Electric Potential (III)

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Electric Potential (III)

• - Fields
• Potential
• Conductors

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Potential and Continuous Charge Distributions

• We can use two completely different methods
• Or, Find from Gausss Law, then

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Potential and Electric Field

• Since

therefore, we have (in Cartesian
coords)
Hence
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• Ex 1a Given V3x2yy2yz, find E.

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• Ex 1b Given V(10/r2)sin?cos? (spherical coords)
• a) find E.
• b) find the work done in moving a 10µC charge
  from A(1, 30o, 120o) to B(4, 90o, 60o).

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• There are many coordinate systems that can be
  used
• Bipolarcylindrical, bispherical, cardiodal,
  cardiodcylindrical, Cartesian, casscylindrical,
  confocalellip, confocalparab, conical,
  cylindrical, ellcylindrical, ellipsoidal,
  hypercylindrical, invcasscylindrical,
  invellcylindrical, invoblspheroidal,
  invprospheroidal, logcoshcylindrical,
  logcylindrical, maxwellcylindrical,
  oblatespheroidal, paraboloidal, paracylindrical,
  prolatespheroidal, rosecylindrical, sixsphere,
  spherical, tangentcylindrical, tangentsphere, and
  toroidal.

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• Ex 2 Find the potential of a finite line charge
  at P,
• AND the y-component of the electric
  field at P.

P
r
d
dq
x
L
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Solution
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Example The Electric Potential of a Dipole
y
a
a
x
P
q
-q
Find a) Potential V at point P along the
x-axis. b) What if xgtgta ?
c) Find E.
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Solution
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Example Find the potential of a uniformly
charged sphere of radius R,
inside and out.
R
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Uniformly Charged Sphere,radius R
E
r
R
V
r
R
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Example Recall that the electric field inside a
solid conducting sphere with charge Q on its
surface is zero. Outside the sphere the field is
the same as the field of a point charge Q (at
the center of the sphere). The point charge is
the same as the total charge on the sphere.
Find the potential inside and outside the sphere.
Q
R
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• Solution (solid conducting)
• Inside (rltR), E0, integral of zero constant,
  so Vconst
• Outside (rgtR), E is that of a point charge,
  integral gives
• VkQ/r

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Solid Conducting Sphere,radius R
E
r
R
V
r
R
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Quiz
A charge Q is placed on a spherical conducting
shell. What is the potential (relative to
infinity) at the centre?
Q

1. keQ/R1
2. keQ/R2
3. keQ/ (R1 - R2)
4. zero

R1
R2
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Calculating V from Sources

1. Point source

(note V?0 as r? )
or
ii) Several point sources
(Scalar)
iii) Continuous distribution
OR I. Find from Gausss Law (if
possible) II. Integrate,
(a line integral)