College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 16

Electrical Energy and Capacitance - all with Video Answers

Educators

Chapter Questions

Problem 1

A uniform electric field of magnitude $375 \mathrm{~N} / \mathrm{C}$ pointing in the positive $x$ -direction acts on an electron, which is initially at rest. After the electron has moved $3.20 \mathrm{~cm}$, what
s. (a) the work done by the field on the electron, (b) the change in potential energy associated with the electron, and (c) the velocity of the electron?

Sheh Lit Chang

Sheh Lit Chang

University of Washington

Problem 2

A uniform electric field of magnitude $327 \mathrm{~N} / \mathrm{C}$ is directed along the $+y$ -axis. A $5.40-\mu \mathrm{C}$ charge moves from the origin to the point $(x, y)=(-15.0 \mathrm{~cm},-32.0 \mathrm{~cm})$. (a) What is the change in the potential energy associated with this charge? (b) Through what potential difference did the charge move?

Sheh Lit Chang

University of Washington

Problem 3

A potential difference of $90 \mathrm{mV}$ exists between the inner and outer surfaces of a cell membrane. The inner surface is negative relative to the outer surface. How much work is required to eject a positive sodium ion (Na $^{+}$ ) from the interior of the cell?

Sheh Lit Chang

University of Washington

Problem 4

An ion accelerated through a potential difference of $60.0 \mathrm{~V}$ has its potential energy decreased by $1.92 \times 10^{-17} \mathrm{~J} .$ Calculate the charge on the ion.

Sheh Lit Chang

University of Washington

Problem 5

The potential difference between the accelerating plates of a TV set is about $25 \mathrm{kV}$. If the distance between the plates is $1.5 \mathrm{~cm}$, find the magnitude of the uniform electric field in the region between the plates.

Sheh Lit Chang

University of Washington

Problem 6

To recharge a 12-V battery, a battery charger must moye $3.6 \times 10^{5} \mathrm{C}$ of charge from the negative terminal to the positive terminal. How much work is done by the charger? Express your answer in joules.

Sheh Lit Chang

University of Washington

Problem 7

Oppositely charged parallel plates are separated by $5.33 \mathrm{~mm}$. A potential difference of $600 \mathrm{~V}$ exists between the plates. (a) What is the magnitude of the electric field between the plates? (b) What is the magnitude of the force on an electron between the plates? (c) How much work must be done on the electron to move it to the negative plate if it is initially positioned $2.90 \mathrm{~mm}$ from the positive plate?

Salamat Ali

Numerade Educator

Problem 8

(a) Find the potential difference $\Delta V_{e}$ required to stop an electron (called a "stopping potential") moving with an initial speed of $2.85 \times 10^{7} \mathrm{~m} / \mathrm{s}$. (b) Would a proton traveling at the same speed require a greater or lesser magnitude potential difference? Explain. (c) Find a symbolic expression for the ratio of the proton stopping potential and the electron stopping potential, $\Delta V_{p} / \Delta V_{e}$. The answer should be in terms of the proton mass $m_{p}$ and electron mass $m_{e}$.

Ben Nicholson

Numerade Educator

Problem 9

A $74.0-\mathrm{g}$ block carrying a charge $Q=35.0 \mu \mathrm{C}$ is connected to a spring for which $k=78.0 \mathrm{~N} / \mathrm{m}$. The block lies on a frictionless, horizontal surface and is immersed in a uniform electric field of magnitude $E=4.86 \times 10^{4} \mathrm{~N} / \mathrm{C}$ directed as shown in Figure $\mathrm{P} 16.9 .$ If the block is released from rest when the spring is unstretched $(x=0)$, (a) by what maximum distance does the block move from its initial position? (b) Find the subsequent equilibrium position of the block and the amplitude of its motion.
(c) Using conservation of energy, find a symbolic relationship giving the potential difference between its initial position and the point of maximum extension in terms of the spring constant $k$, the amplitude $A$, and the charge $Q$

Sheh Lit Chang

University of Washington

Problem 10

On planet Tehar, the free-fall acceleration is the same as that on the Earth, but there is also a strong downward electric field that is uniform close to the planet's surface. A $2.00-\mathrm{kg}$ ball having a charge of $5.00 \mu \mathrm{C}$ is thrown upward at a speed of $20.1 \mathrm{~m} / \mathrm{s}$. It hits the ground after an interval of $4.10 \mathrm{~s}$. What is the potential difference between the starting point and the top point of the trajectory?

