Transcript Chapter 24: Capacitance and Dielectrics and Ch. 26

Chapters 24 and 26.4-26.5
1
Capacitor
++++++++++
++++++++++++++
+q
Potential difference=V
-q
-------------------------
Any two conductors separated by either an insulator or vacuum for a
capacitor
The “charge of a capacitor” is the absolute value of the charge on one of
conductors.
q
 constant
This constant is called the “capacitance” and
V
is geometry dependent. It is the “capacity” for holding charge at a
constant voltage
2
Units
1 Farad=1 F= 1 C/V
 Symbol:

Indicates positive
potential
3
Interesting Fact

When a capacitor has reached full
charge, q, then it is often useful to think
of the capacitor as a battery which
supplies EMF to the circuit.
4
Simple Circuit
+q
H
-q
L
Initially,
After
S is
H closed,
& L =0 H=+q
L=-q
S
i
-i
5
Recalling Displacement Current
Maxwell thought of the capacitor as a flow device, like a resistor so a
“displacement current” would flow between the plates of the capacitor like this
-q
+q
i
i
id
This plate induces a negative charge here
Which means the positive charge carriers are moving here
and thus a positive current moving to the right
6
If conductors had area, A

Then current density would be

Jd=id/A
7
Calculating Capacitance
Calculate the E-field in terms of charge
and geometrical conditions
 Calculate the voltage by integrating the
E-field.
 You now have V=q*something and since
q=CV then 1/something=capacitance

8
Parallel plates of area A and
distance, D, apart
  q
 E  dA 
Area, A
0
EA 
q
0
E
q
A 0
    
V    E  ds   E  ds  ED
f
i

q
V  ED and E 
A 0
V
++++++++++
++++++++++++++
Distance=D
-------------------------
qD q

A 0 C
A 0
1

C
qD
D
A 0
9
Coaxial Cable—Inner conductor of radius a
and thin outer conductor radius b
  q
 E  dA 
0
E 2rL  
q
0
E
q
2rL  0
b
   
V    E  ds   E  drrˆ  
f
i

q
dr
2L  0 r
a
q
b
ln  
2L  0  a 
2L  0  C
1

1
b
b
ln   ln  
2L  0  a 
a
C  20
L
b
ln  
a
V
-q
+q
10
Spherical Conductor—Inner conductor radius
A and thin outer conductor of radius B
  q
 E  dA 
0


q
E 4r   E 
0
4r 2  0
2
q


B
   
V    E  ds   E  drrˆ  
f
i

q
dr
2


4


r
0
A
q  1 1 
q  B A
V
  


40  B A  40  AB 
1
 AB 
 40 
C
1  B A
 B  A


40  AB 
11
Isolated Sphere of radius A
q  1 1 
V
   Let B  
40  B A 
q
V
40 A
C  40 A
12
Capacitors in Parallel
i1
i
E
i2
i3
E
C1
q1
C2
q2
C3
q3
i=i1+i2+i3 implies q=q1+q2+q3
i
Ceq
E
q
CeqV  C1V  C2V  C3V
Ceq  C1  C2  C3
13
Capacitors in Series
i
i
C1
By the loop rule,
E=V1+V2+V3
q
E
E
C2
q
E
q
Ceq
C3
q
E
q
q
q


C1 C2 C3
Ceq
q
q
q
q



Ceq C1 C2 C3
1
1
1
1



Ceq C1 C2 C3
14
Energy Stored in Capacitors
d Work 
V
d charge 
Work   Vdq
q
And V 
C
q
1 2
Work   dq 
q
C
2C
Or
1
Work  CV 2
2
Technically, this is the potential to
do work or potential energy, U
U=1/2 CV2 or U=1/2 q2/C
Recall Spring’s Potential Energy
U=1/2 kx2
15
Energy Density, u

u=energy/volume
 Assume parallel
plates at right
 Vol=AD
 U=1/2 CV2
C
0 A
D
1 0 A 2
u
V
2 AD D
1 V2 1
u  0 2  0E2
2 D
2
Area, A
++++++++++
++++++++++++++
Distance=D
-------------------------
Volume wherein energy resides
16
Dielectrics
Area, A
++++++++++
++++++++++
++++++++++
++++++++++++++
++++++++++++++
++++++++++++++
---------------------------------------------------------------------------
Insulator
Distance=D
Voltage at which the insulating material allows current flow
(“break down”) is called the breakdown voltage
1 cm of dry air has a breakdown voltage of 30 kV (wet air
less)
17
The capacitance is said to increase because we can put more
voltage (or charge) on the capacitor before breakdown.

