Remember: Exam this Thursday, Feb 12 at the regular class time.
Please bring at least two sharpened pencils – the exams are not to
be done in pen!
It is open book, open note.
Don’t forget your calculator!
Definitions
• Voltage—potential difference between two points in space (or a
circuit)
• Capacitor—device to store energy as potential energy in an E
field
• Capacitance—the charge on the plates of a capacitor divided by
the potential difference of the plates C = q/V
• Farad—unit of capacitance, 1F = 1 C/V. This is a very large unit
of capacitance, in practice we use F (10-6) or pF (10-12)
Definitions cont
• Electric circuit—a path through which charge can flow
• Battery—device maintaining a potential difference V
between its terminals by means of an internal
electrochemical reaction.
• Terminals—points at which charge can enter or leave a
battery
Capacitors
• A capacitor consists of two conductors called plates which get equal
but opposite charges on them
• The capacitance of a capacitor C = q/V is a constant of
proportionality between q and V and is totally independent of q and
V
• The capacitance just depends on the geometry of the capacitor, not q
and V
• To charge a capacitor, it is placed in an electric circuit with a source
of potential difference or a battery
CAPACITANCE AND CAPACITORS
Capacitor: two conductors separated by
insulator and charged by opposite and
equal charges (one of the conductors can be
at infinity)
Used to store charge and electrostatic
energy
Superposition / Linearity: Fields, potentials and potential
differences, or voltages (V), are proportional to charge
magnitudes (Q)
C
Q
V
(all taken positive, V-voltage between plates)
Capacitance C (1 Farad = 1 Coulomb / 1 Volt) is
determined by pure geometry (and insulator properties)
1 Farad IS very BIG: Earth’s C < 1 mF
Calculating Capacitance
1.
2.
Put a charge q on the plates
Find E by Gauss’s law, use a surface such that
 
qenc
 E  dA  EA 
0
3.
Find V by (use a line such that V = Es)
 
V   E  ds  Es
4.
Find C by
q
C
V
Parallel plate capacitor
Energy stored in a capacitor is related to the E-field between the plates
Electric energy can be regarded as stored in the field itself.
This further suggests that E-field is the separate entity that may exist alongside
charges.
density   charge Q / area S
E

Q

;
 0  0S
C
V  Ed 
 0S
Qd
 0S
Generally, we find the potential difference
Vab between conductors for a certain
charge Q
Point charge potential difference ~ Q
d
This is generally true for all capacitances
Capacitance configurations
Cylindrical capacitor
Spherical Capacitance
b
dr
1 1
V  keQ  2  keQ(  )
r
a b
a
b
dr
Q b
V  2ke    2ke ln( )
r
l
a
a
C
ab
C
ke (b  a)
l
With b  , C  a / ke -
b
2ke ln( )
a
capacitance of an individual sphere
Definitions
• Equivalent Capacitor—a single capacitor that has the same
capacitance as a combination of capacitors.
• Parallel Circuit—a circuit in which a potential difference applied
across a combination of circuit elements results in the potential
difference being applied across each element.
• Series Circuit—a circuit in which a potential difference applied
across a combination of circuit elements is the sum of the
resulting potential differences across each element.
Capacitors in Series
Q
Q
Vac  V1  ; Vcb  V2 
C1
C2
Total voltage V  V1  V2
Equivalent
1 V
1
1
 

C Q C1 C2
Capacitors in Parallel
Total charge
Q  Q1  Q2
Equivalent
C
Q
 C1  C2
V
Example: Voltage before and after
Initially capacitors are charged by
the same voltage but of opposite polarity :
Q1i  C1Vi ; Q2i  C2Vi
Total charge Q  Q1i  Q2i  Q1 f  Q2 f
Equivalent
C  C1  C2
Q C1  C2
Voltage after : V f  
Vi
C C1  C2
Energy Storage in Capacitors
Electric Field Energy
Electric potential energy stored = amount of work done to charge the capacitor
i.e. to separate charges and place them onto the opposite plates
V
Q
C
To transfer charge dq between
conductors, work dW=Vdq
Q
Q
q
Q2
Total work W   V ( q)dq   dq 
C
2C
0
0
Q2 1
CV 2
Stored energy U 
 QV 
2C 2
2
Charged capacitor – analog to stretched/compressed spring
Capacitor has the ability to hold both charge and energy
CV 2 ( 0 S / d )( Ed )2  0 E 2
uE 