Ben Nicholson

Numerade Educator

Problem 11

An electron is at the origin. (a) Calculate the electric potential $V_{A}$ at point $A, x=0.250 \mathrm{~cm} .$ (b) Calculate the electric potential $V_{B}$ at point $B, x=0.750 \mathrm{~cm} .$ What is the potential difference $V_{B}-V_{A} ?$ (c) Would a negatively charged particle placed at point $A$ necessarily go through this same potential difference upon reaching point $B ?$ Explain.

Salamat Ali

Numerade Educator

Problem 12

Two point charges are on the $y$ -axis. A $4.50-\mu \mathrm{C}$ charge is located at $y=1.25 \mathrm{~cm}$, and a $-2.24-\mu \mathrm{C}$ charge is located at $y=-1.80 \mathrm{~cm} .$ Find the total electric potential at (a) the origin and (b) the point having coordinates $(1.50 \mathrm{~cm}, 0)$.

Sheh Lit Chang

University of Washington

Problem 13

(a) Find the electric potential, taking zero at infinity, at the upper right corner (the corner without a charge) of the rectangle in Figure P16.13. (b) Repeat if the $2.00-\mu \mathrm{C}$ charge is replaced with a charge of $-2.00 \mu \mathrm{C}$.

Salamat Ali

Numerade Educator

Problem 14

Three charges are situated at corners of a rectangle as in Figure P16.13. How much energy would be expended in moving the $8.00-\mu \mathrm{C}$ charge to infinity?

Sheh Lit Chang

University of Washington

Problem 15

Two point charges $Q_{1}=+5.00 \mathrm{n} \mathrm{C}$ and $Q_{2}=-3.00$
$\mathrm{n} \mathrm{C}$ are separated by $35.0 \mathrm{~cm} .$ (a) What is the electric potential at a point midway between the charges? (b) What is the potential energy of the pair of charges? What is the significance of the algebraic sign of your answer?

Salamat Ali

Numerade Educator

Problem 16

A point charge of $9.00 \times 10^{-9} \mathrm{C}$ is located at the origin. How much work is required to bring a positive charge of $3.00 \times 10^{-9} \mathrm{C}$ from infinity to the location $x=30.0 \mathrm{~cm} ?$

Sheh Lit Chang

University of Washington

Problem 17

The three charges in Figure $P 16.17$ are at the vertices of an isosceles triangle. Let $q=7.00 \mathrm{nC}$ and calculate the electric potential at the midpoint of the base.

Salamat Ali

Numerade Educator

Problem 18

An electron starts from rest $3.00 \mathrm{~cm}$ from the center of a uniformly charged sphere of radius $2.00 \mathrm{~cm}$. If the sphere carries a total charge of $1.00 \times 10^{-9} \mathrm{C}$, how fast will the electron be moving when it reaches the surface of the sphere?

Sheh Lit Chang

University of Washington

Problem 19

A proton is located at the origin, and a second proton is located on the $x$ -axis at $x=6.00 \mathrm{fm}\left(1 \mathrm{fm}=10^{-15} \mathrm{~m}\right)$.
(a) Calculate the electric potential energy associated with this configuration. (b) An alpha particle (charge $=2 e$, mass $=6.64 \times 10^{-27} \mathrm{~kg}$ ) is now placed at $(x, y)=(3.00$, 3.00) $\mathrm{fm}$. Calculate the electric potential energy associated with this configuration. (c) Starting with the threeparticle system, find the change in electric potential energy if the alpha particle is allowed to escape to infinity while the two protons remain fixed in place. (Throughout, neglect any radiation effects.) (d) Use conservation of energy to calculate the speed of the alpha particle at infinity. (e) If the two protons are released from rest and the alpha particle remains fixed, calculate the speed of the protons at infinity.

