PHYS 222
Worksheet 9 – Dielectrics
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Alek Jerauld
PHYS 222
Dr. Paula Herrera-Siklódy
2/6/12
Useful Equations
Q2 1
1
 CV 2  QV
2C 2
2
Ck  kC0


 ind   0 1 1 
k

U
  k 0
Stored energy in capacitor. Needs external work to charge it
Capacitance with dielectric material. k > 1, k = 1 vacuum
Charge density induced on dielectric
Permittivity of a dielectric
Related Problems
1) A parallel-plate air capacitor is made by using two plates 16 cm square, spaced 4.7 mm
apart. It is connected to a 12-V battery. (Book 24.50)
(a) What is the capacitance?
C  0
(0.16)2
A
 (8.85)(1012 )
 48.2 pF
d
4.7(103 )
(b) What is the charge on each plate?
Q  CV  48.2(1012 )(12)  0.578 nC
(c) What is the energy stored in the capacitor?
CV 2
u
 3.47 nJ
2
2) Two capacitors are connected parallel to each other and connected to the battery with
voltage 28 V. Let C1 = 9.5 µF, C2 = 4.6 µF be their capacitances. Suppose the charged
capacitors are disconnected from the source and from each other, and then reconnected to
each other with plates of opposite sign together. By how much does the energy of the
system decrease? (Book 24.56)
ui 
Q 2 (Q1  Q2 )2 (C1V  C2V )2 (C1  C2 )V 2



2C 2(C1  C2 )
2(C1  C2 )
2
Q 2 (C1V  C2V )2
uf 

2C
2(C1  C2 )
(C1V  C2V )2 (C1  C2 )V 2
u  u f  ui 

 4.9(103 ) J
2(C1  C2 )
2
3) A parallel-plate capacitor has the space between the
plates filled with two slabs of dielectric, one with
constant K1 and one with constant K2. Each plate has
area A. What is the capacitance?
This capacitor can be split into two parallel capacitors
each with a different dielectric constant. The area of
each of these capacitors will be A/2 while the distance
remains the same as before:
A
2d
A
C2  K 2 0
2d
C1  K1 0
Ceq  C1  C2  K1 0
A
A
A
 K 2 0
  K1  K 2   0
2d
2d
2d
4) A parallel-plate capacitor with only air between the plates is charged by connecting it to a
battery. The capacitor is then disconnected from the battery, without any of the charge
leaving the plates. A voltmeter reads 46.0 V when placed across the capacitor. When a
dielectric is inserted between the plates, completely filling the space, the voltmeter reads
15.0 V. (Book 24.65)
(a) What is the dielectric constant of this material?
Once the battery is unhooked from the capacitor, the capacitor is now a stand-alone
capacitor, thus the charge is conserved:
Q0
V0
Q
 0
Vf
C0 
Ck
Ck  kC0
Q Q
V 46
 k 0  0  k  0   3.07
V0 V f
V f 15
(b) What will the voltmeter read if the dielectric is now pulled partway out so it fills only onethird of the space between the plates?
2A 2
 C
3d 3 0
1A 1
C2  k 0
 kC
3d 3 0
2 1 
Ceq  C0   k 
3 3 
Q
Q0
V 0 
Ceq


C0  2  1 k 
3 3 
Q0
V0
46
 V0  V 

 27.2V
C0
2 1  2 1

  k    3.07 
3 3  3 3

C1   0
5) A parallel-plate capacitor has the space between the
plates filled with two slabs of dielectric, one with constant K1
and one with constant K2 (the figure ). Each slab has
thickness d/2, where d is the plate separation. What is the
capacitance? (Book 24.71)
The electric field for capacitor with area A:
E0 
Q
0 A
When the capacitor is stand-alone:
E0
Q

K1 K1 0 A
E
Q
E2  0 
K 2 K 2 0 A
E1 
V    Edl
Qd
d

2 2 K1 0 A
Qd
d
 Vbc  E2 
2 2 K 2 0 A
 Vab  E1
Vac  Vab  Vbc 
C
 1
Qd
Qd
1  Qd

  
2K1 0 A 2 K 2 0 A  K1 K 2  2 0 A
2 AK K
Q
Q

 0 1 2
Vac  1
d ( K1  K 2 )
 Qd

 1 
K
K 2  2 0 A
 1