Equipotentials and Energy

1
Equipotentials and Energy
Electric Potential
2
Last time Potential energy and potential

• Potential energy stored in a static charge
  distribution
• work we do to assemble the charges
• Electric potential energy of a charge in the
  presence of a set of source charges
• potential energy of the test charge equals the
  potential from the sources times the test charge
  U qV
• Electrostatic potential at any point in space
• sources give rise to V(r)

allows us to calculate the potential V
everywhere if we know the electric field E
(define VA 0 somewhere)
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Conservation of Energy

• The Coulomb force is a CONSERVATIVE force (i.e.,
  the work done by it on a particle which moves
  around a closed path returning to its initial
  position is ZERO.)

• The total energy (kinetic electric potential)
  is then conserved for a charged particle moving
  under the influence of the Coulomb force.

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Lecture 6, ACT 2
Two test charges are brought separately to the
vicinity of a positive charge Q.
2A
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Lecture 6, ACT 2

• Two test charges are brought separately to the
  vicinity of positive charge Q.
• charge q is brought to pt A, a distance r from
  Q.
• charge 2q is brought to pt B, a distance 2r from
  Q.
• Compare the potential energy of q (UA) to that of
  2q (UB)

2A

• The potential energy of q is proportional to
  Qq/r.
• The potential energy of 2q is proportional to
  Q(2q)/(2r).
• Therefore, the potential energies UA and UB are
  EQUAL!!!

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Lecture 6, ACT 2
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Today

• Calculating Electric Potentials
• Charge Spherical Shell
• Example electric potential of a charged sphere
• Equipotentials and conductors
• Electrical Breakdown
• Appendices
• A. Electric potential calculation examples
• B. Method of images to determine charge on
  conductors

8
Potential from charged spherical shell (See
Appendix A for a more complex example)

• Potential
• r gt a

• r lt a

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What does the result mean?

• This is the plot of the radial component of the
  electric field of a charged spherical shell

Er
Notice that inside the shell, the electric field
is zero. Outside the shell, the electric field
falls off as 1/r2. The potential for r gt a is
given by the integral of Er. This integral is
simply the area underneath the Er curve.
R
a
r
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Lecture 6, ACT 1
1

• A point charge Q is fixed at the center of an
  uncharged conducting spherical shell of inner
  radius a and outer radius b.
• What is the value of the potential Va at the
  inner surface of the spherical shell?

(b)
(a)
(c)
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Lecture 6, ACT 1
Eout

• A point charge Q is fixed at the center of an
  uncharged conducting spherical shell of inner
  radius a and outer radius b.
• What is the value of the potential Va at the
  inner surface of the spherical shell?

1
a
Q
b
(b)
(a)
(c)
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Preflight 6
Two spherical conductors are separated by a large
distance. They each carry the same positive
charge Q. Conductor A has a larger radius than
conductor B.
A
B
2) Compare the potential at the surface of
conductor A with the potential at the surface of
conductor B.
a) VA gt VB b) VA VB
c) VA lt VB
13
Potential from a charged sphere
Last time
Er
(where )

• The electric field of the charged sphere has
  spherical symmetry.
• The potential depends only on the distance from
  the center of the sphere, as is expected from
  spherical symmetry.
• Therefore, the potential is constant along a
  sphere which is concentric with the point
  charge. These surfaces are called
  equipotentials.
• Notice that the electric field is perpendicular
  to the equipotential surface at all points.

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Equipotentials

• Defined as The locus of points with the same
  potential.
• Example for a point charge, the equipotentials
  are spheres centered on the charge.