Sheh Lit Chang

University of Washington

Problem 20

A proton and an alpha particle (charge $=2 e$, mass $=6.64 \times 10^{-27} \mathrm{~kg}$ ) are initially at rest, separated by $4.00 \times 10^{-15} \mathrm{~m} .$ (a) If they are both released simultaneously, explain why you can't find their velocities at infinity using only conservation of energy.
(b) What other conservation law can be applied in this case? (c) Find the speeds of the proton and alpha particle, respectively, at infinity.

Ben Nicholson

Numerade Educator

Problem 21

A small spherical object carries a charge of $8.00 \mathrm{nC}$. At what distance from the center of the object is the potential equal to $100 \mathrm{~V}$ ? $50.0$ V? $25.0 \mathrm{~V}$ ? Is the spacing of the equipotentials proportional to the change in voltage?

Sheh Lit Chang

University of Washington

Problem 22

Starting with the definition of work, prove that the local electric field must be everywhere perpendicular to a surface having the same potential at every point.

Sheh Lit Chang

University of Washington

Problem 23

In Rutherford's famous scattering experiments that led to the planetary model of the atom, alpha particles (having charges of $+2 e$ and masses of $6.64 \times 10^{-27} \mathrm{~kg}$ ) were fired toward a gold nucleus with charge $+79 e$. An alpha particle, initially very far from the gold nucleus, is fired at $2.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$ directly toward the nucleus, as in Figure P16.23. How close does the alpha particle get to the gold nucleus before turning around? Assume the gold nucleus remains stationary.

Salamat Ali

Numerade Educator

Problem 24

Four point charges each having charge $Q$ are located at the corners of a square having sides of length $a$. Find symbolic expressions for (a) the total electric potential at the center of the square due to the four charges and
(b) the work required to bring a fifth charge $q$ from infinity to the center of the square.

Ben Nicholson

Numerade Educator

Problem 25

Consider the Earth and a cloud layer $800 \mathrm{~m}$ above the planet to be the plates of a parallel-plate capacitor. (a) If the cloud layer has an area of $1.0 \mathrm{~km}^{2}=1.0 \times 10^{6} \mathrm{~m}^{2}$, what is the capacitance? (b) If an electric field strength greater than $3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}$ causes the air to break down and conduct charge (lightning), what is the maximum charge the cloud can hold?

Sheh Lit Chang

University of Washington

Problem 26

(a) When a 9.00-V battery is connected to the plates of a capacitor, it stores a charge of $27.0 \mu \mathrm{C}$. What is the value of the capacitance? (b) If the same capacitor is connected to a $12.0-V$ battery, what charge is stored?

Sheh Lit Chang

University of Washington

Problem 27

An air-filled parallel-plate capacitor has plates of area $2.30 \mathrm{~cm}^{2}$ separated by $1.50 \mathrm{~mm}$. The capacitor is connected to a $12.0-V$ battery.
(a) Find the value of its capacitance. (b) What is the charge on the capacitor? (c) What is the magnitude of the uniform electric field between the plates?

Salamat Ali

Numerade Educator

Problem 28

(a) How much charge is on each plate of a $4.00-\mu \mathrm{F}$ capacitor when it is connected to a $12.0-\mathrm{V}$ battery? (b) If this same capacitor is connected to a $1.50-\mathrm{V}$ battery, what charge is stored?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 29

An air-filled capacitor consists of two parallel plates, each with an area of $7.60 \mathrm{~cm}^{2}$ and separated by a distance of $1.80 \mathrm{~mm}$. If a $20.0-\mathrm{V}$ potential difference is applied to these plates, calculate (a) the electric field between the plates, (b) the capacitance, and (c) the charge on each plate.

