Physics 212 Lecture 8, Slide 1

Physics 212 Lecture 8

Today's Concept:

Capacitors

Capacitors in a circuits, Dielectrics, Energy in capacitors

Physics 212 Lecture 8, Slide 2

C V

+Q -Q

Q

Q

Battery has moved charge Q from one plate to the other.

C V Q=VC

Simple Capacitor Circuit

8

Physics 212 Lecture 8, Slide 3

V Q1 = C1V Q2 = C2V

C2 C1

Parallel Capacitor Circuit

Key point: V is the same for both capacitors

Key Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2)

Ctotal = C1 + C2

Qtotal

Qtotal

14

Physics 212 Lecture 8, Slide 4

Series Capacitor Circuit

Q

Q

Q

+Q

+Q

-Q

-Q

V

C1

C2

V1

V2

Q V

Q

Key point: Q is the same for both capacitors

Also: V = V1 + V2

Q=VCtotal

Key point: Q = VCtotal = V1C1 = V2C2

Q/Ctotal = Q/C1 + Q/C2

1 Ctotal

1 C1

1 C2

+ = 17

Physics 212 Lecture 8, Slide 5

Checkpoint 1

C C

C

C C

1/Ctotal = 1/C + 1/C

= 2/C

Ctotal = C/2 Ctotal = C Ctotal = 2C

Which has lowest total capacitance:

18

Physics 212 Lecture 8, Slide 6

Ctotal = C

Which has lowest total capacitance ?

C

C

C

C

C

Cleft = C/2 Cright = C/2

Ctotal = Cleft + Cright

Ctotal = C

20

Physics 212 Lecture 8, Slide 7

Similar to Checkpoint 3

C1

C2

C3

V0

V1 Q1

V2

Q2

V3 Q3

Ctotal

Which of the following is NOT necessarily true: A) V0 = V1

B) Ctotal > C1

C) V2 = V3

D) Q2 = Q3

E) V1 = V2 + V3

24

Physics 212 Lecture 8, Slide 8

Checkpoint 3

25

1. See immediately: Q2 = Q3 (capacitors in series)

23 2 3 1 1 1

1 1 1 1 1 5

2 3 6C C C C C C

23 1

6

5C C

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.

Compare Q1, Q2, and Q3.

A. Q1 > Q3 > Q2 B. Q1 > Q2 > Q3 C. Q1 > Q2 = Q3 D. Q1 = Q2 = Q3 E. Q1 < Q2 = Q3

X X

2. How about Q1 vs. Q2 and Q3? Calculate C23 first. 1 1 0

23 2 3 23 0 1 0

6

5

Q C V

Q Q Q C V C V

Physics 212 Lecture 8, Slide 9

Charge capacitor – store a logical “1”

Discharge capacitor – store a logical “0”

Row select : close the switch

1 Gigabyte ~ 1010 cells

Column select : charge or discharge C

Physics 212 Lecture 8, Slide 10

In Prelecture 7 we calculated the work done to move charge Q from one plate to another:

This is potential energy waiting to be used…

Energy in a Capacitor

+Q

V

-Q

U = 1/2QV = 1/2CV2

Since Q = VC

=1/2Q2/C

C

26

Physics 212 Lecture 8, Slide 11

By adding a dielectric you are just making a

new capacitor with larger capacitance (factor of k)

Dielectrics

C0 V Q0=VC0

C1=kC0

V Q1=VC1

11

Physics 212 Lecture 8, Slide 12

Capacitor with dielectric

k V

C1 = k C

V1 = V Q1 = C1V1

= k CV = kQ

k C1 = k C

Q1 = Q V1 = Q1/C1

= Q/kC = V /k

If connected to a battery V stays constant.

Isolated capacitor: total Q stays constant.

Physics 212 Lecture 8, Slide 13

Checkpoint 4a

33

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

Consider first the effect on the capacitance, then, does Q or V stay constant?

Physics 212 Lecture 8, Slide 14

Effects of dielectric

U1/U0 = k

k=2

C0 V

C1

V

U = 1/2CV2 Compare using

Potential Energy goes UP 35

A) U1 < U0 B) U0 = U1 C) U1 > U0

Two identical parallel plate capacitors are connected to identical batteries. Then a dielectric is inserted between the plates of capacitor C1. Compare the energy stored in the two capacitors.

