W04D1 Electric Potential and Gauss

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W04D1 Electric Potential and Gauss
LawEquipotential Lines
Todays Reading Assignment Course Notes
Sections 3.3-3.4, 4.4-4.6. 4.10.5
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Announcements
Exam One Thursday Feb 28 730-930 pm Room
Assignments (See Stellar webpage
announcements) Review Tuesday Feb 26 from 9-11
pm in 26-152 PS 3 due Tuesday Tues Feb 26 at 9
pm in boxes outside 32-082 or 26-152
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Outline

• Continuous Charge Distributions
• Review E and V
• Deriving E from V
• Using Gausss Law to find V from E
• Equipotential Surfaces

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Continuous Charge Distributions
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Continuous Charge Distributions
Break distribution into infinitesimal charged
elements of charge dq. Electric Potential
difference between infinity and P due to dq.
Superposition Principle Reference Point
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Group Problem
Consider a uniformly charged ring with total
charge Q. Find the electric potential difference
between infinity and a point P along the
symmetric axis a distance z from the center of
the ring.
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Group Problem Charged Ring
Choose
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Electric Potential and Electric Field
Set of discrete charges Continuous
charges If you already know electric field
(Gauss Law) compute electric potential
difference using
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Using Gausss Law to findElectric Potential from
Electric Field
If the charge distribution has a lot of symmetry,
we use Gausss Law to calculate the electric
field and then calculate the electric potential V
using
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Group Problem Coaxial Cylinders
A very long thin uniformly charged cylindrical
shell (length h and radius a) carrying a
positive charge Q is surrounded by a thin
uniformly charged cylindrical shell (length h and
radius a ) with negative charge -Q , as shown in
the figure. You may ignore edge effects. Find
V(b) V(a).
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Worked Example Spherical Shells

• These two spherical shells have equal but
  opposite charge.
• Find
• for the regions
• b lt r
• a lt r lt b
• 0 lt r lt a
• Choose

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Electric Potential for Nested Shells
From Gausss Law
r
Use
Region 1 r gt b
No field ? No change in V!
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Electric Potential for Nested Shells
Region 2 a lt r lt b
r
Electric field is just a point charge. Electric
potential is DIFFERENT surroundings matter
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Electric Potential for Nested Shells
Region 3 r lt a
r
Again, potential is CONSTANT since E 0, but
the potential is NOT ZERO for r lt a.
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Group Problem Charge Slab
Infinite slab of thickness 2d, centered at x 0
with uniform charge density . Find
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Deriving E from V
A (x,y,z), B(x?x,y,z)
Ex Rate of change in V with y and z held
constant
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Deriving E from V
If we do all coordinates
Gradient (del) operator
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Concept Question E from V
Consider the point-like charged objects arranged
in the figure below. The electric potential
difference between the point P and infinity and
is
From that can you derive E(P)?

1. Yes, its kQ/a2 (up)
2. Yes, its kQ/a2 (down)
3. Yes in theory, but I dont know how to take a
   gradient
4. No, you cant get E(P) from V(P)

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Concept Question Answer E from V
4. No, you cant get E(P) from V(P)

• The electric field is the gradient (spatial
  derivative) of the potential. Knowing the
  potential at a single point tells you nothing
  about its derivative.
• People commonly make the mistake of trying to do
  this. Dont!

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Group Problem E from V

• Consider two point like charged objects with
  charge Q located at the origin and Q located at
  the point (0,a).
• Find the electric potential V(x,y)at the point P
  located at (x,y).
• Find the x-and y-components of the electric field
  at the point P using

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Concept Question E from V
The graph above shows a potential V as a function
of x. The magnitude of the electric field for x
gt 0 is

1. larger than that for x lt 0
2. smaller than that for x lt 0
3. equal to that for x lt 0

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Concept Question Answer E from V
Answer 2. The magnitude of the electric field
for x gt 0 is smaller than that for x lt 0

• The slope is smaller for x gt 0 than x lt 0
• Translation The hill is steeper on the left
  than on the right.

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Concept Question E from V
The above shows potential V(x). Which is true?

1. Ex gt 0 is positive and Ex lt 0 is positive
2. Ex gt 0 is positive and Ex lt 0 is negative
3. Ex gt 0 is negative and Ex lt 0 is negative
4. Ex gt 0 is negative and Ex lt 0 is positive

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Concept Question Answer E from V
Answer 2. Ex gt 0 is positive and Ex lt 0 is
negative

• E is the negative slope of the potential,
  positive on the right, negative on the left,
• Translation Downhill is to the left on the
  left and to the right on the right.

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Group Problem E from V
A potential V(x,y,z) is plotted above. It does
not depend on x or y. What is the electric field
everywhere? Are there charges anywhere? What
sign?
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Equipotentials
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Topographic Maps
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Equipotential Curves Two Dimensions
All points on equipotential curve are at same
potential. Each curve represented by V(x,y)
constant
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Direction of Electric Field E
E is perpendicular to all equipotentials
Constant E field
Point Charge
Electric dipole
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Direction of Electric Field E
E is perpendicular to all
equipotentials Field of 4 charges
Equipotentials of 4 charges
http//web.mit.edu/viz/EM/visualizations/electrost
atics/InteractingCharges/zoo/zoo.htm
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Properties of Equipotentials

• E field lines point from high to low potential
• E field lines perpendicular to equipotentials
• E field has no component along equipotential
• The electrostatic force does zero work to move a
  charged particle along equipotential