iClicker Quiz
(1) I have completed at least 50% of the reading and studyguide assignments associated with the lecture, as indicated on the course schedule.
a) True b) False

Pieter van Musschenbroek (1692-1761) – University of Leyden (Holland)
+
http://physics.kenyon.edu/EarlyApparatus/Electricity/Condenser/Condenser.html
http://s.bourdreux.free.fr/cabinet_Sigaud/chronologie/musschenbroek.htm
“I would like to tell you about a new but terrible experiment, which I advise you never to try yourself, nor would I, who experienced it and survived by the grace of God, do it again for all the kingdom of France.”

William Watson (1715-1787) www.wikipedia.com
http://www.howstuffworks.com

http://www.codecheck.com/cc/BenAndTheKite.html

Quasi-uniform field between two parallel-plates
 

E



V
Capacitance is like flexibility:
Stretch a spring x

1 k
F
Q

C V
Charge a capacitor
How much charge does one Volt buy you?
If the capacitance is high, one Volt goes a long ways.
The SI unit of capacitance is the Farad = Coulomb/Volt.

The parallel-plate capacitor
Potential difference: V
E
E



0

( Q /

0
A )
V
 
0
 d
E
 ds

E d

Qd

0
A

C

Q
V


0
A d
Given A and d , if we now double the voltage, the capacitance will
(1) increase by a factor of two,
(2) decrease by a factor of two,
(3) stay the same.
If V doubles, then I also doubles. C doesn’t depend on V or I .

iClicker Quiz
(1) I have completed at least 50% of the reading and studyguide assignments associated with the lecture, as indicated on the course schedule.
a) True b) False

E
 kQ r
V
 
0
 d
E
 ds
  a
 b kQ dr r
2
 kQ
1 a
1 b
 kQ
 ab a

C

Q
V

1 k b ab
 a

4

0 b ab
 a

  
E
 d A

EA

E ( 2
 rL )

Q

0

E

Q / L
2
 
0 r
V
   a b
E
 ds
 
Q
2

/

L
0
 a b dr r

Q
2

/

L
0 ln b a

C / L

Q / L
V
 ln
2
 
0
 


C eq

Q tot
V

Q
1

Q
2
V

Q
1
V

Q
2
V

C
1

C
2
C eq is larger than either C
1 or C
2
.


C eq

Q
V tot

V
1
Q

V
2

V
Q
1
1

V
2
Q



1
C
1

1
C
2



1
C eq is smaller than smallest of C
1 and C
2
.

C
C
C
C

C C
C
C C
C
2 C

C /2
C
C /2

a
Network capacitance depends on which two points you choose to measure between.
C
1 C
2
C
5 b Easy: C ab

( C
1
|| C
2
)

( C
3
|| C
4
)

C
5
C
3
C
4 a
C
1
C
3
C
5
C
2
C
4 b Much harder (needs matrix methods)
C ab

(C3
C3 (C4 C5


C4) C5
C2 (C4


C1 (C3
C5))


C4

C5)
C1 (C4 (C3


C2
C5)
(C3


C4

C2 (C3

C5)
C4

C5)) b

Energy storage in Capacitors
Dielectrics
Dipoles

Electric potential energy stored in a capacitor
Potential difference: V
Q

CV
Charging: initial Q and V are zero.
E dU
 dQ V

( CdV ) V

CVdV
The first dQ is easy. But as V increases, each dQ is a little more expensive.
U
  dU

1
2
CV
2 
1
2
Q
2
C

Potential difference: V
E u

Energy
Volume

1
2
CV
2
Ad

1
2
(

0
A / d ) V
2
Ad

1
2

0
V d
2

1
2

0
E
2

E
Force between the plates of a parallel-plate capacitor?
C


0
A x
U

1
2
CV
2 
1 Q
2
2 C
V
Quiz : The two plates of a capacitor should experience a force that is:
(1) attractive (2) repulsive (3) zero.
Remember that F =
 dU / dx , which means that the force will act to lower the electrical potential energy.
x
Need to hold either Q or V constant!
constant Q
 attractive constant V
 repulsive

d
τ 




 i
τ i
  i r i

F i r
1

F
1
 r
2

F
2 r q
1
(
 q E r
1

 r
2
) r
2


E q E p

E
Rotation aligns the moment vector ( p ) with the field?
Which way does torque vector point in this figure?
Use the RHR.


U


 



2 /
τ  d
θ  




2 / pE
 p

E sin(

) d




2 /
τ
 d

 pE cos(

)


0
   
0
,
 
1
Dielectric materials contain polar molecules that can align with an external field. The field produced by these dipoles gets added to the external field.
water

Many types of capacitors
Tubular : most common low Q and low V
Oil : high V
Electrolytic : common polarizeed large Q at small V

Ultracapacitors (a.k.a. supercapacitors)
Huge internal A . Very small d between + and

Extraordinarily-high Q and u , but low V http://en.wikipedia.org/wiki/electric_double-layer_capacitor

Adding a dielectric to any situation

0
   
0
C ppc

A
 d

A

0 d u

1
2

E
2 
1
2

0
E
2
  
E
 dA

Q


Q

0

Q

C

V
A

0 d
E

V d
U

1
2
CV
2 
1
2
Q
2
C u

U vol

1
2

0
E
2
Constant Q : How do ( A , d ,

) affect V , E , U and u ?
  