2 Sd
2 Sd
2
Density of energy (energy/volume)
Energy is conserved in the E-field
Applications of Capacitors: Energy Storage
Z-machine for controlled nuclear fusion
Sandia National Labs
P ~ 1014 Watt
T ~ 2 109 K
In real life we want to store more charge at lower voltage,
hence large capacitances are needed
Increased area, decreased separations, “stronger”
insulators
Electronic circuits – like a shock absorber in the car, capacitor smoothes power
fluctuations
Response on a particular frequency – radio and TV broadcast and receiving
Undesirable properties – they limit high-frequency operation
Example: Transferring Charge and Energy Between Capacitors
Switch S is initially open
1) What is the initial charge Q0?
2) What is the energy stored in C1?
3) After the switch is closed
what is the voltage across each
capacitor? What is the charge on each? What is the total energy?
a)
Q0  C1V0
b)
1
U i  Q0V0
2
c) when switch is closed, conservation of charge
Q1  Q2  Q0
Capacitors become connected in parallel V 
C1V0
C1  C2
1
1
d) U f  Q1V  Q2V  Ui Where had the difference gone?
2
2
It was converted into the other forms of energy (EM radiation)
Definitions
• Dielectric—an insulating material placed between plates of a
capacitor to increase capacitance.
• Dielectric constant—a dimensionless factor  that determines
how much the capacitance is increased by a dielectric. It is a
property of the dielectric and varies from one material to another.
• Breakdown potential—maximum potential difference before
sparking
• Dielectric strength—maximum E field before dielectric breaks
down and acts as a conductor between the plates (sparks)
Most capacitors have a non-conductive material (dielectric) between the conducting
plates. That is used to increase the capacitance and potential across the plates.
Dielectrics have no free charges and they do not conduct electricity
Faraday first established this
behavior
Capacitors with Dielectrics
•
Advantages of a dielectric include:
1. Increase capacitance
2. Increase in the maximum operating voltage. Since dielectric
strength for a dielectric is greater than the dielectric strength
for air
Emax di  Emax air  Vmax di  Vmax air
•
3. Possible mechanical support between the plates which
decreases d and increases C.
To get the expression for anything in the presence of a
dielectric you replace o with o
k 0 S
C
;
d
V  Ed  E decreases: E  E0 / k
Field inside the capacitor became smaller – why?
We know what happens to the conductor in the electric field
Field inside the conductor E=0
outside field did not change
Potential difference (which is the
integral of field) is, however, smaller.

V  ( d  b)
o
C
0 A
d [1  b / d ]
There are polarization (induced)
charges
– Dielectrics get polarized
Properties of Dielectrics
Redistribution of charge – called polarization
K
C
dielectric constant of a material
C0
We assume that the induced charge is directly
proportional to the E-field in the material
E
E0
K
when Q is kept constant
V
V0
K
In dielectrics, induced charges do not exactly
compensate charges on the capacitance plates
E0 

;
0
  K 0

E

1
u   E2
2
E
  i
0


1

 i   1  
K
Induced charge density
Permittivity of the dielectric material
E-field, expressed through charge density  on the conductor plates
(not the density of induced charges) and permittivity of the dielectric 
(effect of induced charges is included here)
Electric field density in the dielectric
Example: A capacitor with and without dielectric
Area A=2000 cm2
d=1 cm; V0 = 3kV;
After dielectric is inserted, voltage V=1kV
Find; a) original C0 ; b) Q0 ; c) C d) K e) E-field