The “dielectric strength” of vacuum is 1


Dry air is 1.00059
So we can replace, our old capacitance, Cair,
by a capacitance based on the dielectric
strength, k, which is

Cnew=k*Cair

An example is the white dielectric material in
coaxial cable, typically polyethylene (k=2.25)
or polyurethane (k=3.4)
 Dielectric strength is dependent on the
frequency of the electric field
18
Induced Charge and Polarization in
Dielectrics
E
++++++++++++++++++++++
Note that the
charges have
separated or
polarized
-- - - - - - - - - - - - - - - - - Ei
+++++++++++++++++++
-----------------------
E0
k

E0 
0
E0

0  i
 
k 0 0

1
 i   1  
 k
ETotal=E0-Ei
19
Permittivity of the Dielectric


k0
For real materials, we
define a “D-field” where


D=k0E
 
 D  da  q freeenclosed
 

For these same
H  ds i freeenclosed   D

materials, there can be a
t
 
magnetization based on
 D   D  da
the magnetic
susceptibility, c, :

H= cm0B
20
Capacitor Rule

For a move through a capacitor in the
direction of current, the change in
potential is –q/C

If the move opposes the current then the
change in potential is +q/C.
move
Vaa-Vbb= -q/C
+q/C
Va
i
Vb
21
RC Circuits
Initially, S is
open so at t=0,
i=0 in the
resistor, and the
charge on the
capacitor is 0.
 Recall that
i=dq/dt

R
A
S
B
V
C
22
Switch to A

Start at S (loop
clockwise) and use
the loop rule
R
A
S
B
q
 iR   V  0
C
q
V  iR 
C
dq q
V R

dt C
V
C
23
An Asatz—A guess of the solution
My ansatz : q p  A  Be

t
RC
t
dq p
B  RC


e
dt
RC
dq q
V R

dt C
B
A  Be
V  e

C
C
at t  0, q  0
t

RC
q p 0   A  Be

0
RC

t
RC
A
 V  or A  CV
C
0
CV  B  0
B  CV
q (t )  CV (1  e

t
RC
)
24
Ramifications of Charge

At t=0, q(0)=CV-CV=0
 At t=∞, q=CV (indicating fully charged)
 What is the current between t=0 and the time
when the capacitor is fully charged?
t





d
d 
i  q (t ) 
CV 1  e RC  

dt
dt 


t
t
CV  RC V  RC
i
e
 e
RC
R
25
Ramifications of Current


At t=0, i(0)=V/R (indicates full current)
At t=∞, i=0 which indicates that the current
has stopped flowing.

Another interpretation is that the capacitor has
an EMF =V and thus
R
A
S
Circuit after a
very long time
B
V
~V
26
Voltage across the resistor and
capacitor
Potential across resistor, VR

V  
VR  iR   e RC  Ve RC
R
t
t
R
A
S
Potential across
capacitor, VC
B
V
C
t



RC
CV 1  e 
q


VC  
C
C
t



VC  V 1  e RC 


At t=0, VC=0 and VR=V
At t=∞, VC=V and VR=0
27
RC—Not just a cola
RC is called the “time constant” of the
circuit
 RC has units of time (seconds) and
represents the time it takes for the
charge in the capacitor to reach 63% of
its maximum value
 When RC=t, then the exponent is -1 or
e-1
 t=RC

28
Switch to B

The capacitor is fully
charged to V or
q=CV at t=0
S
CV
B
q
 iR   0
V
C
dq
q
R

dt
C
dq
1
t

dt  ln q   
k
q
RC
RC
q (t )  Ae

t
RC
C
t
and
R
A
A  RC
i (t )  
e
RC
If q  CV at t  0, q(0)  Ae0
A  CV
29
Ramifications
At t=0, q=CV and i=-V/R
 At t=∞, q=0 and i=0 (fully discharging)
 Where does the charge go?


The charge is lost through the resistor
30
Three Connection Conventions For
Schematic Drawings
Connection
Between Wires

A

B

C
No Connection
31
Ground Connectors
Equivalently
32
Household Wiring
“hot” or black
“return”/ “neutral”
or white
“ground” or green
Single Phase
Rated 20 A (NW-14)
Max V 120 VAC
Normally, the “return”
should be at 0 V w.r.t.
ground
In THEORY, but sometimes no!
33
The Death of Little Johnny
A short develops
between the hot lead
and the washer case
hot
Little Johnny
X X
Washer
Uhoh! It leaks!
neutral
RG
RG=∞,
=0, then
IfIf R
then
G
Johnny
is
dead!
Johnny is safe
120V
RLittle
Johnny
RG
34
Saving Little Johnny
A short develops
between the hot lead
and the washer case
Little Johnny
hot
Washer
Uhoh! It leaks!
neutral
RG
No Path to Johnny!
120V
RLittle
Johnny
RG
35