Physics 2112
Unit 8: Capacitors
Today’s Concept:



Capacitors in a circuits
Dielectrics
Energy in capacitors
Unit 8, Slide 1
Where we are…….
Unit 8, Slide 2
Simple Capacitor Circuit
Q
V
Direction of arrows is
opposite of the direction of
electron motion
V
C
C
Sorry. It’s
historical. There is
nothing I can do.
Q = VC
Q
This “Q” really means that the battery has
moved charge Q from one plate to the other,
so that one plate holds +Q and the other -Q.
Electricity & Magnetism Lecture 8, Slide 3
Parallel Capacitor Circuit
Qtotal
V
C2
C1
Q1 = C1V
Q2 = C2V
Qtotal
Key point: V is the same for both capacitors
Key Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2)
Ctotal = C1 + C2
Electricity & Magnetism Lecture 8, Slide 4
Series Capacitor Circuit
Q = VCtotal
Q
C1
V1
V
Q
V
C2
V2
Q
Key point: Q is the same for both capacitors
Key point: Q = VCtotal = V1C1 = V2C2
Also: V = V1 + V2
Q/Ctotal = Q/C1 + Q/C2
1
1
1
=
+
C2
Ctotal C1
Electricity & Magnetism Lecture 8, Slide 5
Capacitor Summary
Series
Parallel
C1
C1
Wiring
Voltage
Current
C2
Every electron that goes
through one must go
through the other
Different for each capacitor.
Vtotal = V1 + V2
Same for each capacitor
Itotal = I1 = I2
Decreases
Capacitance 1/C = 1/C + 1/C
eq
1
2
C2
Each resistor on a
different wire.
Same for each capacitor.
Vtotal = V1 = V2
Different for each capacitor
Itotal = I1 + I2
Increases
Ceq = C1 + C2
Electricity & Magnetism Lecture 9, Slide 6
Example 8.1 (Capacitors in Series)
Given the circuit to the left:
C1 =3uF
V =12V
C2 =9uF
What is Q1?
What is V1?
 Conceptual Idea:
CV = Q
 Plan:
•
•
•
Find equivalent capacitance
Use knowledge that in series Q1 = Q2 = Qtot
Find V using Q = CV
Unit 8, Slide 7
Example 8.2 (Capacitors in Parallel)
Given the circuit to the left:
C2 =9uF
V =12V
C1 =3uF
What is Q1?
What is Q2?
 Conceptual Idea:
CV = Q
 Plan:
•
•
•
Find equivalent capacitance
Use knowledge that in parallel V1 = V2
Find Q using Q = CV
Unit 8, Slide 8
CheckPoint: Three Capacitor Configurations
The three configurations shown below are constructed using identical capacitors.
Which of these configurations has lowest total capacitance?
C
B
A
C
C
C
C
C
Ctotal = C
1/Ctotal = 1/C + 1/C
Ctotal = 2C
= 2/C
Ctotal = C/2
Electricity & Magnetism Lecture 8, Slide 9
CheckPoint: Two Capacitor Configurations
The two configurations shown below are constructed using identical capacitors. Which
of these configurations has the lowest overall capacitance?
B
A
C
C
C
C
C
Cleft = C/2
Ctotal = C
A.
B.
C.
Cright = C/2
Ctotal = Cleft + Cright
C
=C
total
A
B
Both configurations have the same
capacitance
Electricity & Magnetism Lecture 8, Slide 10
CheckPoint: Capacitor Network
A circuit consists of three unequal capacitors C1, C2, and C3 which are connected to a
battery of voltage V0. The capacitance of C2 is twice that of C1. The capacitance of C3 is
three times that of C1. The capacitors obtain charges Q1, Q2, and Q3.
Q2
C2
Compare Q1, Q2, and Q3.
A. Q1 > Q3 > Q2
V0
B. Q1 > Q2 > Q3
C. Q1 > Q2 = Q3
D. Q1 = Q2 = Q3
E. Q1 < Q2 = Q3
V2
C1
V1
Q1
V3
C3
Q3
Unit 8, Slide 11
Example 8.3 (Capacitor Network)
C3 =12uF
Given the circuit to the left:
C1 =3uF
C2 =11uF
What is Q1?
V =12C
What is Q3?
C5 =9uF
C4 =6uF
 Conceptual Idea:
CV = Q
Find V at each capacitor and then Q .
 Plan:
•
•
•
•
Break circuit down into elements that are in parrallel or in series.
Find equivalent capacitance
Work backwards to find DV across each one
Fine Q using Q = CV
Unit 8, Slide 12
Example 8.1 (Capacitor Network)
C3
C4
C3
C5
C1234
C2
C1
C2
C1
C34
C5
C12345
C5
Unit 8, Slide 13
Energy in a Capacitor
In Prelecture 7 we calculated the work done to move charge Q from one plate to
another:
C
+Q
V
-Q
U = 1/2QV
= 1/2CV2
= 1/2Q2/C
Since Q = VC
This is potential energy waiting to be used…
Electricity & Magnetism Lecture 8, Slide 14
Messing with Capacitors
If connected to a battery V stays constant
V1 = V
V
C1 = k C
Q1 = C1V1
= k CV = k Q
If isolated then total Q stays constant
Q1 = Q
C1 = k C
V1 = Q1/C1
= Q/k C = V /k
Electricity & Magnetism Lecture 8, Slide 15
Dielectrics
C1 = k C0
C0
V
Q0 = VC0
Q1 = VC1
V
By adding a dielectric you are just making a
new capacitor with larger capacitance (factor of k)
Electricity & Magnetism Lecture 8, Slide 16
CheckPoint: Capacitors and Dielectrics 1
Two identical parallel plate capacitors are given the same
charge Q, after which they are disconnected from the battery.
After C2 has been charged and disconnected, it is filled with a
dielectric.
Compare the voltages of the two
capacitors.
A. V1 > V2
B. V1 = V2
C. V1 < V2
Electricity & Magnetism Lecture 8, Slide 17
CheckPoint: Capacitors and Dielectrics 2
Two identical parallel plate capacitors are given the
same charge Q, after which they are disconnected
from the battery. After C2 has been charged and
disconnected, it is filled with a dielectric.
Compare the potential energy
stored by the two capacitors.
A. U1 > U2
B. U1 = U2
C. U1 < U2
Electricity & Magnetism Lecture 8, Slide 18
CheckPoint: Capacitors and Dielectrics 3
The two capacitors are now connected to
each other by wires as shown. How will
the charge redistribute itself, if at all?
A. The charges will flow so that the charge on C1 will become equal to
the charge on C2.
B. The charges will flow so that the energy stored in C1 will become
equal to the energy stored in C2
C. The charges will flow so that the potential difference across C1 will
become the same as the potential difference across C2.
D. No charges will flow. The charge on the capacitors will remain what
it was before they were connected.
Electricity & Magnetism Lecture 8, Slide 19
Example 8.4 (Partial Dielectric)
C0
V
x0
V
An air-gap capacitor,
having capacitance
k C0 and width x0 is
connected to a
x0/4
battery of voltage V.
A dielectric (k ) of width x0/4 is inserted into the gap as shown.
What is Qf, the final charge on the capacitor?
Conceptual Analysis:
C
Q
V
Strategic Analysis:
 Think of new capacitor as two capacitors in parallel
 Calculate new capacitance C
 Apply definition of capacitance to determine Q
Electricity & Magnetism Lecture 8, Slide 20
Calculation
k
k
A1 =3/4 Ao
A2 =1/4 Ao
Electricity & Magnetism Lecture 8, Slide 21