lecture 8 - u of i · physics 212 lecture 8, slide 2 c v +q -q q q battery has moved charge q from...
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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)