workshops:colombia10:colombia10integrals
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Transcript workshops:colombia10:colombia10integrals
Vector Integrals
and Electrostatics
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Corinne Manogue
Tevian Dray
7 October 2010
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7 October 2010
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Curvilinear Coordinates
7 October 2010
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Curvilinear Basis Vectors
7 October 2010
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Curvilinear Basis Vectors
7 October 2010
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The Vector Differential
7 October 2010
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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
7 October 2010
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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.
7 October 2010
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Potential Due to a Ring of Charge
7 October 2010
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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.
7 October 2010
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Steady Current
7 October 2010
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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.
7 October 2010
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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.
7 October 2010
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Potential Due to Charged Disk
z
7 October 2010
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
7 October 2010
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Superposition
• Superposition for solutions of linear
differential equations:
1 q
V
4 0 r
V (r )
7 October 2010
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
7 October 2010
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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
7 October 2010
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
7 October 2010
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
7 October 2010
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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.
7 October 2010
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Steady Current
7 October 2010
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