
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Capacitance measures the ability to store energy
and charge
Charge a capacitor then disconnect battery/source
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No more charges can move onto the capacitor: charge is
constant
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When moving plates (i.e., changing capacitance) potential
difference changes
Change capacitance of a capacitor while leaving the
battery/source connected
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charges can move onto or off the capacitor: charge is
changing
–
Potential difference stays constant
PHYS272 - Spring 15 - von Doetinchem - 204
Electric field energy
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Charge a capacitor by moving electrons from one plate to the other: work
against the electric field between the plates
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Energy is stored in the field in the region between the plates
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Energy per unit volume (energy density):
Does not depend on geometry!
PHYS272 - Spring 15 - von Doetinchem - 205
Electric field energy
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Not only valid for parallel-plate capacitors
→ true for any electric field configuration in vacuum
Vacuum can have electric fields and is in this sense
not really empty
Two interpretations:
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energy is property of an electric field
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or a shared property of all charges creating the field
PHYS272 - Spring 15 - von Doetinchem - 206
Eletric field energy
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Magnitude of electric field to store 1J in a volume of
1m3
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Relation between electric field and energy density
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Dry air can sustain ~3MV/m without breaking down
PHYS272 - Spring 15 - von Doetinchem - 207
Dielectrics
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Most capacitors have non-conducting material
between them: dielectric
Helps separating two oppositely charged conductors
Prevents ionization of the air and allows for higher
potential build up before discharges occur → store
higher energies
PHYS272 - Spring 15 - von Doetinchem - 208
Dielectrics
http://www.youtube.com/watch?v=e0n6xLdwaT0
PHYS272 - Spring 15 - von Doetinchem - 209
Dielectrics
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constant voltage:
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separation between plates
widened:
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plates closer together:
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electrometer shows charge flowing off of the plates
electroscope shows no change in voltage
charge flows back onto the plates
fixed amount of charge onto left plate:
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separation is widened
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Plexiglas inserted between the plates:
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electroscope shows a rising voltage (charge stays constant)
voltage drops
When Plexiglas is removed
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voltage rises back up again → charge is still there
PHYS272 - Spring 15 - von Doetinchem - 210
Induced charge and polarization
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When dielectric is inserted and charge
is kept constant
→ potential difference drops
→ electric field drops
→ surface charge density drops, but
not the charge
Redistribution of charges in dielectric
occurs: polarization
For a given charge the potential
difference is reduced by a factor K:
K=Cdielectric/Cvacuum
K is always larger than unity
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Kair=1.0006
KWater=80.4
poor conductor, but good solvent
→ ions dissolve in water → current flow
PHYS272 - Spring 15 - von Doetinchem - 211
Induced charge and polarization
PHYS272 - Spring 15 - von Doetinchem - 212
Induced charge and polarization
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High capacitances can be reached with a dielectric
with high K value
PHYS272 - Spring 15 - von Doetinchem - 213
Capacitor with and without dielectric
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A capacitor is charged and disconnected from the
battery
After that a dielectric is moved in between the plates
PHYS272 - Spring 15 - von Doetinchem - 214
Capacitor with and without dielectric
0
PHYS272 - Spring 15 - von Doetinchem - 215
Capacitor with and without dielectric
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Electric potential energy stored in a capacitor is reduced when
inserting a dielectric into the capacitor
The fringing field at the edges of the capacitor exerts force on the
dielectric, effectively trying to pull in the dielectric
→ the work the electric field does is taken from the stored energy
PHYS272 - Spring 15 - von Doetinchem - 216
Additional material
PHYS272 - Spring 15 - von Doetinchem - 217
Z machine
http://www.youtube.com/watch?v=TVaIvAPMd_g
PHYS272 - Spring 15 - von Doetinchem - 218
Cylindrical capacitor
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Important property of parallel-plate, spherical, an cylindrical
capacitors: capacitance just depends on dimensions
PHYS272 - Spring 15 - von Doetinchem - 219
Energy stored in a capacitor
PHYS272 - Spring 15 - von Doetinchem - 220
Energy stored in a capacitor
PHYS272 - Spring 15 - von Doetinchem - 221