1. Physics 2112 Unit 8: Capacitors
Today’s Concept:
Capacitors in a circuits
Dielectrics
Energy in capacitors

2. Where we are…….

3. Simple Capacitor Circuit
Direction of arrows is opposite of the direction of electron motion
Sorry. It’s historical. There is nothing I can do. Q C V
Q = VC V C
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.

4. Parallel Capacitor Circuit
Qtotal C1 C2 V
Q1 = C1V
Q2 = C2V
Key point: V is the same for both capacitors
Key Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2)
Ctotal = C1 + C2

5. Series Capacitor CircuitQ
Q = VCtotal C1 C2 V1 V2 V V 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
Ctotal C1 C2 + =

6. Capacitor Summary Series Parallel Wiring Voltage Current Capacitance
Every electron that goes through one must go through the other
Each resistor on a different wire.
Wiring
Same for each capacitor.
Vtotal = V1 = V2
Voltage
Different for each capacitor.
Vtotal = V1 + V2
Same for each capacitor
Itotal = I1 = I2
Different for each capacitor
Itotal = I1 + I2
Current
Decreases
1/Ceq = 1/C1 + 1/C2
Increases
Ceq = C1 + C2
Capacitance

7. Example 8.1 (Capacitors in Series)
Given the circuit to the left:
What is Q1?
What is V1?
C1 =3uF
V =12V
C2 =9uF
Conceptual Idea:
Plan:
Find equivalent capacitance
Use knowledge that in series Q1 = Q2 = Qtot
Find V using Q = CV

8. Example 8.2 (Capacitors in Parallel)
Given the circuit to the left:
What is Q1?
What is Q2?
C2 =9uF
C1 =3uF
V =12V
Conceptual Idea:
Plan:
Find equivalent capacitance
Use knowledge that in parallel V1 = V2
Find Q using Q = CV

9. CheckPoint: Three Capacitor Configurations
The three configurations shown below are constructed using identical capacitors. Which of these configurations has lowest total capacitance? B C C A
1/Ctotal = 1/C + 1/C
= 2/C
Ctotal = C
Ctotal = 2C
Ctotal = C/2

10. CheckPoint: Two Capacitor Configurations
The two configurations shown below are constructed using identical capacitors. Which of these configurations has the lowest overall capacitance? B C A C
Cright = C/2
Cleft = C/2
Ctotal = C
Ctotal = Cleft + Cright
Ctotal = C A B
Both configurations have the same capacitance

11. 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. C1 C2 C3 V0 V1 Q1 V2 Q2 V3 Q3
Compare Q1, Q2, and Q3.
Q1 > Q3 > Q2
Q1 > Q2 > Q3
Q1 > Q2 = Q3
Q1 = Q2 = Q3
Q1 < Q2 = Q3

12. Example 8.3 (Capacitor Network)
C3 =12uF
Given the circuit to the left:
What is Q1?
What is Q3?
C1 =3uF
C2 =11uF
V =12C
C5 =9uF
C4 =6uF
Conceptual Idea:
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

13. Example 8.1 (Capacitor Network)

14. Energy in a Capacitor +Q C V U = 1/2QV = 1/2CV2 -Q = 1/2Q2/C
In Prelecture 7 we calculated the work done to move charge Q from one plate to another: C +Q V
U = 1/2QV
= 1/2CV2
Since Q = VC
= 1/2Q2/C -Q
This is potential energy waiting to be used…

15. Messing with Capacitors
If connected to a battery V stays constant If isolated then total Q stays constant V
V1 = V
Q1 = C1V1
= k CV = k Q
C1 = k C
Q1 = Q
V1 = Q1/C1
= Q/k C = V /k
C1 = k C

16. Dielectrics
C1 = k C0 V
Q1 = VC1 C0 V
Q0 = VC0
By adding a dielectric you are just making a new capacitor with larger capacitance (factor of k)

17. 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.
V1 > V2
V1 = V2
V1 < V2

18. 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.
U1 > U2
U1 = U2
U1 < U2

19. 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?
The charges will flow so that the charge on C1 will become equal to the charge on C2.
The charges will flow so that the energy stored in C1 will become equal to the energy stored in C2
The charges will flow so that the potential difference across C1 will become the same as the potential difference across C2.
No charges will flow. The charge on the capacitors will remain what it was before they were connected.

20. Example 8.4 (Partial Dielectric)V C0 x0
An air-gap capacitor, having capacitance C0 and width x0 is connected to a battery of voltage V. V k
x0/4
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:
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.
Strategic Analysis:
Think of new capacitor as two capacitors in parallel
Calculate new capacitance C
Apply definition of capacitance to determine Q

21. Calculation A1 =3/4 Ao A2 =1/4 Ao k k
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.