C
 
V

E

U
 u

A
 
C
 
V

E

U
 u
 d
 
C
 
V

E

U
 u

Constant V : How do ( A , d ,

) affect Q , E , U and u ?
  
C
 
Q

E

U
 u

A
 
C
 
Q

E

U
 u
 d
 
C
 
Q

E

U
 u


Q

C

V
A

0 d
E

V d
U

1
2
CV
2 
1
2
Q
2
C u

U vol

1
2

0
E
2
Quiz: Double A at constant Q . What happens to U ?
(1) doubled (2) quadrupled (3) unchanged (4) halved (5) quartered
Quiz: Double
 at constant V . What happens to E ?
(1) doubled (2) quadrupled (3) unchanged (4) halved (5) quartered

Crossed capacitor problem
V
1
+ +
  C
1
C
2
+ +
 
V
2
Q
1

C
1
V
1
Q
2

C
2
V
2
V '

Q
C '
'

Q
1
C
1

Q
2

C
2
+ +
  C
1
Q
1

C
1
V
1
C
2
+ +
 
Q
2

C
2
V
2
C
1
C
2
+ +
  C
1
Q
1

C
1
V
1
C
2

+ +
Q
2

C
2
V
2
Q '
1

C
1
V ' Q '
2

C
2
V '

Suppose you do the same thing without crossing them.
V
1
+ +
  C
1
C
2
+ +
 
V
2
Q
1

C
1
V
1
Q
2

C
2
V
2
V '

Q
C '
'

Q
1
C
1

Q
2

C
2
+ +
  C
1
Q
1

C
1
V
1
C
2
+ +
 
Q
2

C
2
V
2
C
1
C
2
+ +
  C
1
Q
1

C
1
V
1
C
2
+ +
 
Q
2

C
2
V
2
Q '
1

C
1
V ' Q '
2

C
2
V '

Crossed capacitor problem: special case where V
1
= V
2
V
+ +
  C
1
C
2
+ +
 
Q
1

C
1
V Q
2

C
2
V
V '

Q '
C '

Q
1
C
1

Q
2

C
2

V


C
1
C
1

C
2

C
2


+ +
  C
1
Q
1

C
1
V
C
2
+ +
 
Q
2

C
2
V
C
1
C
2
Q '
1

C
1
V ' Q '
2

C
2
V '
+ +
  C
1
Q
1

C
1
V
C
2

+ +
Q
2

C
2
V

6V
+
ΔV
+ + + + +
- - - - -
+ + + + +
- - - - -

6V
+
ΔV
+ + + + +
- - - - -
+ + + + +
- - - - -

C
2
C
2
C
2
C
1
C
4
C
5
C
3
C
3
C
5

The electrical potential energy stored in a capacitor is given by
U=(1/2)CV^2. It is also given by U=(1/2C)Q^2. Is the energy really proportional to C or to 1/C, or is neither really true?
● i don't totally understand the question but i do know that both formulas are correct and therefore it would seem that U is proportional to C
● neither is really true. imagine a small capacitor and also imagine an infinitely large capacitor the is no charge on either of them there is no potential energy stored on either though one has an infinitely large capacitance.
●
Really, both are true depending on which equation you use and which variables you set as constants. The reason this occurs is because these equations are equivalent, by the relationship C=Q/V.
Therefore, depending on whether you set voltage or charge as a constant, you can really look at energy as proportional to either C or
1/C.
●
Energy is actually proportional to the charge (Q) and potential

Q tot

Q
1

Q
2
C

Q tot
V

Q
1

Q
2
V

Q
1
V

Q
2
V

C
1

C
2

V tot

V
1

V
2
C

Q
V tot

V
1
Q

V
2

V
1
Q
1

V
2
Q



1
C
1

1
C
2



1

d
τ 




 i
τ i
  i r i

F i r
1

F
1
 r
2

F
2 r q
1
(
 q E r
1

 r
2
) r
2


E q E p

E


U


 



2 /
τ  d
θ  




2 / pE
 p

E sin(

) d




2 /
τ
 d

 pE cos(

)

d) e) f) g) a) b) c)
Let the i unit vector point to the right. Let j point towards the board. Let k point up. Let an upward-pointing E-field be present in the room. A dipole moment directed along the j – i direction will experience a torque along what direction.
j i k
– i
– j
– k none of these k i

In an electrically polarized material, one obtains extra surface charge per volt, and therefore, an increase in capacitance.

0
   
0
,
 
1
  
E
 dA

Q


Q

0 and C ppc

A
 d

A

0 d

Energy density in an electric field
U

1
2
CV
2

1 Q
2
2 C u

Energy
Volume

1
2
CV
2
Ad

1
2
(

0
A / d ) V
2
Ad

1
2

0
V d
2

1
2

0
E
2