Transcript workshops:colombia10:colombia10integrals

Vector Integrals
and Electrostatics
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Corinne Manogue
Tevian Dray
7 October 2010
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Curvilinear Coordinates
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Curvilinear Basis Vectors
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Curvilinear Basis Vectors
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The Vector Differential
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The Vector Differential
dr  dx xˆ  dy yˆ  dz zˆ
dr  dr rˆ  rd ˆ  dz zˆ
dr  dr rˆ  rd ˆ  rd  ˆ
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The Ring
• Consider a very thin ring of charge with
radius R and total charge Q.
• Find the electrostatic potential due to this
ring everywhere in space.
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Potential Due to a Ring of Charge
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The Ring
• Consider a very thin ring of charge with
radius R and total charge Q.
• Find the electric field due to this ring
everywhere in space.
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Steady Current
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The Spinning Ring
• Consider a very thin ring of charge with
radius R and total charge Q. The ring is
rotating about its axis with period T.
• Find the vector potential due to this ring
everywhere in space.
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An Example
• Typical of EARLY upper-division work for
physics majors and many engineers.
• Solution requires:
– many mathematical strategies,
– many geometrical and visualization strategies,
– only one physics concept.
• Demonstrates different use of language.
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Potential Due to Charged Disk
z
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What is the electrostatic
potential at a point, on
axis, above a uniformly
charged disk?
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One Physics Concept
• Coulomb’s Law:
1 q
V
4 0 r
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Superposition
• Superposition for solutions of linear
differential equations:
1 q
V
4 0 r
 V (r ) 
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1
4 0

 ( r )da 
r  r
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Chopping and Adding
z
r 2  z 2
Integrals involve
chopping up a part of
space and adding up a
physical quantity on
each piece.
r
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Computational Skill
• Can the students set-up and do the integral?
V (r ) 
1
4 0


4 0
2

4 0
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
  r  da 
r  r
2 R

0 0

dr  r  d 
r 2  z 2
R2  z2  z

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Constants vs. Variables
• Which of these symbols are constants and
which are variables?

V ( r,  , z ) 
4 0
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2

0
R d 
2

r  R  2rR cos(   )  z
2
2
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Limits (Far Away)
2
V (r ) 
4 0

R2  z2  z


2 
R2

 z 1 2  z 

4 0 
z

2   1 R 2


 z 1 
2
4 0   2 z
 
 z
 
1  R 2

4 0 z
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The Spinning Ring - Limits
• Approximate this vector potential near the
center of the ring, in the plane of the ring.
• Approximate this vector potential near the
center of the ring, along the z-axis.
• Approximate this vector potential far from
the ring, in the plane of the ring.
• Approximate this vector potential far from
the ring, along the z-axis.
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Steady Current
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