Download Physics 2102 Lecture: 06 THU 11 FEB

Physics 2113
Jonathan Dowling
Physics 2113
Lecture: 17 WED 25 FEB
Capacitance II
Capacitors in Parallel: V=Constant
• An ISOLATED wire is an equipotential
surface: V=Constant
• Capacitors in parallel have SAME
potential difference but NOT ALWAYS
same charge!
V = VAB = VA –VB
Q1
A
• VAB = VCD = V
• Qtotal = Q1 + Q2
• CeqV = C1V + C2V
• Ceq = C1 + C2
C
C1
VA
Q2
VB
C2
VC
VD
B
D
V = VCD = VC –VD
• Equivalent parallel capacitance =
sum of capacitances
Cparallel = C1 + C2
PAR-V (Parallel: V the Same)
Qtotal
V=V
Ceq
Capacitors in Series: Q=Constant
•
Q1 = Q2 = Q = Constant
•
VAC = VAB + VBC
Isolated Wire:
Q=Q1=Q2=Constant
Q1
SERI-Q: Series Q the Same
Q
Q Q
= +
Ceq C1 C2
1
Cseries
B
A
1
1
= +
C1 C2
SERIES:
• Q is same for all capacitors
• Total potential difference = sum of V
Q2
C1
C
C2
Q = Q 1 = Q2
Ceq
Capacitors in Parallel and in Series
• In parallel :
Cpar = C1 + C2
Vpar = V1 = V2
Qpar = Q1 + Q2
Q1
C1
Qeq
Q2
C2
Ceq
• In series :
1/Cser = 1/C1 + 1/C2
Vser = V1 + V2
Qser= Q1 = Q2
Q1
Q2
C1
C2
Example: Parallel or Series?
Parallel: Circuit Splits Cleanly in Two (Constant V)
What is the charge on each capacitor?
• Qi = CiV
• V = 120V on ALL Capacitors (PAR-V)
• Q1 = (10  F)(120V) = 1200  C
• Q2 = (20  F)(120V) = 2400  C
• Q3 = (30  F)(120V) = 3600  C
Note that:
• Total charge (7200  C) is shared
between the 3 capacitors in the ratio
C1:C2:C3 — i.e. 1:2:3
C1=10  F
C2=20  F
C3=30  F
120V
Cpar = C1 + C2 + C3 = (10 + 20 + 30)mF = 60mF
Example: Parallel or Series
Series: Isolated Islands (Constant Q)
What is the potential difference across each capacitor?
• Q = CserV
• Q is same for all capacitors (SERI-Q)
• Combined Cser is given by:
1
1
1
1
=
+
+
Cser (10mF) (20mF) (30mF)
C1=10mF
C2=20mF
C3=30mF
120V
• Ceq = 5.46  F (solve above equation)
• Q = CeqV = (5.46  F)(120V) = 655  C
• V1= Q/C1 = (655  C)/(10  F) = 65.5 V
• V2= Q/C2 = (655  C)/(20  F) = 32.75 V
• V3= Q/C3 = (655  C)/(30  F) = 21.8 V
Note: 120V is shared in the
ratio of INVERSE
capacitances i.e.
(1):(1/2):(1/3)
(largest C gets smallest V)
Example: Series or Parallel?
Neither: Circuit Simplification Needed!
In the circuit shown, what is
the charge on the 10 F
capacitor?
