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Relation between field and potential
E = -V
2
∫E.dl = -(V2-V1) along any path
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• Set reference point 1 at infinity where V1 = 0
• Use linear path from infinity to desired point 2, coordinate r
r
V(r) = -∫E.dl
∞
1
Equation satisfied by potential
 xE=0
 . E = r/e0
E = -V
2V = - r/e0
Poisson’s equation
Laplace’s equation
Free space
2V = 0
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Continuous distribution of charges
E=
rv(R’) R’
 ________
 4pe0|R’|3
d3 R ’
R‘
rv(R’)
V = ________
d3 R ’
 4pe0|R’|
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Various examples
Charged Disc
V=?
Charged line
Charged
Sheet
Charged
Ball
Charged
Hollow
Ball
Calculate V directly, or from E obtained through Gauss’ Law
What do equipotentials and field lines look like?
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Equipotentials
Familiar examples
Equipotentials
Connect pts. with same V
E = -V runs perpendicular to it
Point charge
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How to draw equipotentials?
Point Dipole
R >> d
p. R
V = _________
4pe0R2
Note 1/R2 !
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Point Dipole
p. R
V = _________
R >> d
4pe0R2
Note 1/R2 !


E = -V = -RV/R – (q/R)V/q


p(2Rcosq + qsinq)
_____________
4pe0R3
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So dipoles annihilate each other, thereby countering
the field that separated and created them in the first
place. In other words, they conspire to produce a
polarizing field.
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Electric Field Applet
http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html9/ 30/ 12 10:37
Electric Field Applet
Done.
1
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Potential Gradient
Equipotential Lines
Electric Field Lines
Initial Velocity
Wall Boundary
System Charge: - 13.0
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Electric Field Applet
http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html9 / 30/ 12 10:37 P
Electric Field Applet
Done.
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Ref resh
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Potential Gradient
Equipotential Lines
Electric Field Lines
Initial Velocity
Wall Boundary
System Charge: - 13.0
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Conductors are equipotentials
• Conductor  Static Field inside zero (perfect screening)
• Since field is zero, potential is constant all over
• E is perpendicular to the conducting surface
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Images
Charge above
Ground plane
(fields perp. to surface)
Compare with
field of a Dipole!
Equipotential on metal enforced by the image
So can model as
Charge
+
Image
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Field lines near a conductor
+
+
+
--+ -- -- +
+ + +
Equipotentials bunch up here
 Dense field lines
Principle of operation
of a lightning conductor
Plot potential, field lines
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How much charge can we store on a metal?
We can calculate the voltage on a metal
for a given charge. Conversely, we can calculate
the charge we need to store to create a given
voltage on a metal.
How would we quantify the charge that is needed
to create 1 volt on a metal?
The ‘Capacitance’
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Capacitance
Capacity to store charge
C = Q/V
a
b
L
Q = Ll
E = l/2pe0r
V = -(l/2pe0)ln(r/a)
C = 2pe0L/ln(b/a)
Dimension e0 x L
(F/m) x m
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Capacitance
Capacity to store charge
C = Q/V
A
d
Q = Ars
E = rs/e0
V = Ed
C = e0A/d
(F/m) x m
Increasing area increases Q and decreases C
Increasing separation increases V and decreases Q
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Capacitance
Capacitor microphone – sound vibrations move a
diaphragm relative to a fixed plate and change C
Tuning  rotate two cylinders and vary degree of
overlap with dielectric  change C
Changing C changes resonant frequency of RL circuit
Increasing area increases Q and decreases C
Increasing separation increases V and decreases Q
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Increasing C
with a dielectric
+ +
+
- -
+
-
+
+
-
e/e0 = er
C  erC
+
-
+
+
-
bartleby.com
To understand this, we need to see how dipoles operate
They tend to reduce voltage for a given Q
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Dipoles Screen field
+
+
- -P +
- (opposing +
-polarization+
- Field) +
+
+
E=(D-P)/e0
D
(unscreened
Field)
Thus the unscreened external field D gets reduced
to a screened E=D/e by the polarizing charges
For every free charge creating the D field from a
distance, a fraction (1-1/er) bound charges screen D to E=D/e
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Effect on Maxwell equations:
Reduction of E
Point charge
in free space
.E = rv/e0
Point charge
in a medium
.E = rv/e0er
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