HW7 Solutions
Notice numbers may change randomly in your assignments and
you may have to recalculate solutions for your specific case.
Tipler 24.P.021
(a) Find the energy stored in a 20.00 nF capacitor when it is
charged to 5.00 µC.
(b) How much additional energy is required to increase the
charge from 5.00 to 10.00 µC?
Solution:
Tipler 24.P.025
A parallel-plate capacitor with plates of area 500.00 cm2 and is
connected across the terminals of a battery. After some time
has passed, the capacitor is disconnected from the battery.
When the plates are then moved 0.40 cm farther apart, the
charge on each plate remains constant but the potential
difference between the plates increases by 100 V.
(a) What is the magnitude of the charge on each plate?
(b) Do you expect the energy stored in the capacitor to
increase, decrease, or remain constant as the plates are
moved this way? Explain your answer.
(c) Support your answer to Part (b) by determining the change
in stored energy in the capacitor due to the movement of the
plates.
Solution:
Tipler 24.P.034
For the circuit shown in the figure below, (V = 10.0 V)
(a) Find the equivalent capacitance between the terminals
(b) Find the charge stored on the positively charged plate of
each capacitor.
(c) Find the voltage across each capacitor
(d) Find the total stored energy.
Solution:
Tipler 24.P.035
Five identical capacitors of capacitance C0 are connected in a
so-called bridge network as shown in the figure below.
(a) What is the equivalent capacitance between points a and
b?
(b) Find the equivalent capacitance between points a and b if
the capacitor at the center is replaced by a capacitor that has
a capacitance of 10C0.
Solution:
Tipler 24.P.045
Three concentric, thin, long conducting cylindrical shells have
radii of 2.00 mm, 6.00 mm, and 7.00 mm. The space between
the shells is filled with air. The innermost and outermost
cylinders are connected at one end by a conducting wire. Find
the capacitance per unit length of this configuration.
Solution:
Tipler 24.P.051
An isolated conducting sphere of radius R has a charge Q
distributed uniformly over its surface. Find the distance R' from
the center of the sphere such that half the total electrostatic
energy of the system is associated with the electric field
beyond that distance. (Use k, Q, and R as necessary.)
Solution:
Tipler 24.P.079
A parallel combination of two identical 2.0 µF parallel-plate
capacitors is connected to a 100.0 V battery. The battery is
then removed and the separation between the plates of one of
the capacitors is doubled. Find the charge on each of the
capacitors.
Solution:
Tipler 24.P.064
A parallel-plate capacitor has plates separated by a distance
d. The capacitance of this capacitor is C0 when no dielectric is
in the space between the plates. However, the space between
the plates is completely filled with two different dielectrics. One
dielectric has a thickness 1/4 d and a dielectric constant κ1,
and the other dielectric has a thickness 3/4 d and a dielectric
constant κ2. Find the capacitance of this capacitor.
Solution:
Problems for Practice
Tipler 24.P.028
Three capacitors are connected in a triangle as shown in the
figure. Find an expression for the equivalent capacitance
between points a and c in terms of the three capacitance
values.
Solution:
Tipler 24.P.067
The membrane of the axon of a nerve cell is a thin cylindrical
shell of radius r = 1.00 x 10-5 m, length L = 10.00 cm, and
thickness d = 10.00 nm. The membrane has a positive charge
on one side and a negative charge on the other, and acts as a
parallel-plate capacitor of area A = 2πrL and separation d. Its
dielectric constant is about κ = 3.00.
(a) Find the capacitance of the membrane.
(b) If the potential difference across the membrane is 70.00
mV, find the charge on each side of the membrane.
(c) Find the electric field through the membrane.
Solution:
Tipler 24.P.088
You are an intern at an engineering company that makes
capacitors used for energy storage in pulsed lasers. Your
manager asks your team to construct a parallel-plate, air-gap
capacitor that will store 80 kJ of energy.
(a) What minimum volume is required between the plates of
the capacitor?
(b) Suppose you have developed a dielectric that has a
dielectric strength of 3.00 108 V/m and has a dielectric
constant of 5.40. What volume of this dielectric, between the
plates of the capacitor, is required for it to be able to store 80
kJ of energy?
Solution:
Tipler& Llewellyn 4.P.06
A gold foil of thickness 0.80 µm is used in a Rutherford experiment to scatter α
particles with energy 7.0 MeV.
(a) What fraction of the particles will be scattered at angles greater than 90
degrees?
(b) What fraction will be scattered at angles between 45 and 71 degrees?
Solution:
2
(a)
The fraction scattered is f = ! (b(" )) nt , with
n = ! N A M = 5.90 " 10 28 atoms m 3 and b(! ) =
kq" Q
!
cot . As
2
m" v
2
2(79)ke2
90! 2(79)(1.44eV "10 #9 m)
b(90 ) =
cot
=
= 1.625 $ 10 #14 m ,
6
2K!
2
2(7 $ 10 eV)
!
f = ! (1.625 " 10 #14 )2 (5.90 " 10 #28 )(0.8 " 10 #6 ) = 3.92 " 10 #5
(b)
The fraction scattered in this range are
f = ! (b(45! ))2 nt " ! (b(71! ))2 nt = 1.51 # 10 "4 .
Tipler& Llewellyn 4.P.10
What energy α particle would be needed just to reach the surface of a Pb
nucleus if its radius is 5.21 fm?
Solution:
The distance of closest approach is rd =
kq! Q
. As Pb has Z=82, the kinetic
K!
energy is K! =
kq! Q 2Z(1.44eV "10 #9 m)
=
= 45.3MeV .
rd
5.21 $ 10 #15 m
Tipler& Llewellyn 4.P.10
(a) Compute the radius of the n=7 orbit in hydrogen.
(b) Compute the radius of the n=7 orbit of singly ionized helium (He+) ,
which is hydrogenlike.
Solution:
n 2 a0
.
Z
2
(a) The n=7 orbit of hydrogen is r7 = 7 a0 = 2.6nm .
7 2 a0
= 1.3nm .
(b) The n=7 orbit of singly ionized helium is r7 =
2
The Bohr orbits are given by rn =