Physics 212 Lecture 8, Slide 15

Checkpoint 4b

33

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

Consider first the effect on the capacitance, then, does Q or V stay constant?

Physics 212 Lecture 8, Slide 16

Checkpoint 4c

Q:

2

2

1

1

C

Q

C

Q 2

2

11 Q

C

CQ

U: 2

121

1 VCU 2

221

2 VCU 2

2

11 U

C

CU

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.

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.

V must be the same !

Physics 212 Lecture 8, Slide 17

Calculation

An air-gap capacitor, having capacitance C0 and width x0 is connected to a 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?

QC

V

k V

C0

x0

V

x0/4

What changes when the dielectric added?

(A) Only C (B) only Q (C) only V (D) C and Q (E) V and Q

Adding dielectric changes the physical capacitor C changes

V does not change and C changes Q changes

38

Physics 212 Lecture 8, Slide 18

Calculation

k V

C0

x0

V

(A) Vleft < Vright (B) Vleft = Vright (C) Vleft > Vright

How does potential difference change ?

k Vleft Vright

The conducting plate is an equipotential !

x0/4

40

Physics 212 Lecture 8, Slide 19

Calculation

Consider capacitor to be two capacitances, C1 and C2, in parallel:

k V

C0

x0

V

(A) C1 = C0 (B) C1 = 3/4C0 (C) C1 = 4/3C0 (D) C1 = 1/4C0

What is C1 ?

In general. For parallel plate capacitor: C = e0A/d

k = k C1

C2

A = 3/4A0

d = d0 C1 = 3/4 (e0A0/d0)

C1 = 3/4C0

x0/4

43

Physics 212 Lecture 8, Slide 20

Calculation

k V

C0

x0

V

(A)C2 = kC0 (B) C2 = 3/4 kC0 (C) C2 = 4/3 kC0 (D) C2 = 1/4 kC0

What is C2 ?

k

In general. For parallel plate capacitor filled with dielectric: C = ke0A/d

= k C1

C2

A = 1/4A0

d = d0 C = ¼(ke0A0/d0)

C1 = 3/4C0

C2 = 1/4 kC0

x0/4

45

Physics 212 Lecture 8, Slide 21

Calculation

k V

C0

x0

V

(A) (B) (C)

What is C?

k

C = parallel combination of C1 and C2: C = C1 + C2

= k C1

C2

C2 = 1/4 kC0

C

C = C0 (3/4 + 1/4 k)

x0/4

C1 = 3/4C0

46

1 2C C C 1

1 2

1 1C

C C

1 2C C Ck

Physics 212 Lecture 8, Slide 22

Calculation

k V

C0

x0

V

What is Q?

k = k C1

C2 C

Q VC

x0/4

C2 = 1/4 kC0 C1 = 3/4C0

C = C0 (3/4 + 1/4 k)

50

0

3 1

4 4fQ VC k

Physics 212 Lecture 8, Slide 23

Different Problem An air-gap capacitor, having capacitance C0 and width x0 is connected to a battery of voltage V and then battery is disconnected.

A dielectric (k) of width 1/4x0 is inserted into the gap as shown.

What is Vf, the final voltage on the capacitor? Q

k V

C0

x0

V

1/4x0

Q stays same: no way to add or subtract

(A) Vf < V (B) Vf = V (C) Vf > V

Q = Q0 = C0V

Q0

We know C: (property of capacitor) C = C0 (3/4 + 1/4 k)

Vf = Q/C = V/(3/4 + 1/4 k)

Physics 212 Lecture 8, Slide 24

Different Problem

k V

C0

x0

V

1/4x0

Q0

How did energy stored in capacitor change when dielectric inserted?

(A) U increased (B) U stayed same (C) U decreased

U = ½ Q2/C Q remained same C increased

U decreased

An air-gap capacitor, having capacitance C0 and width x0 is connected to a battery of voltage V and then battery is disconnected. A dielectric (k) of width 1/4x0 is inserted into the gap as shown. What is Vf, the final voltage on the capacitor?

Vf = Q/C = V/(3/4 + 1/4 k)