Physics 272
September 16
Fall 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Fall 14 - von Doetinchem - 163
Electric potential
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Electric potential energy:
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Electric force is conservative
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Work done by an electric force is represented by the
change in potential energy
Electric potential:
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Potential energy per unit charge
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Potential difference between two points equals the amount
of work to move a test charge between those points.
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Potential difference between two points is given by the line
integral along the electric field
Equipotential lines are lines of constant potential.
Electric field lines and equipotential lines are
perpendicular.
Phys272 - Fall 14 - von Doetinchem - 164
Potential gradient
invert integration limits
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Drop integral signs and just look at the infinitesimal change:
Electric field
component
along l
projection of electric
field along path
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Vector electric field can be calculated from derivatives of the scalar electric potential
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Potential gradient points towards the most rapid change in position.
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The shortest way to the next equipotential line is perpendicular to the old line:
→ electric field perpendicular to equipotential lines
Absolute value of potential is not important for electric field, only the local change.
Phys272 - Fall 14 - von Doetinchem - 166
Capacitance and dielectrics
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Capacitor stores electric potential and electric
charge
Capacitor: just insulate two conductors (with same
amount of negative and positive charge)
Work must be done to
move charges through
the resulting potential
→ stored electric
potential energy
Source: http://de.wikipedia.org/wiki/Kondensator_%28Elektrotechnik%29
Phys272 - Fall 14 - von Doetinchem - 176
Capacitance and dielectrics
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Applications:
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flashs (capacitor is very quickly discharged)
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Frequency dependent resistance in alternating current
circuits
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electronic devices (decoupling capacitors, filtering, etc.)
Capacitor has a certain capacitance depending on
its properties: size, shape, material
Capacitance increases when using an insulating
material between the negative and positive side
(polarization)
Electric field can be seen as a store-house of
electric potential energy
Phys272 - Fall 14 - von Doetinchem - 177
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
Electric field is proportional to the stored charge (the same is true for the
potential difference)
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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 - Fall 14 - von Doetinchem - 178
Calculating capactiance: capacitors in vacuum
Gauss' law:
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Nothing between oppositely charged conductors
condenser microphone:
capacitance changes due to flexible plate moved by sound waves → current flow
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 - Fall 14 - von Doetinchem - 179
Spherical capacitor
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Outer sphere makes no contribution to the field
between the sphere
Phys272 - Fall 14 - von Doetinchem - 180
Cylindrical capacitor
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Important property of parallel-plate, spherical, an cylindrical
capacitors: capacitance just depends on dimensions
Phys272 - Fall 14 - von Doetinchem - 181
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 - Fall 14 - von Doetinchem - 184
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 - Fall 14 - von Doetinchem - 185
Equivalent capacitance
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Make a drawing of the arrangement
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Identify groups of parallel and series connections
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Series connection: capacitors have same charge,
different potential difference
Parallel connection: same potential difference,
different charge
Phys272 - Fall 14 - von Doetinchem - 186
Capacitor network
Phys272 - Fall 14 - von Doetinchem - 187
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
Discharging of capacitor: electric field between does work
Phys272 - Fall 14 - von Doetinchem - 192
Energy storage in capacitors and electric-field energy
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Potential energy of uncharged capacitor set to zero
Capacitance measures the ability to store energy
and charge
Charging of capacitor: charge increases and energy
increases
Less work is required to transfer charge if
capacitance is higher
Phys272 - Fall 14 - von Doetinchem - 193
Z machine
http://www.youtube.com/watch?v=TVaIvAPMd_g
Phys272 - Fall 14 - von Doetinchem - 194
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 - Fall 14 - von Doetinchem - 195
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 - Fall 14 - von Doetinchem - 196
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 - Fall 14 - von Doetinchem - 197
Energy stored in a capacitor
Phys272 - Fall 14 - von Doetinchem - 198
Energy stored in a capacitor
Phys272 - Fall 14 - von Doetinchem - 199