The electric field is always perpendicular to an
equipotential surface!
Why??
Along the surface, there is NO change in V (its
an equipotential!) Therefore, We can conclude
then, that is zero. If the dot product
of the field vector and the displacement vector
is zero, then these two vectors are
perpendicular, or the electric field is always
perpendicular to the equipotential surface.
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Electric Dipole Equipotentials
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Electric Fish

• Some fish have the ability to produce detect
  electric fields
• Navigation, object detection, communication with
  other electric fish
• Strongly electric fish (eels) can stun their
  prey

Black ghost knife fish
Dipole-like equipotentials More info Prof. Mark
Nelson, Beckman Institute, UIUC
-Electric current flows down the voltage
gradient -An object brought close to the fish
alters the pattern of current flow
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Conductors

• Claim
• The surface of a conductor is always an
  equipotential surface (in fact, the entire
  conductor is an equipotential).
• Why??
• If surface were not equipotential, there would
  be an electric field component parallel to the
  surface and the charges would move!!

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Preflight 6
3) The two conductors are now connected by a
wire. How do the potentials at the conductor
surfaces compare now ?
a) VA gt VB b) VA VB c) VA
lt VB
4) What happens to the charge on conductor A
after it is connected to conductor B ?
a) QA increases b) QA decreases c) QA doesnt
change
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Sparks

• High electric fields can ionize nonconducting
  materials (dielectrics)
• Breakdown can occur when the field is greater
  than the dielectric strength of the material.
• E.g., in air,

What is ?V?
Ex.
Note High humidity can also bleed the charge off
? reduce ?V.
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What does grounding do?1. Acts as an infinite
source or sink of charge.2. The charges
arrange themselves in such a way as to minimize
the global energy (e.g., E?0 at infinity, V?0 at
infinity).3. Typically we assign V 0 to
ground.
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Followup Question
Two charged balls are each at the same potential
V. Ball 2 is twice as large as ball 1.

• As V is increased, which ball will induce
  breakdown first?

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Energy Units
MKS U QV 1 coul-volt 1 joule
for elementary particles (e, p, ...) 1
eV 1.6x10-19 joules
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Charge on Conductors?

• How is charge distributed on the surface of a
  conductor?
• KEY Must produce E0 inside the conductor and E
  normal to the surface .
• Spherical example (with little
  off-center charge)

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Lecture 6, ACT 3
An uncharged spherical conductor has a weirdly
shaped cavity carved out of it. Inside the
cavity is a charge -q. How much charge is on the
cavity wall?
3A
(a) Less thanlt q (b) Exactly q
(c) More than q
3B
How is the charge distributed on the cavity wall?
(a) Uniformly (b) More charge closer to q (c)
Less charge closer to -q
3C
How is the charge distributed on the outside of
the sphere?
(a) Uniformly (b) More charge near the cavity (c)
Less charge near the cavity
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Lecture 6, ACT 3
An uncharged spherical conductor has a weirdly
shaped cavity carved out of it. Inside the
cavity is a charge -q. How much charge is on the
cavity wall?
3A
(a) Less thanlt q (b) Exactly q
(c) More than q
By Gauss Law, since E0 inside the conductor,
the total charge on the inner wall must be q
(and therefore -q must be on the outside surface
of the conductor, since it has no net charge).
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Lecture 6, ACT 3
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Lecture 6, ACT 3
3C
How is the charge distributed on the outside of
the sphere?
(a) Uniformly (b) More charge near the
cavity (c) Less charge near the cavity
As in the previous example, the charge will be
uniformly distributed (because the outer surface
is symmetric). Outside the conductor the E field
always points directly to the center of the
sphere, regardless of the cavity or
charge. Note this is why your radio, cell
phone, etc. wont work inside a metal building!
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Conductors versus InsulatorsCharges move
to Charges cannot cancel electric field move
at all in the conductor E0 ?
equipotential Charge distribution surface on
insulator unaffected by external
fieldsAll charge on surface Charge can sit
inside(Appendix B describes method of
images to find the surface charge distribution
on a conductor only for your reading pleasure!)
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Charge on Conductor Demo

• How is the charge distributed on a non-spherical
  conductor?? Claim largest charge density at
  smallest radius of curvature.
• 2 spheres, connected by a wire, far apart
• Both at same potential

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Equipotential Example

• Field lines more closely spaced near end with
  most curvature higher E-field
• Field lines to surface near the surface (since
  surface is equipotential).
• Near the surface, equipotentials have similar
  shape as surface.
• Equipotentials will look more circular
  (spherical) at large r.

