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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 16

Electrical Energy and Capacitance

Educators

AI

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 is (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?

Salamat A.
Numerade Educator

Problem 2

A proton is released from rest in a uniform electric field of magnitude 385 $\mathrm{N} / \mathrm{C}$ . Find (a) the electric force on the proton, (b) the acceleration of the proton, and (c) the distance it travels in 2.00$\mu \mathrm{s}$ .

Ben N.
Numerade Educator

Problem 3

A potential difference of $90,0 \mathrm{mV}$ 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?

Salamat A.
Numerade Educator

Problem 4

Cathode ray tubes (CRTs) used in old-style televisions have been replaced by modern LCD and LED screens. Part of the CRT included a set of accelerating plates separated by a distance of about 1.50 $\mathrm{cm} .$ If the potential difference across the plates was 25.0 $\mathrm{kV}$ , find the magnitude of the electric field in the region between the plates.

Ben N.
Numerade Educator

Problem 5

A constant electric field accelerates a proton from rest through a distance of 2.00 $\mathrm{m}$ to a speed of $1.50 \times 10^{5} \mathrm{m} / \mathrm{s}$ . (a) Find the change in the proton's kinetic energy. (b) Find the change in the system's electric potential energy. ( $\mathrm{c} )$ Calculate the magnitude of the electric field.

Salamat A.
Numerade Educator

Problem 6

A point charge $q=+40.0 \mu \mathrm{C}$ moves from $A$ to $B$ separated by a distance $d=0.180 \mathrm{m}$ in the presence of an external electric field $\overrightarrow{\mathrm{E}}$ of magnitude 275 $\mathrm{N} / \mathrm{C}$ directed toward the right as in Figure $\mathrm{P} 16.6 .$ Find (a) the electric force exerted on the charge, (b) the work done by the electric force, (c) the change in the electric potential energy of the charge, and (d) the potential difference between $A$ and $B$ .

Ben N.
Numerade Educator

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 A.
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, $\Delta V_{p} / \Delta V_{c} .$ The answer should be in terms of the proton mass $m_{p}$ and electron mass $m_{c} .$

Ben N.
Numerade Educator

Problem 9

An ionized oxygen molecule $\left(\mathrm{O}_{2}^{+}\right)$ at point $A$ has charge $+e$ and moves at $2.00 \times 10^{3} \mathrm{m} / \mathrm{s}$ in the positive $x$ -direction. A constant electric force in the negative $x$ -direction slows the molecule to a stop at point $B$ , a distance of 0.750 $\mathrm{mm}$ past $A$ on the $x$ -axis. Calculate (a) the $x$ -component of the electric field and (b) the potential difference between points $A$ and $B$ .

Salamat A.
Numerade Educator

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 -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 s. What is the potential difference between the starting point and the top
point of the trajectory?

Ben N.
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 $V_{B}-V_{A} ?(\mathrm{c})$ Would a negatively charged particle placed at point $A$ necessarily go through this same potential difference upon reaching point $B$ ? Explain.

Salamat A.
Numerade Educator

Problem 12

The two charges in Figure $\mathrm{P} 16.12$ are separated by $d=2.00$ $\mathrm{cm} .$ Find the electric potential at (a) point $A$ and $(\mathrm{b})$ point $B$ , which is halfway between the
charges.

Ben N.
Numerade Educator

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 $\mathrm{P} 16.13$ . (b) Repeat if the $2.00-\mu \mathrm{C}$ charge is replaced with a charge of $-2.00 \mu \mathrm{C}$ .

Salamat A.
Numerade Educator

Problem 14

Three charges are situated at corners of a rectangle as in Figure $\mathrm{P} 16.13$ . How much work must an external agent do to move the $8.00-\mu \mathrm{C}$ charge to infinity?

Ben N.
Numerade Educator

Problem 15

Two point charges $Q_{1}=+5.00 \mathrm{nC}$ and $Q_{2}=$ $-3.00 \mathrm{nC}$ 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 A.
Numerade Educator

Problem 16

Three identical point charges each of charge $q$ are located at the vertices of an equilateral triangle as in Figure $\mathrm{P} 16.16$ . The distance from the center of the triangle to each vertex is a. (a) Show that the electric field at the center of the triangle is zero. (b) Find a symbolic expression for the electric
potential at the center of the triangle. (c) Give a physical explanation of the fact that the electric
potential is not zero, yet the electric field is zero at the center.

Ben N.
Numerade Educator

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 A.
Numerade Educator

Problem 18

$A \quad$ positive point charge $q=$ $+2.50 \mathrm{nC}$ is located at $x=1.20 \mathrm{m}$ and
a negative charge of $-2 q=-5.00 \mathrm{nC}$ is located at the origin as in Figure $\mathrm{P} 16.18$ . (a) Sketch the electric potential versus $x$ for points along the $x$ -axis in the range $-1.50 \mathrm{m} < x < 1.50 \mathrm{m} .$ (b) Find a symbolic expression for the potential on the $x$ -axis at an arbitrary point $P$ between the two charges. (c) Find the electric potential at $x=0.600 \mathrm{m}$ . (d) Find the point along the $x$ -axis between the two charges where the electric potential is zero.