5 F
• The two 5 F capacitors are in
parallel
• Replace by 10 F
• Then, we have two 10 F capacitors
in series
• So, there is 5V across the 10  F
capacitor of interest by symmetry
• Hence, Q = (10 F )(5V) = 50 C
10 F
5 F
10V
10  F
10 F
10V
Energy U Stored in a Capacitor
• Start out with uncharged capacitor
• Transfer small amount of charge dq
from one plate to the other until
charge on each plate has
magnitude Q
• How much work was needed?
dq
Q
Q
q
Q
CV
U = ò Vdq = ò dq =
=
C
2C
2
0
0
2
2
Energy Stored in Electric Field of Capacitor
• Energy stored in capacitor: U = Q2/(2C) = CV2/2
• View the energy as stored in ELECTRIC FIELD
• For example, parallel plate capacitor: Energy DENSITY =
energy/volume = u =
2
2
æ
ö
Q
e
Q
e
E
Q
Q
0
0
=
ç
÷
u=
=
=
=
2CAd 2æç e 0 A ö÷ Ad 2e 0 A2
2 çè e 0 A ÷ø
2
2
2
2
è d ø
volume = Ad
General
expression for
any region with
vacuum (or air)
Dielectric Constant
• If the space between capacitor plates is
filled by a dielectric, the capacitance
DIELECTRIC
INCREASES by a factor 
• This is a useful, working definition for
dielectric constant.
• Typical values of  are 10–200 but it is
always greater than 1!
+Q –Q
C =  0 A/d
The  and the constant
 o are both called dielectric
constants. The  has no units
(dimensionless).
Trick: Just substitute  o for o
in all the previous formulas!
Atomic View
Emol
Molecules set up
counter E field Emol that
somewhat cancels out
capacitor field Ecap.
This avoids sparking
(dielectric breakdown)
by keeping field inside
dielectric small.
Ecap
Hence the bigger the
dielectric constant the
more charge you can
store on the capacitor.
Example: Battery Connected —
Voltage V is Constant but Charge Q Changes
• Capacitor has charge Q, voltage V
• Battery remains connected while dielectric
slab is inserted.
• Do the following increase,
dielectric
decrease or stay the same:
slab
– Potential difference?
– Capacitance?
– Charge?
– Electric field?
Example: Battery Connected —
Voltage V is Constant but Charge Q Changes
• Initial values:
capacitance =
C; charge = Q; potential difference = V;
electric field = E;
• Battery remains connected
• V is FIXED; Vnew = V (same)
• Cnew =  C (increases)
• Qnew = ( C)V =  Q (increases).
dielectric
slab
• Since Vnew = V, Enew = V/d=E (same)
Energy stored? u=0E2/2 => u= 0E2/2 = E2/2
increases
Example: Battery Disconnected —
Voltage V Changes but Charge Q is Constant
• Capacitor has charge Q, voltage V
• Battery remains is disconnected then
dielectric slab is inserted.
• Do the following increase,
decrease or stay the same:
– Potential difference?
– Capacitance?
– Charge?
– Electric field?
dielectric
slab
Example: Battery Disconnected —
Voltage V Changes but Charge Q is Constant
• Initial values:
capacitance =
C; charge = Q; potential difference = V;
electric field = E;
• Battery remains disconnected
• Q is FIXED; Qnew = Q (same)
• Cnew =  C (increases)
• Vnew = Q/Cnew = Q/( C) (decreases).
dielectric
slab
• Since Vnew < V, Enew = Vnew/d = E/
(decreases)
Energy stored? u = e E 2 / 2
0
® ke 0 E
2
new
/ 2 = ke 0 ( E / k ) / 2
= e 0 E 2 / (k 2 ) decreases
2
Summary
• Any two charged conductors form a capacitor.
• Capacitance : C= Q/V
• Simple Capacitors:
Parallel plates: C = 0 A/d
Spherical : C = 4 0 ab/(b-a)
Cylindrical: C = 2 0 L/ln(b/a)
• Capacitors in series: same charge, not necessarily equal potential;
equivalent capacitance 1/Ceq=1/C1+1/C2+…
• Capacitors in parallel: same potential; not necessarily same charge;
equivalent capacitance Ceq=C1+C2+…
• Energy in a capacitor: U=Q2/2C=CV2/2; energy density u=0E2/2
• Capacitor with a dielectric: capacitance increases C’=κC