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Summary

• If we know the electric field everywhere, we can
  calculate the potential, e.g.,
• The place where V0 is arbitrary (often at
  infinity)
• Physically, DV is what counts
• Equipotential surfaces are surfaces where the
  potential is constant
• Conductors are equipotentials
• Breakdown can occur if the electric field
  exceeds the dielectric strength
• Next time ? capacitors

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Appendix A Electrical potential examples
Calculate the potential V(r) at the point shown
(rlta)
uncharged conductor
c
b
a
solid sphere with total charge Q
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Calculating Electric Potentials
Calculate the potential V(r) at the point shown
(r lt a)

• Where do we know the potential, and where do we
  need to know it?

V0 at r ? ...
we need r lt a ...

• Determine E(r) for all regions in between these
  two points

• Determine DV for each region by integration

... and so on ...

• Check the sign of each potential difference DV

(from the point of viewof a positive charge)
DV gt 0 means we went uphillDV lt 0 means we
went downhill
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Calculating Electric Potentials
Calculate the potential V(r) at the point shown
(r lt a)

• Look at first term

• Line integral from infinity to c has to be
  positive, pushing against a force

Line integral is going in which is just the
opposite of what usually is done - controlled by
limits

• Whats left?

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Calculating Electric Potentials
Calculate the potential V(r) at the point shown
(r lt a)

• Look at third term

c
b
a

• Line integral from b to a, again has to be
  positive, pushing against a force

Line integral is going in which is just the
opposite of what usually is done - controlled by
limits

• Whats left?

Previous slide we have calculated this already
36
Calculating Electric Potentials
Calculate the potential V(r) at the point shown
(r lt a)

• Look at last term

• Line integral from a to r, again has to be
  positive, pushing against a force.
• But this time the force doesnt vary the same
  way, since r determines the amount of source
  charge

This is the charge that is inside r and sources
field

• Whats left to do?
• ADD THEM ALL UP!
• Sum the potentials

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Calculating Electric Potentials
Calculate the potential V(r) at the point shown
(r lt a)

• Add up the terms from I, III and IV

I
III
IV
An adjustment to account for the fact that the
conductor is an equipotential, DV 0 from c ? b
The potential difference from infinity to a if
the conducting shell werent there
38
Calculating Electric Potentials
Summary
The potential as a function of r for all 4
regions is
I r gt c II b lt r lt c
III a lt r lt b IV r lt a
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Lets try some numbers
Q 6m C a 5cm b 8cm c 10cm
I r gt c V(r 12cm) 449.5 kV
II b lt r lt c V(r 9cm) 539.4 kV
III a lt r lt b V(r 7cm) 635.7 kV
IV r lt a V(r 3cm) 961.2 kV
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Appendix B FYI Induced charge distribution on
conductor via method of images

• Consider a source charge brought close to a
  conductor

• Charge distribution induced on conductor by
  source charge

• Induced charge distribution is real and sources
  E-field so that the total is zero inside
  conductor!
• resulting E-field is sum of field from source
  charge and induced charge distribution
• E-field is locally perpendicular to surface

• With enough symmetry, can solve for s on
  conductor
• how? Gauss Law

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Appendix B (FYI) Induced charge distribution on
conductor via method of images

• Consider a source charge brought close to a
  planar conductor

• Charge distribution induced on conductor by
  source charge
• conductor is equipotential
• E-field is normal to surface
• this is just like a dipole

-

• Method of Images for a charge (distribution) near
  a flat conducting plane
• reflect the point charge through the surface and
  put a charge of opposite sign there
• do this for all source charges
• E-field at plane of symmetry - the conductor
  surface determines s.