Ben N.
Numerade Educator

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 three-particle 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 twe protons are released from rest and the alpha particle remains fixed, calculate the speed of the protons at infinity.

Salamat A.
Numerade Educator

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 N.
Numerade Educator

Problem 21

A tiny sphere of mass 8.00$\mu g$ and charge $-2.80 \mathrm{nC}$ is initially at a distance of 1.60$\mu \mathrm{m}$ from a fixed charge of $+8.50 \mathrm{nC}$ . If the 8.00 -mg sphere is released from rest, find (a) its kinetic energy when it is 0.500$\mu \mathrm{m}$ from the fixed charge and (b) its
speed when it is 0.500$\mu \mathrm{m}$ from the fixed charge.

Salamat A.
Numerade Educator

Problem 22

The metal sphere of a small Van de Graaff generator illustrated in Figure 15.23 has a radius of 18 $\mathrm{cm} .$ When the electricfield at the surface of the sphere reaches $3.0 \times 10^{6} \mathrm{V} / \mathrm{m}$ , the air breaks down, and the generator discharges. What is the maximum potential the sphere can have before breakdown occurs?

Ben N.
Numerade Educator

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 $\mathrm{P} 16.23 .$ How close does the alpha particle get to the gold nucleus before turning around? Assume the gold nucleus remains stationary.

Salamat A.
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 N.
Numerade Educator

Problem 25

Calculate the speed of (a) an electron and (b) a proton with a kinetic energy of 1.00 electron volt (eV). (c) Calculate the average translational kinetic energy in eV of a $3.00 \times 10^{2}-\mathrm{K}$
ideal gas particle. (Recall from Topic 10 that $\frac{1}{2} m v^{2}=\frac{3}{2} k_{\mathrm{B}} T . )$

Salamat A.
Numerade Educator

Problem 26

An electric field does $1.50 \times 10^{3} \mathrm{eV}$ of work on a carbon nucleus of charge $9.61 \times 10^{-19} \mathrm{C}$ . Find the change in the nucleus' (a) electric potential and (b) electric potential energy in joules.

Ben N.
Numerade Educator

Problem 27

An alpha particle, which has charge $3.20 \times 10^{-19} \mathrm{C},$ is moved from point $A$ , where the electric potential is $3.60 \times 10^{3} \mathrm{J} / \mathrm{C}$ , to point $B$ , where the electric potential is $5.80 \times 10^{3} \mathrm{J} / \mathrm{C}$ . Calculate the work done by the electric field on the alpha particle in electron volts.

Salamat A.
Numerade Educator

Problem 28

In the classical model of a hydrogen atom, an electron orbits a proton with a kinetic energy of $+13.6 \mathrm{eV}$ and an electric potential energy of $-27.2 \mathrm{eV} .(\mathrm{a})$ Use the kinetic energy to calculate the classical orbital speed. (b) Use the electric potential energy to calculate the classical orbital radius.

Ben N.
Numerade Educator

Problem 29

Consider the Earth and a cloud layer $8.0 \times 10^{2} \mathrm{m}$ 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?

Salamat A.
Numerade Educator

Problem 30

(a) When a $9.00-\mathrm{V}$ 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? $(\mathrm{b})$ If the same capacitor is connected to a 12.0 $\mathrm{N}$ battery, what charge is stored?

Ben N.
Numerade Educator

Problem 31

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 $12.0-\mathrm{V}$ 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 A.
Numerade Educator

Problem 32

Air breaks down and conducts charge as a spark if the electric field magnitude exceeds $3.00 \times 10^{6} \mathrm{V} / \mathrm{m} .$ (a) Determine the maximum charge $Q_{\mathrm{max}}$ that can be stored on an air-filled parallelel-plate capacitor with a plate area of $2.00 \times 10^{-4} \mathrm{m}^{2}$ (b) A 75.0$\mu \mathrm{F}$ air-filled parallel-plate capacitor stores charge $Q_{\text { max }}$ . Find the potential difference across its plates.

Ben N.
Numerade Educator

Problem 33

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 A.
Numerade Educator

Problem 34

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 $10^{-10} \mathrm{m}=1$ A. Express the plate separation in angstroms.

Ben N.
Numerade Educator

Problem 35

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 A.
Numerade Educator

Problem 36

A small object with a mass of $350 . \mu \mathrm{g}$ 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?

Ben N.
Numerade Educator

Problem 37

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.

Salamat A.
Numerade Educator

Problem 38

Two capacitors, $C_{1}=5.00 \mu \mathrm{F}$ and $C_{2}=12.0 \mu \mathrm{F},$ are connected in parallel, and the resulting combination is connected to a $9.00-\mathrm{V}$ battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c) the charge stored on each capacitor.

Ben N.
Numerade Educator

Problem 39

Find (a) the equivalent capacitance of the capacitors in Figure $P 16.39,(b)$ the charge on each capacitor, and $(c)$ the potential difference across each capacitor.

Salamat A.
Numerade Educator

Problem 40

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 N.
Numerade Educator

Problem 41

For the system of capacitors shown in Figure $P 16.41,$ find $(\text { a })$ the equivalent capacitance of the system, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.