Salamat Ali

Numerade Educator

Problem 30

A 1-megabit computer memory chip contains many $60.0 \times 10^{-15}-\mathrm{F}$ capacitors. Each capacitor has a plate area of $21.0 \times 10^{-12} \mathrm{~m}^{2}$. Determine the plate separation of such a capacitor. (Assume a parallel-plate configuration.) The diameter of an atom is on the order of $10^{-10} \mathrm{~m}=1 \AA$. Express the plate separation in angstroms.

Ben Nicholson

Numerade Educator

Problem 31

A parallel-plate capacitor with area $0.200 \mathrm{~m}^{2}$ and plate separation of $3.00 \mathrm{~mm}$ is connected to a $6.00-\mathrm{V}$ battery. (a) What is the capacitance? (b) How much charge is stored on the plates? (c) What is the electric field between the plates? (d) Find the magnitude of the charge density on each plate. (e) Without disconnecting the battery, the plates are moved farther apart. Qualitatively, what happens to each of the previous answers?

Salamat Ali

Numerade Educator

Problem 32

A small object. with a mass of $350 \mathrm{mg}$ carries a charge of $30.0 \mathrm{nC}$ and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plates are separated by $4.00 \mathrm{~cm}$. If the thread makes an angle of $15.0^{\circ}$ with the vertical, what is the potential difference between the plates?

Sheh Lit Chang

University of Washington

Problem 33

Given a $2.50-\mu \mathrm{F}$ capacitor, a $6.25-\mu \mathrm{F}$ capacitor, and a $6.00-\mathrm{V}$ battery, find the charge on each capacitor if you connect them (a) in series across the battery and (b) in parallel across the battery.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 34

Find the equivalent capacitance of a $4.20-\mu \mathrm{F}$ capacitor and an $8.50-\mu \mathrm{F}$ capacitor when they are connected (a) in series and (b) in parallel.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 35

Find (a) the equivalent capacitance of the capacitors in Figure $\mathrm{P} 16.35$, (b) the charge on each capacitor, and
(c) the potential difference across each capacitor.

Salamat Ali

Numerade Educator

Problem 36

Two capacitors give an equivalent capacitance of $9.00 \mathrm{pF}$ when connected in parallel and an equivalent capacitance of $2.00 \mathrm{pF}$ when connected in series. What is the capacitance of each capacitor?

Ben Nicholson

Numerade Educator

Problem 37

For the system of capacitors shown in Figure $\mathrm{P} 16.37$, find
(a) the equivalent capacitance of the system, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.

Salamat Ali

Numerade Educator

Problem 38

GP Consider the combination of capacitors in Figure P16.38. (a) Find the equivalent single capacitance of the two capacitors in series and redraw the diagram (called diagram 1) with this equivalent capacitance. (b) In diagram 1 find the equivalent capacitance of the three capacitors in parallel and redraw the diagram as a single battery and single capacitor in a loop. (c) Compute the charge on the single equivalent capacitor. (d) Returning to diagram 1, compute the charge on each individual capacitor. Does the sum agree with the value found in part (c)? (e) What is the charge on the $24.0-\mu \mathrm{F}$ capacitor and on the $8.00-\mu \mathrm{F}$ capacitor? (f) Compute the voltage drop across the $24.0-\mu \mathrm{F}$ capacitor and $(\mathrm{g})$ the $8.00-\mu \mathrm{F}$ capacitor.

Sheh Lit Chang

University of Washington

Problem 39

Find the charge on each of the capacitors in Figure $\mathrm{P} 16.39 .$

Sheh Lit Chang

University of Washington

Problem 40

A $10.0-\mu \mathrm{F}$ capacitor is fully charged across a $12.0 \mathrm{-V}$ batLery. The capacitor is then disconnected from the battery and connected across an initially uncharged capacitor with capacitance $C$. The resulting voltage across each capacitor is $3.00 \mathrm{~V}$. What is the value of $C$ ?

Sheh Lit Chang

University of Washington

Problem 41

A $25.0-\mu \mathrm{F}$ capacitor and a $40.0-\mu \mathrm{F}$ capacitor are charged by being connected across separate $50.0-\mathrm{V}$ batteries.
(a) Determine the resulting charge on each capacitor.
(b) The capacitors are then disconnected from their batteries and connected to each other, with each negative plate connected to the other positive plate. What is the final charge of each capacitor, and what is the final potential difference across the $40.0-\mu \mathrm{F}$ capacitor?

