Physics 272
January 30
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Spring 14 - von Doetinchem - 140
Summary
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General form of Gauß's law:
The total electric flux through a closed surface
is equal to the total (net) electric charge inside
the surface, divided by ε 0.
Phys272 - Spring 14 - von Doetinchem - 141
Summary
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In electrostatics the electric field inside a solid
conductor is 0, excess charges only on surface
Charge in isolated cavity of neutral conductor:
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Excess charge in conductor can only be on the surface
Phys272 - Spring 14 - von Doetinchem - 142
Summary
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Potential energy decreases if a charged particle
moves in the direction of the electric field
If the displacement of a positive charge is in the
direction of the electric field the work is positive
Phys272 - Spring 14 - von Doetinchem - 143
Summary
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Electric potential energy of two point charges: q0 moves in field of q
Work only depends on radial displacement along the force
→ does not depend on exact path, can move without loss/increase perpendicular to force
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Summary
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total potential energy is the algebraic sum:
Phys272 - Spring 14 - von Doetinchem - 145
Interpreting electric potential energy
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Version A:
Potential energy difference equals the work done by
the electric force when the particle moves between
two points.
Version B:
Potential energy difference is the work that must be
done by an external force to move the particle
slowly (no kinetic energy) between two points
against the electric force.
Different signs, be careful.
Phys272 - Spring 14 - von Doetinchem - 146
Electric potential
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Describe potential energy on
a “per unit charge” basis
(like the electric field describes
force per unit charge)
Determination of electric field is
often easier by using the potential
Source: http://de.wikipedia.org/wiki/Alessandro_Volta
Alessandro Volta
1745-1825
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Potential energy and potential are scalars
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Potential difference in circuits is often called voltage
Phys272 - Spring 14 - von Doetinchem - 147
Calculating electric potential
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Potential of a continuous charge distribution:
Potential is zero at points that are infinitely far away
from all the charges creating the potential
Electric potential at a certain point is the
potential energy that would be associated with a
unit charge placed at that point.
Phys272 - Spring 14 - von Doetinchem - 148
Finding electric potential from electric field
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If electric field is known or can be found easily:
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Potential does not depend on th 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 14 - von Doetinchem - 149
Electron volts
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Electric field unit:
1V/m = 1 Volt/meter = 1N/C = 1 Newton/Coulomb
Useful: electron charge → electron volt
unit of energy
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For instance, very important in particle physics and
to describe processes in atoms
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Benefits of electric potential
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Potential calculation is sometimes easier than the electric
field calculation
We will come back to calculate the vector of the electric
field from the scalar potential
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Electric force and electric potential
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Electric force and electric potential
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Calculating electric potential
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Make a drawing
Charge distribution known: scalar sum or integral over
distribution
If we know the electric field as function of the position or can
determine it by, e.g., Gauss's law
–
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define potential to be zero at a convenient place,
Whenever possible try using an energy approach (scalar)
rather than a dynamics approach (vector)
Make a small graph of the result → does it make sense?
Phys272 - Spring 14 - von Doetinchem - 156
Charged conducting sphere
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Electric field outside a sphere is the same as for a point
charge
Electric field inside the conducting sphere is zero
→ no work is done if a test charge moves inside the sphere
→ potential inside the sphere is the same as on the surface
Phys272 - Spring 14 - von Doetinchem - 158
Van de Graaf generator
http://www.youtube.com/watch?v=sy05B32XTYY
Phys272 - Spring 14 - von Doetinchem - 159
Ionization and corona discharge
Source: http://en.wikipedia.org/wiki/Corona_discharge
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In air: charge cannot build up indefinitely
→ electric field can become so strong that neutral air transforms into ionized air
where charges can move
→ air becomes conducting
→ charges from the sphere go into the air
→ maximum limit for charge depends on radius
small wires or sharp objects allow easier discharge
Lightning bolt rod has a blunt end to allow substantial charge buildup
→ sharp ends would allow a smaller charge buildup
Phys272 - Spring 14 - von Doetinchem - 160
Potential of a line of charge
Phys272 - Spring 14 - von Doetinchem - 162
Equipotential surfaces
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Equipotential lines
are similar to field
lines
→ they help us to
visualize a potential
Similar to a
topographic map: line
corresponds to same
potential
Electric field lines
Equipotential lines
Charge can be moved
around this potential line
Source: http://de.wikipedia.org/wiki/%C3%84quipotentialfl%C3%A4che
without exerting electric force
→ force must be perpendicular to equipotential line
→ field lines are perpendicular to potential lines
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Lines are closer to each other for steeper gradients
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Equipotential lines cannot intersect
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Electric field is generally not constant over an equipotential line
Phys272 - Spring 14 - von Doetinchem - 163
Equipotentials and conductors
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Electric force is conservative → it is not possible to
do work on a test charge like that:
Electric field lines are perpendicular to surface of
conductor.
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Every potential difference inside a conductor is 0
→ surface must be at a constant potential
In the electrostatic case the surface of a
conductor is always an equipotential surface.
Phys272 - Spring 14 - von Doetinchem - 164
Charge on surface of cavity with no charge inside cavity?
Phys272 - Spring 14 - von Doetinchem - 166
Potential gradient
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Vector electric field can be calculated from 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 - Spring 14 - von Doetinchem - 168
Potential gradient
Phys272 - Spring 14 - von Doetinchem - 169
Potential and field of a point charge
unit vector
Phys272 - Spring 14 - von Doetinchem - 171
Review
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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.
The electric field can be calculated from the potential
gradient of the electric potential.
Phys272 - Spring 14 - von Doetinchem - 172
Discussion
1) It is easy to produce a potential difference of several thousand volts between your body and the
floor by scuffing your shoes across a nylon carpet. When you touch a metal doorknob, you get a
mild shock. Yet contact with a power line of comparable voltage would probably be fatal. Why is
there a difference?
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Potential is high on your body, but the amount of net charge is low
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The power line delivers much more electrons and the electrical energy is much higher.
2) A high-voltage DC power line falls on a car, so the entire metal body of the car is at a potential of
10,000V with respect to the ground. What happens to the occupants when they are sitting in the
car? What happens when they step out?
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Everything is on the same potential when the persons sit in the car → no potential
gradient
Stepping out while being in contact with the car is dangerous: many electrons will
move into the ground following a high potential gradient
3) When a thunderstorm is approaching, sailors at sea sometimes observe a phenomenon called St.
Elmo's fire, a bluish flickering light at the tips of the masts. Why is the effect pronounced when the
masts are wet?
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Static electricity can place a net charge on the ship (did you ever notice your hairs
standing up during an oncoming thunderstorm?)
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Light can appear as a discharge from the sharpest points like the top of the masts
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Seawater conducts well due to the salt and guides the charges to the discharge point
Phys272 - Spring 14 - von Doetinchem - 174
Capacitance and dielectrics
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Capactior 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
Applications: flashs, electronic devices
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 - Spring 14 - von Doetinchem - 175
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 - Spring 14 - von Doetinchem - 176
Calculating capactiance: capacitors in vacuum
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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 (µF, 10-6)
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radio tuning unit: 10-100 picofarad (pF, 10 -12)
Phys272 - Spring 14 - von Doetinchem - 177