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 =

(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 =