Sheh Lit Chang

University of Washington

Problem 42

(a) Find the equivalent capacitance between points a and $b$ for the group of capacitors connected as shown in Figure $\mathrm{P} 16.42$ if $C_{1}=5.00 \mu \mathrm{F}, C_{2}=10.00 \mu \mathrm{F}$, and $C_{3}=$
$2.00 \mu \mathrm{F} .$ (b) If the potential between points $a$ and $b$ is $60.0 \mathrm{~V}$, what charge is stored on $C_{\mathrm{g}}^{?}$

Sheh Lit Chang

University of Washington

Problem 43

A $1.00-\mu \mathrm{F}$ capacitor is charged by being connected across a $10.0-\mathrm{V}$ battery. It is then disconnected from the battery and connected across an uncharged $2.00-\mu \mathrm{F}$ capacitor. Determine the resulting charge on each capacitor.

Salamat Ali

Numerade Educator

Problem 44

Find the equivalent capacitance between points $a$ and $b$ in the combination of capacitors shown in Figure P16.44.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 45

A $12.0-\mathrm{V}$ battery is connected to a $4.50-\mu \mathrm{F}$ capacitor. How much energy is stored in the capacitor?

Salamat Ali

Numerade Educator

Problem 46

Two capacitors, $C_{1}=18.0 \mu \mathrm{F}$ and $C_{2}=36.0 \mu \mathrm{F}$, are connected in series, and a $12.0-V$ battery is connected across them. (a) Find the equivalent capacitance, and the energy contained in this equivalent capacitor. (b) Find the energy stored in each individual capacitor. Show that the sum of these two energies is the same as the energy found in part (a). Will this equality always be true, or does it depend on the number of capacitors and their capacitances? (c) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy as in part (a)? Which capacitor stores more energy in this situation, $C_{1}$ or $C_{2} ?$

Ben Nicholson

Numerade Educator

Problem 47

A parallel-plate capacitor has capacitance $3.00 \mu \mathrm{F}$. (a) How much energy is stored in the capacitor if it is connected to a $6.00-\mathrm{V}$ battery? (b) If the battery is disconnected and the distance between the charged plates doubled, what is the energy stored? (c) The battery is subsequently reattached to the capacitor, but the plate separation remains as in part (b). How much energy is stored? (Answer each part in microjoules.)

Salamat Ali

Numerade Educator

Problem 48

A certain storm cloud has a potential difference of $1.00 \times 10^{8} \mathrm{~V}$ relative to a tree. If, during a lightning storm, $50.0 \mathrm{C}$ of charge is transferred through this potential difference and $1.00 \%$ of the energy is absorbed by the tree, how much water (sap in the tree) initially at $30.0^{\circ} \mathrm{C}$ can be boiled away? Water has a specific heat of $4186 \mathrm{~J} / \mathrm{kg}^{\circ} \mathrm{C}$, a boiling point of $100^{\circ} \mathrm{C}$, and a heat of vaporization of $2.26 \times 10^{6} \mathrm{~J} / \mathrm{kg}$

Sheh Lit Chang

University of Washington

Problem 49

The voltage across an air-filled parallel-plate capacitor is measured to be $85.0 \mathrm{~V}$. When a dielectric is inserted and completely fills the space between the plates as in Figure $16.24$, the voltage drops to $25.0 \mathrm{~V}$. (a) What is the dielectric constant of the inserted material? Can you identify the dielectric? (b) If the dielectric doesn't completely fill the space between the plates, what could you conclude about the voltage across the plates?

Sheh Lit Chang

University of Washington

Problem 50

A parallel-plate capacitor in air has a plate separation of $1.50 \mathrm{~cm}$ and a plate area of $25.0 \mathrm{~cm}^{2}$. The plates are charged to a potential difference of $2.50 \times 10^{2} \mathrm{~V}$ and disconnected from the source. The capacitor is then immersed in distilled water. Determine (a) the charge on the plates before and after immersion, (b) the capacitance and potential difference after immersion, and
(c) the change in energy stored in the capacitor due to immersion. Assume the distilled water is an insulator.

