Download Lecture 06 - Purdue Physics

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2 0
R 
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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Lectures 6 & 7: Chapter 23
Electric Potential
Q V
40 R
Q
40 r
R
Definitions
R
C
r
Examples
R
B
r
B
q
r
A
A
Path independence
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Equipotential
surfaces
6
From Mechanics (PHYS 172)
• Energy
Kinetic Energy: associated with the state of motion
Potential Energy: associated with the configuration of
the system
• Conservative Forces:
Work done by a conservative force is independent of path
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From Mechanics (PHYS 172)
WTOT  DK
• Work
F
dr
W>0
Object speeds up ( DK > 0 )
W<0
Object slows down (DK < 0 )
F
dr
or
F
dr
F
dr
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W0
Constant speed (DK  0 )
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Electric Potential Energy
When an electrostatic force acts between two or
more charges within a system, we can assign an
Electric Potential Energy:
+
+
+
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+
F
Dx
If a Coulomb force does negative work
Potential energy increases
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Example: Electric Potential Energy
What is the change in electrical potential energy of a released electron in
the atmosphere when the electrostatic force from the near Earth’s
electric field (directed downward) causes the electron to move vertically
upwards through a distance d?
1. DU of the electron is related to the work done
on it by the electric field:
2. Work done by a constant force on a particle
undergoing displacement:
3. Electrostatic Force and Electric Field are related:
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Example: Electric Potential Energy
What is the change in electrical potential energy of a released electron in
the atmosphere when the electrostatic force from the near Earth’s
electric field (directed downward) causes the electron to move vertically
upwards through a distance d?
1. DU of the electron is related to the work done
on it by the electric field:
Key Idea:
DU  W
2. Work done by a constant force on a particle
undergoing displacement:
Key Idea:
E
Fe
d
W  F d
3. Electrostatic Force and Electric Field are
related:
Key Idea:
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F  qE
W  qE  d  qEd cos  qEd cos180  qEd
Electric potential decreases as electron rises.
DU  W  qEd
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Electric Potential versus Electrical Potential
Energy
Electric Potential is a property of an electric field and is
measured in J/C or V
Electric Potential Energy is an energy of system
consisting of the charged object and the external
electric field, and is measured in Joules.
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Potential & Electric Fields
The electric field points in the direction in which the
potential decreases most rapidly.
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Example: Potential Difference
*independent of path
(a) What is DV moving directly from point i to
point f?
i
c
f
(b) What is DV moving from point i to point c to point f?
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Question:
• A single charge ( Q = -1C) is fixed at the
origin. Define point A at x = + 5m and point B
at x = +2m.
– What is the sign of the potential difference
between A and B?
– Where VBA = VB-VA
(a) VBA < 0
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(b) VBA = 0
-Q
B
´
-1C
A
´
x
(c) VBA > 0
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Potential due to a point charge:
Find V in space around a charged particle relative to
the zero potential at infinity:
+
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V(r) versus r for a positive charge at r = 0
V(r) to 
kq
r
r
For a point charge
V(r  0)  
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Electrical Potential Energy
Push q0 “uphill” and its electrical potential
energy increases according to
kq0 q
U
r
The work required to move q0
initially at rest at  is
kq0 q
W
.
r
Work per unit charge is
kq
V .
r
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Demos:
+
+
+
R
+
+
+
+
+
+
R +
+
+
+
+
+
+
+
+
+
V(r)
+
kQ
R
kQ
V (R) 
R
Gauss’ law says the sphere looks like a point
charge outside R.
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Demo
DU  q DV
charge flow
+
kQ
V
R
+
+
R
+
+
+
+
+
+
+
+
Also try an
elongated neon bulb.
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Get energy out
r1
r2
DV  V (r1 )  V (r2 )
across fluorescent
light bulb
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Potential due to a Group of Point Charges
Find the Potential at the center of the square.
N
qn
V(r)  Vn (r) 

4 o n1 rn
n1
d = 1.3 m
q1 = +12 nC
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q2 = -24 nC
+
-
+
+
q3 =+31 nC
1
N
q3 =+17 nC
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Electrical Potential Energy of a System of Point
Charges
U of a system of fixed point charges equals W done by an
external agent to assemble the system by bringing each
charge in from infinity.
+
+
If q1 & q2 have the same sign, we must do positive work to push against
mutual repulsion.
If q1 & q2 have opposite signs, we must do negative work against mutual
attraction
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Calculating the Electric Field from the Potential
Field
 V ˆ V ˆ V ˆ
E  V   
i+
j
k
y z 
x
 E points in the direction in which
the potential decreases most rapidly.
From this we see that
V
V
V
Ex  
, Ey  
, and Ez  
.
x
y
z
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Potential Energy of an Electric Dipole
Potential energy can be associated with the orientation of
an electric dipole in an electric field.
  p  E     pE sin 
0
0
90
90
U  W     d   pE sin  d
U   pE cos + U 0
Choose U 0 (  900 )  0  U   pE cos   p  E
U is least =0 U=-pE
U is greatest =180 U=pE
U =0 when =90