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Fields, forces, work, and potential
+
+
Electric forces and work
Electric potential energy
and electric potential
Tues. Oct. 7, 2008
Physics 208 Lecture 11
1
Potential from electric field
dV  E  d

V  Vo

d
d
E

d
V  Vo  E d

V=Vo
 V  Vo  E d

dV largest in direction of E-field.
dV smallest (zero)
 perpendicular to E-field
Tues. Oct. 7, 2008
Physics 208 Lecture 11
2
Equipotential lines


Lines of constant potential
In 3D, surfaces of constant potential
Tues. Oct. 7, 2008
Physics 208 Lecture 11
3
Topographic map

Each lines is
constant elevation
Same as constant
gravitational potential
gh (energy = mgh)

Height interval between lines constant

Tues. Oct. 7, 2008
Physics 208 Lecture 11
4
Electric field from potential

Said before that dW  Fext  ds  FCoulomb  ds 
dV  E  ds


Spell out the vectors:
 for
This works
dV  E x dx  E y dy  E z dz
dV
dV
dV
Ex  
, Ey  
, Ez  
dx
dy
dz

Usually written

Tues. Oct. 7, 2008
dV dV dV 
E  V   ,
,

 dx dy dz 
Physics 208 Lecture 11
5
Quick Quiz
Suppose the electric potential is constant
everywhere. What is the electric field?
A) Positive
B) Negative
C) Increasing
D) Decreasing
E) Zero
Tues. Oct. 7, 2008
Physics 208 Lecture 11
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Electric Potential - Uniform Field
dV  E  ds
 VB  VA 

B
B
B
 E  ds   Exˆ  ds
A
   Edx  E
A
B
A
 dx  E x
B
 xA 
A
B

A

Constant E-field corresponds to linearly
decreasing (in direction of E) potential
Here V depends only on x, not on y
x
Tues. Oct. 7, 2008
Physics 208 Lecture 11
7
Check of basic cases

Previous quick quiz: uniform potential
corresponds to zero electric field
E  V  constant   0

Linear potential corresponds to constant
electric field
 



E  V  Ex    Ex, Ex, Ex  Exˆ
y
z 
x
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Physics 208 Lecture 11
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Potential ( V ) of spherical conductor




What is V of spherical conductor relative to infinity?
Charge on surface  spherical charge shell
Gauss’ law  E = keQ / r2 in the radial direction
V is work / Coulomb to bring point charge from ∞
V R  V  
E
R
 E  ds




R
ds

Q
E dr   k 2 dr
r
R


Q
Q
 k
k
rR
R
Tues. Oct. 7, 2008
E  ds  E dr  Edr


Physics 208 Lecture 11
9
Quick quiz
Previous result says conducting sphere of radius R carrying
charge Q is at a potential kQ/ R
Two conducting spheres of diff radii connected by long
conducting wire. What is approximately true of Q1, Q2?

R1
Q1
Q2
R2
A) Q2>Q1
B) Q2<Q1
C) Q2=Q1
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Physics 208 Lecture 11
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Connected spheres

Since both must be at the same potential,
kQ1 kQ2
R2

 Q2  Q1
R1
R2
R1
Smaller radius sphere
has smaller charge
Surface charge densities?
Q
R1




1
2
2
4R
R2
Electric field?


R
E   E 2  1 1
o
R2
Tues. Oct. 7, 2008
Smaller sphere has larger
surface charge density
Local E-field bigger at more sharply
curved (smaller R) regions
Physics 208 Lecture 11
11
Varying E-fields on conductor



Larger electric fields near smaller radii surfaces.
Large electric fields at sharp points,
Strong fields can ionize air atoms.
Tues. Oct. 7, 2008
Physics 208 Lecture 11
12
Potential and charge


Have shown that a conductor has an electric
potential, and that potential depends on its charge
For a charged conducting sphere:
+ + +
+
+
+
+
+
+
+ +
Tues. Oct. 7, 2008
Q k
V R V   k  Q
R R
Electric potential proportional
to total charge
Physics 208 Lecture 11
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Quick Quiz
Consider this conducting object. When it has total
charge Qo, its electric potential is Vo. When it has
charge 2Qo, its electric potential
A. is Vo
B. is 2Vo
C. is 4Vo
D. depends on shape
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Physics 208 Lecture 11
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Capacitance

Electric potential of any conducting object
proportional to its total charge.
1
V Q
C
C = capacitance
  Large capacitance: need lots of charge to change potential


Small capacitance: small charge can change potential.
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Physics 208 Lecture 11
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Capacitors

Where did the charge come from?


Usually transferred from another conducting
object, leaving opposite charge behind
A capacitor consists of two conductors


Conductors generically called ‘plates’
Charge transferred between plates


Plates carry equal and opposite charges
Potential difference between plates
proportional to charge transferred Q
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Physics 208 Lecture 11
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Definition of Capacitance

Same as for single conductor
1
V  Q
C


but V = potential difference between plates
Q = charge transferred between plates



SI unit of capacitance is farad (F) = 1 Coulomb / Volt
This is a very large unit: typically use
mF = 10-6 F, nF = 10-9 F, pF = 10-12 F
Tues. Oct. 7, 2008
Physics 208 Lecture 11
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How was charge transferred?


Battery has fixed electric potential difference
across its terminals
Conducting plates connected to battery
terminals by conducting wires.

Vplates = Vbattery across plates

Electrons move

V

from negative battery terminal to -Q plate

from +Q plate to positive battery terminal
This charge motion requires work

The battery supplies the work
Q  CV
Tues. Oct. 7, 2008
Physics 208 Lecture 11
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Work done to charge a capacitor


Requires work to transfer charge dq from one plate:
q
dW  Vdq  dq
C
Total work = sum of incremental work
Q

W

0

q
Q2
dq 
C
2C
Work done stored as potential energy in capacitor
Q2 1
1
2

U
 QV  CV 
2C 2
2
Tues. Oct. 7, 2008
Physics 208 Lecture 11
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Example: Parallel plate capacitor
+Q



Charge Q moved from right outer
conductor to left conductor
-Q
inner
Charge only on inner surfaces
Plate surfaces are charge sheets,
each producing E-field
E left  E right   /2o   /2o   /o
Uniform field between plates
d
Tues. Oct. 7, 2008
Physics 208 Lecture 11
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Quick Quiz
Electric field between plates of infinite parallel-plate
capacitor has a constant value /o. What is the
field outside of the plates?
A. /o
B. /2o
C. - /2o
D. /4o
E. 0
Tues. Oct. 7, 2008
Physics 208 Lecture 11
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What is potential difference?
Potential difference = V+-V= - (work to move charge q
from + plate to plate) / q
 qEd/q
 d 
V  Ed  d /o  Q 
o A 

 d 
V  V  V  Q 
o A 
Tues. Oct. 7, 2008
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Q
Physics 208 Lecture 11
E
d
-
-Q
22
What is the capacitance?
 d 
V  V  V  Q 
o A 
V  Q /C
C
o A
d
-Q
This is a geometrical factor
+Q
Energy stored in parallel-plate capacitor
1
1 o A
1
2
2
U  CV  
Ed  Ado E 2
2
2 d
Energy density
Tues. Oct. 7, 2008
2
1
U /Ad  o E 2
2
Physics 208 Lecture 11
d
23