Sheh Lit Chang

University of Washington

Problem 51

Determine (a) the capacitance and (b) the maximum voltage that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of $175 \mathrm{~cm}^{2}$ and an insulation thickness of $0.0400 \mathrm{~mm}$.

Sheh Lit Chang

University of Washington

Problem 52

A commercial capacitor is constructed as in Figure $16.26 \mathrm{a}$. This particular capacitor is made from a strip of aluminum foil separated by two strips of paraffin-coated paper. Each strip of foil and paper is $7.00 \mathrm{~cm}$ wide. The foil is $0.00400 \mathrm{~mm}$ thick, and the paper is $0.0250 \mathrm{~mm}$ thick and has a dielectric constant of $3.70 .$ What length should the strips be if a capacitance of $9.50 \times 10^{-8} \mathrm{~F}$ is desired

Sheh Lit Chang

University of Washington

Problem 53

A model of a red blood cell portrays the cell as a shperical capacitor, a positively charged liquid sphere of surface area A separated from the surrounding negatively charged fluid by a membrane of thickness $t$. Tiny electrodes introduced into the interior of the cell show a potential difference of $100 \mathrm{mV}$ across the membrane. The membrane's thickness is estimated to be $100 \mathrm{~nm}$ and has a dielectric constant of $5.00$. (a) If an average red blood cell has a mass of $1.00 \times 10^{-12} \mathrm{~kg}$, estimate the volume of the cell and thus find its surface area. The density of blood is $1100 \mathrm{~kg} / \mathrm{m}^{3}$.
(b) Estimate the capacitance of the cell by assuming the membrane surfaces act as parallel plates. (c) Calculate the charge on the surface of the membrane. How many electronic charges does the surface charge represent?

Linda Winkler

Numerade Educator

Problem 54

Three parallel-plate capacitors are constructed, each having the same plate spacing $d$ and with $C_{1}$ having plate area $A_{1}, C_{2}$ having area $A_{2}$, and $C_{3}$ having area $A_{3}$. Show that the total capacitance $C$ of the three capacitors connected in parallel is the same as that of a capacitor having plate spacing $d$ and plate area $A=A_{1}+A_{2}+A_{3}$.

Sheh Lit Chang

University of Washington

Problem 55

Three parallel-plate capacitors are constructed, each having the same plate area $A$ and with $C_{1}$ having plate spacing $d_{1}, C_{2}$ having plate spacing $d_{2}$, and $C_{3}$ having plate spacing $d_{3}$. Show that the total capacitance $C$ of the three capacitors connected in series is the same as a capacitor of plate area $A$ and with plate spacing $d=d_{1}+d_{2}+d_{3}$.

Salamat Ali

Numerade Educator

Problem 56

For the system of four capacitors shown in Figure $\mathrm{P} 16.37$, find (a) the total energy stored in the system and
(b) the energy stored by each capacitor. (c) Compare the sum of the answers in part (b) with your result to part (a) and explain your observation.

Sheh Lit Chang

University of Washington

Problem 57

A parallel-plate capacitor with a plate separation $d$ has a capacitance $C_{0}$ in the absence of a dielectric. A slab of dielectric material of dielectric constant $\kappa$ and thickness $d / 3$ is then inserted between the plates as in Figure P16.57. Show that the capacitance of this partially filled capacitor is given by
$$
C=\left(\frac{3 \kappa}{2 k+1}\right) C_{0}
$$
(Hint: Treat the system as two capacitors connected in series, one with dielectric in it and the other one empty.)

Sheh Lit Chang

University of Washington

Problem 58

Two capacitors give an equivalent capacitance of $C_{p}$ when connected in parallel and an equivalent capacitance of $C_{s}$ when connected in series. What is the capacitance of each capacitor?

