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Lecture 6-1
Thin sheet of any charge distribution
tiny disk
 E  ( E ' E disk  )
 ( E ' E disk  )
 E disk   E disk 
    

 
n
 2 0  2 0  

 n
0
E L  E ' E disk 
Just to left of disk
E R  E ' E disk 
Just to right of disk

E n 
0
Lecture 6-2
Charges and fields of a conductor
• In electrostatic equilibrium, charges
inside a conductor do not move. Thus, E
= 0 everywhere in the interior of a
conductor.
• Since E = 0 inside, there are no net
charges anywhere in the interior. Net
charges can only be on the surface(s).
 
0
The electric field must be
perpendicular to the surface just
outside a conductor, since, otherwise,
there would be currents flowing along
the surface.
Lecture 6-3
Electrostatic Shielding (Continued)
If you move charge q in the cavity, the
exterior electric fields and the extreior
charge distribution are not affected.
q
Conducting shell electrostatically
shields its exterior from changes
on the inside.
Add Q’
Q’’
+ +
+
+
+
+
If you now add charge Q’ to the
conductor and/or Q’’ on the outside
of the conductor, the interior
electric fields do not change.
Conducting shell electrostatically
shields its interior from changes
on the outside, too.
Lecture 6-4
READING QUIZ 1
IN WHAT DIRECTION CAN YOU MOVE A TEST CHARGE RELATIVE
TO AN ELECTRIC FIELD SO THAT THE ELECTRIC POTENTIAL DOES
NOT CHANGE?
A| Move in the direction of the electric field lines.
B| Move opposite to the direction of the electric field lines.
C| Move from point A in the electric field to point B in
in the electric field along an arbitrary path.
D| Move relative to the electric field along a path which is
everywhere perpendicular to the electric field.
Lecture 6-5
Electric Potential Energy of a Charge in Electric Field
• Coulomb force is conservative
=> Work done by the Coulomb
force is path independent.
dl
• Can associate potential energy
U (r ) to charge q0 at any point r in
space.
It’s energy! A scalar measured in
J (Joules)

dW  q0 E  d l

dU  dW  q0 E  d l
Lecture 6-6
Electric Potential Energy of a Charge (continued)
U  U ( r )  U (i )

r
0



 i qq00EE ddl l
r

i
i is “the” reference point.
Choice of reference point
(or point of zero potential
energy) is arbitrary.
i is often chosen to be
infinitely far away (∞)
dl

dW  q0 E  d l

dU  dW  q0 E  d l
Lecture 6-7
Gravitational vs Electrostatic Potential Energy
U  U (b)  U (a )
a

bb

  FF d dl l
aa
b
mg
qE
Gravity
Coulomb
 mg l
 qE l
(if g, E uniform)
Work done by gravity or the
Coulomb force decreases the
potential energy.
Lecture 6-8
Potential Energy in the Field due to a Point Charge q
From ∞ U (r )  


dl
 qq0 EE  dl
0

PP
q0 q
   k 2 r dl
l
 P
q0 q
   k 2 dl

l
r
r
q0 q
 q0 q 
 k
k

r
 l 
This is also called the potential energy of
the two-charge configuration of q and q0.
What is the work required to
bring q0 in from infinity?
Lecture 6-9
Potential Energy of a Multiple-Charge Configuration
(a)
kq1q2 / d
(b)
q1q3
q2 q3
q1q2
k
k
k
d
d
2d
(c)
q1q3 q2 q4
q3q4
q1q2
k
k
k
k
d
d
d
d
q2 q3
q1q4
k
k
2d
2d
Lecture 6-10
Physics 241 –Warm-up quiz
Three point charges carry the same charge -q. Which of
the following statements is true? Select one of (a) – (e).
-q
-q
-q
A
B
A. An electron would have a higher potential energy
at point A than at point B
B. A proton would have a higher potential energy at
point A than at point B
C. An electron would have a lower potential energy
at point A than at point B
D. The potential energy is the same for an electron
and a proton at point A.
E. The potential energy is the same for a proton at
point A and point B.
Lecture 6-11
Electric Potential
• U(r) of a test charge q0 in electric field generated by
other source charges is proportional to q0 .
• So U(r)/q0 is independent of q0, allowing us to introduce
electric potential V independent of q0.
U ( r )
V ( r ) 
q0
U (r)
V (r) 
q0
• [Electric potential] = [energy]/[charge]
SI units:
J/C = V (volts)
1J
taking the same
reference point
Scalar!
Potential energy difference when 1 C of charge is
moved between points of potential difference 1 V
Lecture 6-12
Potential at P due to a point charge q
From ∞
V (r ) 
U q0 (r )
q0
q
k
r
Lecture 6-13
Electron Volt
• V=U/q is measured in volts => 1 V (volt) = 1 J / 1 C
J N m
V 
 E  m  V

