Download Planar Symmetry: Infinite Sheet of Charge

Planar Symmetry: Infinite Sheet of Charge
Non-Conducting Sheet
s+
+
+
+
+
+
+
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Example: Two Parallel Non-Conducting Sheets
s+
+
+
+
+
+
+
+
+
+
+
+
+
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s-
-
In this example, assume
s+ > s-
Find the electric field to the left of the sheets,
between the sheets and to the right of the sheets.
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Example: Gauss’ Law: Spherical Symmetry
A spherically symmetric charge distribution has a charge
density given by =a/r where a is a constant. Find the electric
field as a function of r.
 o  E  dA  qenc
2

0
0
 o Er 2  d  sin  d   o E 4 r 2  q
2
2
r
4

ar
 o E 4 r 2  q    4 r 2 dr  a 4  dr 
2
0
0 r
r
r
a
E
2 o
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Figure from: http://www.oceanopticsbook.info/view/light_and_radiometry/geometry 6
From Lecture #1: Conductors versus Insulators
• Insulators: material in which electric charges are
“frozen” in place.
• Conductors: material in which electric charges can
move around “freely.
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Conductors
If an excess charge is placed on an isolated conductor, that
amount of charge will move entirely to the surface of the
conductor. None of the excess charge will be found within
the body of the conductor.
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Conductors
Isolator Conductor with a Cavity
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Gauss’ Law: Spherical Symmetry
Demo: 5A-13 No
Internal Field
Electric Field inside and outside a shell
of uniform charge distribution
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Conductors: Shielding from an Electrical Field
• Electric field lines for an
oppositely charged metal
cylinder and metal plate.
Note that:
1. Electric field lines are
perpendicular to the
conductors.
2. There are no electric field
lines inside the cylinder.
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Conductors: Shielding from an Electrical Field
student
sensor
sparks
screened cage
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Van de Graaff
generator
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Conductors
DEMO: 5A-12 Charge within a Conductor
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Conductors
Charge Distribution on a Conductor
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http://cnx.org/content/m42317/latest/Figure_19_07_07a.jpg
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Example:
-100e
A ball of charge -50e lies at the
center of a hollow spherical
metal shell that has a net
charge -100e.
-50e
(1) What is the charge on the shell’s inner surface?
(a) -50e
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(b) 0
(c ) +50e
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Example:
-100e
A ball of charge -50e lies at the
center of a hollow spherical
metal shell that has a net
charge -100e.
-50e
(2) What is the charge on the shell’s outer surface?
(a) -150e
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(b) -50e
(c) +100e
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Question 1.
• A solid conducting sphere is concentric
with a thin conducting shell, as shown.
• The inner sphere carries a charge Q1, and
the spherical shell carries a net charge Q2,
such that Q2 = -3Q1.
How is the charge distributed on the sphere?
Q2
Q1
R1
R2
(A) There is no charge on the sphere.
(B) The charge is uniformly distributed on the outside
surface of the sphere.
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(C) The charge is uniformly distributed throughout the
sphere.
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Question 2.
• A solid conducting sphere is concentric
with a thin conducting shell, as shown.
• The inner sphere carries a charge Q1, and
the spherical shell carries a charge Q2,
such that Q2 = -3Q1.
How is the charge distributed on the
spherical shell?
Q2
Q1
R1
R2
(A) There is no charge on the shell.
(B) The charge is uniformly distributed on the outside
surface of the shell.
(C) The charge is uniformly distributed on the inner and
outer surfaces of the shell.
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Question 3.
• A solid conducting sphere is concentric
with a thin conducting shell, as shown.
• The inner sphere carries a charge Q1, and
the spherical shell carries a charge Q2,
such that Q2 = -3Q1.
Q2
Q1
R1
What is the electric field at r < R1?
A)
E 0
R2
1 Q1
E
B)
4 0 r12
C)
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1 -3Q
E
4 0 r 2
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Question 4.
• A solid conducting sphere is concentric
with a thin conducting shell, as shown.
• The inner sphere carries a charge Q1, and
the spherical shell carries a charge Q2,
such that Q2 = -3Q1.
Q2
Q1
R1
What is the electric field at R1<r < R2?
A)
E 0
R2
1 Q1
E
B)
4 0 r12
C)
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1 -3Q
E
4 0 r 2
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Question 5.
• A solid conducting sphere is concentric
with a thin conducting shell, as shown.
• The inner sphere carries a charge Q1, and
the spherical shell carries a charge Q2,
such that Q2 = -3Q1.
Q2
Q1
R1
What is the electric field at R2<r
A)
1 Q1
E
4 0 r12
B)
1 Q1 + Q2
E
4 0 r12
C)
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1 -3Q1
E
4 0 r 2
R2
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Question 6
• A solid conducting sphere is concentric
with a thin conducting shell, as shown.
• The inner sphere carries a charge Q1, and
the spherical shell carries a charge Q2,
such that Q2 = -3Q1.
Q2
Q1
R1
R2
What happens when you connect the two spheres with a
wire?
(A) The charge is uniformly distributed on the outside
surface of the shell.
(B) There is no charge on the sphere or the shell.
(C) The charge is uniformly distributed on the outer
surfaces of the sphere and the shell.
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Example
Consider the following two topologies:
A)
B)
A solid non-conducting sphere
carries a total charge Q = -3 C
distributed evenly throughout. It is
surrounded by an uncharged
conducting spherical shell.
s2
s1
-Q
E
Same as (A) but conducting shell removed
•Compare the electric field at point X in cases A and B:
(a) EA < EB
(b) EA = EB
(c) EA > EB
•Select a sphere passing through the point X as the Gaussian surface.
•How much charge does it enclose?
•Answer: -Q, whether or not the uncharged shell is present. (The
field at point X is determined only by the objects with NET
CHARGE.)
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Conductors: External Electric Field
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Two Parallel Conducting Sheets
Find the electric field to the left of the sheets,
between the sheets and to the right of the sheets.
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Uniform Charge Density: Summary
Non-conductor
Cylindrical
symmetry
Planar
Spherical
symmetry
r
E
2 0
R2 
E
2 0 r
s
E
2 0
1
Q
E
r
3
4 0 R
1
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Q
E
2
4 0 r
Conductor
E 0
inside

E
2 0 r
s
E
0
outside
inside
E 0
1
Q
E
2
4 0 r
outside
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Summary of Lectures 3, 4 & 5
*Relates net flux, F, of an electric field through a
closed surface to the net charge that is enclosed
by the surface.
 o F   o  E  dA  qenc
*Takes advantage of certain symmetries
(spherical, cylindrical, planar)
*Gauss’ Law proves that electric fields vanish in
conductor extra charges reside on surface
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