Physics 112 Walker, Chapter 20

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Physics 112Walker, Chapter 20

• Electrostatic Potential Energy
• Electrostatic Potential
• Prof. Charles E. Hyde-Wright
• Spring 2003

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Electric Potential and Electric Potential
Energy ? Symbol for electric potential is V ? We
will first define Electric Potential
Energy Symbol is U Scalar quantity (just a
magnitude, not a direction) Unit is Joule (J)
? Electric Potential Energy is an energy of a
charged object in an external electric
field. ? Electric Potential is the property of
the electric field itself, whether or not a
charged object has been placed in it.
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Potential Energy, Work, and Kinetic Energy

• Like the gravitational force, the electrostatic
  force is a
• conservative force.
• We can define an electrical potential energy U
  associated with the electrostatic force.
• Potential Energy is the potential to do
  Mechanical Work
• Work Change in Potential Energy W -DU
  -(Uf-Ui) Total Work change in kinetic energy
• -(Uf-Ui) W Kf-Ki

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Force, Energy, Dynamics

• F ma
• motion of the mass m as a function of time.
• Potential energy
• a charge travels from position ri to rf
• Velocity of a particle as a function of position

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Electrical Energy
The electrostatic force is a conservative
force. We can define an electrical potential
energy U (Joules) associated with the
electrostatic force. As a charge q moves
parallel to a constant electric field E, it
experiences a force FqE. The work done by the
electric field is, WFdqEd. (work is
negative if force F and displacement d are in
opposite directions) The change in the potential
energy is just the negative of the work done
by the field DU - W - qEd
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Change in electric potential energy Move the
particle opposite the direction of force
increase its potential energy
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Electric Potential
It is often convenient to consider not the
potential energy, but rather the potential
difference between two points. The potential
difference between points A and B, (VB -VA ),
is defined as the change in potential energy of
a charge q moved from A to B divided by the
charge Potential is a scalar, NOT a
vector.
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The electric potential, V, decreases as one
moves in the direction of the electric
field. In the case shown here, the
electric field is constant as a result, the
electric potential decreases
uniformly with distance. We have arbitrarily
set the potential equal to zero at the
right-hand plate.
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Units
The potential V is measured in units of
volts 1 V 1 J /C 1 Nm / C With this
definition of the volt, we can express the units
of the electric field as E1 N/C 1
V/m Note potential (V) ¹ potential energy (U)
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Electric Field, Electric Potential Energy, and
Work DU -W -Fd DV DU/q Ed
1 N/C1 V/m
(uniform field)
The zero of potential For calculating physical
quantities it is the difference in potential
which has significance, not the potential
itself. Therefore, we are free to choose as
having zero potential at any arbitrary point
which is convenient. Typical choices are the
earth infinity
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Relate the electric field, the separation of
two points, and the potential difference.
If the bottom plate is at ground (V0) then the
top plate is at potential V Ed. If we follow
the direction of the electric field lines, the
potential V is going down
Since E0 inside a conductor, then every point
inside the conductor is at the same potential V
(not necessarily zero)
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Energy Conservation
A consequence of the fact that electric force is
conservative is that the total energy of an
object is conserved (as long as nonconserative
like friction can be ignored) Expressing the
kinetic energy
Electric potential energy is
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Consider the figure and calculate (a) the
mass of the charge. (b) its kinetic energy at
point A and at point B.
The battery is like a pump, pumping a mass m up a
height h Giving the mass a potential energy
mgh Replace m ? q, g ? E, h ? d The battery
pumps a charge q up to the electrostatic
potential VEd of the top plate, giving the
charge an energy qEd
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Walker Problem 19, pg. 673
A particle with a mass of 3.5 g and charge of
0.045 ?C is realized from rest at point A. The
electric field strength is E1200 V/m. (a)
In which direction will this charge
move? (b) What speed will it have after
moving through a distance of 5.0 cm?
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Point Charges If we define the zero of
potential to be at infinity, then the potential
at a point A which is a distance r from a point
charge q is found to have a potential given
by q A r


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Electrostatic potential of point charge
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Many Charges and Superposition If we wish to
know the potential at a given point in space
which results from all surrounding charges, we
simply add up the potential from each charge
Note that because potential is a scalar,
the summation is less difficult than for the
vector field E.
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Potential and Work For any group of charges, we
can calculate the work necessary to bring the
charges together from infinity. This is the
potential energy associated with a two charge
system q2 Note
that we can also write this as r U q2 V1
q1 V2 q1


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Considering only electrostatic forces, in
general, the work required to move a charge q
from point A to point B is given by W -
q(VB - VA) From this we see that no work is done
to move a charge between two points which have
the same potential.
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Walker Problem 31, pg. 673

• Consider three charges shown in the figure.
• How much work must be done to move the
• 2.7 ?C charge to infinity?
• (b) How much work must be done to move the
• -6.1 ?C charge to infinity?

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The Electron Volt (eV)
It is often convenient to work with a unit of
energy called the electron volt. One electron
volt is defined as the amount of energy an
electron (with charge e) gains when accelerated
through a potential difference of 1 V 1 eV
(1.6 10-19 C)V 1.6 10-19 J
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The Electron Volt (1eV1.610-19 J)Practical Use

• A Battery is an electron pump.
• AA battery (1.5 V), each electron pumped through
  the battery from to - is given a potential
  energy of 1.5eV.
• The electric field in the Jefferson Lab
  accelerator is 5 MV/m.
• After traveling 1 km in this electric field, each
  electron in the beam gains an energy DK e
  (5106 V/m)(1000m) 5 GeV.
• This is ten thousand times larger than the rest
  mass energy (mc20.5 MeV ) of an electron

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Equipotential Surfaces

• A surface in space for which the potential is
  the same everywhere (such as the surface of a
  conductor) is called an equipotential surface.
• No work is required to move a charge on an
  equipotential surface,
• The electric field at every point on an
  equipotential surface is perpendicular to the
  surface.
• Equipotential surfaces are like contour lines on
  a togographic map,
• Electric field lines are lines of steepest
  descent (of the electrostatic potential V).