Ben Nicholson

Numerade Educator

Problem 59

An isolated capacitor of unknown capacitance has been charged to a potential difference of $100 \mathrm{~V}$. When the charged capacitor is disconnected from the battery and then connected in parallel to an uncharged $10.0-\mu \mathrm{F}$ capacitor, the voltage across the combination is measured to be $30.0 \mathrm{~V}$. Calculate the unknown capacitance.

Sheh Lit Chang

University of Washington

Problem 60

Two charges of $1.0 \mu \mathrm{C}$ and $-2.0 \mu \mathrm{C}$ are $0.50 \mathrm{~m}$ apart at two vertices of an equilateral triangle as in Figure $\mathrm{P} 16.60 .$
(a) What is the electric potential due to the $1.0-\mu \mathrm{C}$ charge at the third vertex, point $P ?$ (b) What is the electric potential due to the $-2.0-\mu \mathrm{C}$ charge at $P ?$ (c) Find the total electric potential at $P$. (d) What is the work required to move a $3.0-\mu \mathrm{C}$ charge from infinity to $P$.

Linda Winkler

Numerade Educator

Problem 61

Find the equivalent capacitance of the group of capacitors shown in Figure $\mathrm{P} 16.61 .$

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 62

A spherical capacitor consists of a spherical conducting shell of radius $b$ and charge $-Q$ concentric with a smaller conducting sphere of radius $a$ and charge Q. (a) Find the capacitance of this device. (b) Show that as the radius $b$ of the outer sphere approaches infinity, the capacitance approaches the value $a / k_{e}=4 \pi \epsilon_{0} a$.

Vishal Gupta

Numerade Educator

Problem 63

The immediate cause of many deaths is ventricular fibrillation, an uncoordinated quivering of the heart, as opposed to proper beating. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart will sometimes start organized beating again. A defibrillator is a device that applies a strong elec-

Sheh Lit Chang

University of Washington

Problem 64

When a certain air-filled parallel-plate capacitor is connected across a battery, it acquires a charge of $150 \mu \mathrm{C}$ on each plate. While the battery connection is maintained, a dielectric slab is inserted into, and fills, the region between the plates. This results in the accumulation of an additional charge of $200 \mu \mathrm{C}$ on each plate. What is the dielectric constant of the slab?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 65

Capacitors $C_{1}=6.0 \mu \mathrm{F}$ and $C_{2}=2.0 \mu \mathrm{F}$ are charged as a parallel combination across a $250-\mathrm{V}$ battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor.

Salamat Ali

Numerade Educator

Problem 66

The energy stored in a $52.0-\mu \mathrm{F}$ capacitor is used to melt a $6.00$ -mg sample of lead. To what voltage must the capacitor be initially charged, assuming the initial temperature of the lead is $20.0^{\circ} \mathrm{C}$ ? Lead has a specific heat of $128 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, a melting point of $327.3^{\circ} \mathrm{C}$, and a latent heat of fusion of $24.5 \mathrm{~kJ} / \mathrm{kg}$

Sheh Lit Chang

University of Washington

Problem 67

Metal sphere $A$ of radius $12.0 \mathrm{~cm}$ carries $6.00 \mu \mathrm{C}$ of charge, and metal sphere $\mathrm{B}$ of radius $18.0 \mathrm{~cm}$ carries $-4.00 \mu \mathrm{C}$ of charge. If the two spheres are attached by a very long conducting thread, what is the final distribution of charge on the two spheres?

Salamat Ali

Numerade Educator

Problem 68

An electron is fired at a speed $v_{0}=5.6 \times 10^{6} \mathrm{~m} / \mathrm{s}$ and at an angle $\theta_{0}=-45^{\circ}$ between two parallel conducting plates that are $D=2.0 \mathrm{~mm}$ apart, as in Figure $\mathrm{P} 16.68$. If the voltage difference between the plates is $\Delta V=100 \mathrm{~V}$, determine (a) how close, $d$, the electron will get to the bottom plate and (b) where the electron will strike the top plate.

Linda Winkler

Numerade Educator