POTENTIAL
C
CDIFFERENCES V2 – V1
N V
E  
C m
1 J  1C  1V
1 eV | e | 1V  1.602  1019 C  1V
(electron volt)
• V depends on an arbitrary choice of the reference point.
• V is independent of a test charge with which to measure it.
Lecture 6-14
Potential due to two (source) charges
q1  q2  0
q1
q2
V ( x)  k
k
|x|
|xa|
Lecture 6-15
Potential due to Multiple Source Charges: Example
V ( P) 
q1  q2  q3  q4
k
d/ 2
Dotted line is an equipotential when
q1=12nC, q2= -24nC, q3=31nC, q4=17nC
Lecture 6-16
E from V
We can obtain the electric field E from the potential V by
inverting the integral that computes V from E:

r


r
V (r )   E  d l   ( Ex dx  E y dy  Ez dz )
V
Ex  
x
V
Ey  
y
V
Ez  
z
Expressed as a vector, E is the negative gradient of V


E  V
Lecture 6-17
Lightning
E = 3 x 106 N/C
at electrical
breakdown of air
ΔV on the order
of 109 V
http://micro.magnet.fsu.edu/electromag/java/lightning/index.html
Lecture 6-18
DOCCAM 2
5A-16
PROOF PLANE
Examples (only a PREVIEW)
Q
Point charge Q: V ( r )  k
r
Uniformly charged
sphere:
Q
V
(
r
)

k
r>R
r
r<R
E (r )  k
Q
V
r


r
2
r
r
Q
E (r )  k 2 r
r
Q 
r2 
V (r ) 
3 2 
4 0 2 R 
R 
1
1 Qr
E (r ) 
r
3
4 0 R
V

z
z
(1 
)z
2
2
z
2 0
z R
Charged disk:
V ( z )  ??
Charged sheet:
V ( z )  ??
E ( z )  sgn( z ) 
V ( z )  ??

E (r ) 
r
2 0 r
Charged line:
E( z)  

z
2 0
Lecture 6-20
Physics 241 – 10:30 Quiz 3 September 8, 2011
A spherical shell is uniformly charged with a positive
charge density σ. Which of the following statements is
(are) true? Select one of (a) – (e).
1. An electron would have a higher potential energy
at point A than at point B
2. A proton would have a higher potential energy at
point A than at point B
3. The electric potential is lower at A than at B
4. The electric potential is higher at A than at B
a)
b)
c)
d)
e)
1 and 3 only
1 and 4 only
2 and 3 only
2 and 4 only
None of them
σ
A
B
Lecture 6-21
Physics 241 – 11:30 Quiz 3 September 8, 2011
A sphere is uniformly charged with a negative charge
density. Which of the following statements is (are) true?
Select one of (a) – (e).
1. A proton would have a higher potential energy at
point A than at point B
2. An electron would have a higher potential energy
at point A than at point B
3. The electric potential is lower at A than at B
4. The electric potential is higher at A than at B
a)
b)
c)
d)
e)
1 and 3 only
1 and 4 only
2 and 3 only
2 and 4 only
None of them
-σ
A
B
Lecture 6-22
Physics 241 – 10:30 Quiz 3 January 27, 2011
A sphere is uniformly charged with a negative charge
density. Which of the following statements is (are) true?
Select one of (a) – (e).
1. A proton would have a higher potential energy at
point A than at point B
2. An electron would have a higher potential energy
at point A than at point B
3. The electric potential is lower at A than at B
4. The electric potential is higher at A than at B
a)
b)
c)
d)
e)
1 and 3 only
1 and 4 only
2 and 3 only
2 and 4 only
None of them
+3σ
A
B
Lecture 6-23
Physics 241 – 11:30 Quiz 3 January 27, 2011
A sphere is uniformly charged with a positive surface
charge density. Which of the following statements is
(are) true? Select one of (a) – (e).
1. A proton would have a higher potential energy at
point A than at point B
2. An electron would have a higher potential energy
at point A than at point B
3. The electric potential is lower at A than at B
4. The electric potential is higher at A than at B
a)
b)
c)
d)
e)
1 and 3 only
1 and 4 only
2 and 3 only
2 and 4 only
None of them
+3σ
A
B