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Electric Field Lines and Equipotential Surfaces
Uniform electric field
Point Charge
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Electric Field Lines andEquipotential Surfaces
for two point charges
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Conductors
Recall that the boundary condition for just
outside a conductor is that the electric field
is perpendicular to the surface, and inside the
electric field is zero. If a charge moves along
the surface of a conductor no work is done
because the displacement is always perpendicular
to the field. All points both inside of a
conductor and on the surface are at the same
electric potential.
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Electric charges on the surface of ideal
conductors
Equal potential everywhere on or in
conductor. Charge on conductor is Concentrated at
sharp points Vkq / r charge on surface scales as
One over radius of curvature. Area scales as
radius squared Charge density charge/area
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Capacitance
A capacitor is device that stores the energy
associated with a configuration of charges. In
general, a capacitor consists of 2 conductors,
one with a charge Q and the other with a
charge Q (on the surfaces).
- - - -
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Electric Field, Potential, Charge in a Capacitor
d

• The two plates (area A, separation d) of a
  capacitor have charge Q and Q
• What is the electric field between the plates?
• Draw a Gaussian surface (area a) half in, half
  out of plate.
• Flux through surface is (aE)
• Charge enclosed by surface is Q a/A
• aE Qa/A/e0
• E Q/(A e0)
• Drawing charge Q out of the right plate and onto
  the left plate produces a potential difference V
  between the two plates.
• V Ed Q d/(A e0) Q / C
• C A e0 / d capacitance (units Farad C/V)

• -
• -
• -
• -
• -
• -
• -

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The capacitance C is defined as the ratio of
the magnitude of the charge on either conductor
to the magnitude of the potential difference
between the conductors Capacitance is purely
geometrical. For parallel plate C A e0 /d.
For any geometry, capacitance scales as
Area/distance. e.g. finger and doornob. The unit
of capacitance is the Farad (F) 1 F 1 C/V
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The Parallel-plate Capacitor A common type of
capacitor is the parallel-plate capacitor, made
up simply of two flat plates of area A separated
by a distance d. Its capacitance is given
by where ?0 is a constant called the
permittivity of free space.
?08.85?10-12 C2 / Nm2
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Charge, Voltage and Shocks

• If a spark jumps across a 1mm gap when you reach
  for a doorknob, what was the potential difference
  between you and the knob and how much charge was
  on you?
• Assume your finger and the knob form a parallel
  plate capacitor with area 1 cm2

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Walker Problem 48, pg. 675

• What plate area is required if an air-filled,
  parallel plate capacitor
• with a plate separation of 2.6 mm is to have a
  capacitance of 12 pF?
• (b) What is the maximum voltage that can be
  applied to this capacitor
• without causing dielectric breakdown (Emax
  3MV/m)?

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Dielectrics
A dielectric is an insulating material in which
the individual molecules polarize in proportion
to the the strength of an external electric
field. This reduces the electric field inside
the dielectric by a factor k, called the
dielectric constant.

Capacitance is increased by ?.
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Dielectric Strength

• Dielectrics are insulators charges are not free
  to move (beyond molecular distances)
• Above a critical electric field strength,
  however, the electrostatic forces polarizing the
  molecules are so strong that electrons are torn
  free and charge flows.
• This is called Dielectric Breakdown, and can
  disturb the mechanical structure of the material

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Dielectric Properties of common materials
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Energy Storage in a Capacitor

• Fig. 20-18. The voltage of a capacitor being
  charged is V Q / C

If the capacitor has charge q, and we add a small
charge dq, the electrostatic potential energy of
charge dq is U(dq)V (dq) q / C As we build up
the total charge Q, the average value the voltage
V is Q/(2C). The total potential energy of a
capacitor with charge Q is U Q2/(2C) CV2/2
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Energy Stored in a Capacitor Recall that work
is required to move charges about or charge the
capacitor. The work required to charge a
capacitor with a charge q to a voltage V
is So this must correspond to the energy
stored in the capacitor. Because QCV, this can
be rewritten
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Energy Density in Electric Field
Electric energy density Energy stored in
capacitor / Volume of gap
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Walker, Problem 56. p. 675
What electric field strength would store 10.0 J
of energy in every 1.00 mm3 cube of space?
Energy Density (10.0 J) / (1.00 10-3 m)3
(10.0 J) / (1.00 10-9 m 3)
1.00 10 10 J/m 3
Energy Density in electric field
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Walker Problem 35 p 674 (modified)
Consider the charges in the figure. Find the
electric potential energy of this system of
charges.

• 3
• 1 4

Vij potential at position i from charge j Uij
potential energy of interaction between
charges i and j Uij qi Vij qj Vji (no double
counting)
Utotal U12 U13 U14
U23 U24 U34
0
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Walker, Problem 35, pg. 674
How much external work is required to assemble
this configuration of charge? W Utotal
Add charges one at a time. Charge 1 requires no
work.
Adding charge 2 requires work U12 gt 0. Adding
charge 3 requires work U13U23 lt 0. Adding charge
4 requires work U14U24U34 gt 0. Total work 0
(in this case).
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Walker, Problem 35, pg. 674
Why is total energy 0?
Repulsive energy 24 on diagonal is canceled by
attractive energy 13 on diagonal
Repulsive energy 12 and 14 is canceled by
attractive energy 23 and 34