Salamat A.
Numerade Educator

Problem 42

Consider the combination of capacitors in Figure P16.42. (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 $(\mathrm{c}) ?$ (e) What is the charge on the $24.0-\mu \mathrm{F}$ capacitor and on the $8.00-\mu \mathrm{F}$ capacitor? Compute the voltage drop
across (f) the $24.0-\mu \mathrm{F}$ capacitor and (g) the $8.00-\mu \mathrm{F}$ capacitor.

Ben N.
Numerade Educator

Problem 43

Find the charge on each of the capacitors in Figure $\mathrm{P} 16.43$

Salamat A.
Numerade Educator

Problem 44

Three capacitors are connected to a battery as shown in Figure $\mathrm{P} 16.44$ . Their capacitances are $C_{1}=3 C, C_{2}=C,$ and $C_{3}=5 C$ (a) What is the equivalent capacitance of this set of capacitors? (b) State the ranking of the capacitors according to the charge they store from largest to smallest. (c) Rank the capacitors according to the potential differences across them from largest to smallest. (d) Assume $C_{3}$ is increased. Explain what happens to the charge stored by each capacitor.

Ben N.
Numerade Educator

Problem 45

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 capacttors 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? (c) What is the final potential difference across the $40.0-\mu \mathrm{F}$ capacitor?

Salamat A.
Numerade Educator

Problem 46

(a) Find the equivalent capacitance between points $a$ and $b$ for the group of capacitors connected as shown in Figure P16.46 if $C_{1}=5.00 \mu \mathrm{F}, \quad C_{2}=10.00 \mu \mathrm{F}$ 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 $C_{3} ?$

Ben N.
Numerade Educator

Problem 47

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 A.
Numerade Educator

Problem 48

Four capacitors are connected as shown in Figure $\mathrm{Pl} 6.48$ . (a) Find the equivalent capacitance between points $a$ and $b$ (b) Calculate the charge on each capacitor, taking $\Delta V_{a b}=15.0 \mathrm{V}$

Ben N.
Numerade Educator

Problem 49

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

Salamat A.
Numerade Educator

Problem 50

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 N.
Numerade Educator

Problem 51

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-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 A.
Numerade Educator

Problem 52

Each plate of a 5.00$\mu \mathrm{F}$ capacitor stores 60.0$\mu \mathrm{C}$ of charge. (a)
Find the potential difference across the plates. (b) How much energy is stored in the capacitor?

Ben N.
Numerade Educator

Problem 53

The voltage across an air-filled parallel-plate capacitor is measured to be 85.0 V. When a dielectric is inserted and completely fills the space between the plates as in Figure P16.53, the voltage drops to 25.0 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?

Salamat A.
Numerade Educator

Problem 54

(a) How much charge can be placed on a capacitor with air between the plates before it breaks down if the area of each plate is 5.00 $\mathrm{cm}^{2} ?$ (b) Find the maximum charge if polystyrene is used between the plates instead of air. Assume the dielectric strength of air is $3.00 \times 10^{6} \mathrm{V} / \mathrm{m}$ and that of polystyrene is $24.0 \times 10^{6} \mathrm{V} / \mathrm{m} .$

Ben N.
Numerade Educator

Problem 55

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 cm2 and an insulation thickness of 0.040 0 mm.

Salamat A.
Numerade Educator

Problem 56

A parallel-plate capacitor has plates of area $A=7.00 \times 10^{-2}$ $\mathrm{m}^{2}$ separated by distance $d=2.00 \times 10^{-4} \mathrm{m} .$ (a) Calculate the capacitance if the space between the plates is filled with air. What is the capacitance if the space is filled half with air and half with a dielectric of constant $\kappa=3.70$ as in (b) Figure P16.56a, and (c) Figure $P 16.56 \mathrm{b}$ ? (Hint: In (b) and (c), one of the capacitors is a parallel combination and the other is a series combination.)

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Problem 57

A model of a red blood cell portrays the cell as a spherical 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. mV across the membrane. The membrane’s thickness is estimated to be 100 . 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 $1.10 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ (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?

Salamat A.
Numerade Educator

Problem 58

When a potential difference of $150 . \mathrm{V}$ is applied to the plates of an air-filled parallel-plate capacitor, the plates carry a surface charge density of $3.00 \times 10^{-10} \mathrm{Cm}^{2} .$ What is the spacing between the plates?

Ben N.
Numerade Educator

Problem 59

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 A.
Numerade Educator

Problem 60

For the system of four capacitors shown in Figure Pl6.41, 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.

Ben N.
Numerade Educator

Problem 61

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 P16.61a. Show that the capacitance of this partially filled capacitor is given by
$$
C=\left(\frac{3 \kappa}{2 \kappa+1}\right) C_{0}
$$
Hint: Treat the system as two capacitors connected in series as in Figure $P 16.61 \mathrm{b}$ , one with dielectric in it and the other one empty.

Salamat A.
Numerade Educator

Problem 62

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 N.
Numerade Educator

Problem 63

A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is $2.00 \times 10^{8} \mathrm{V} / \mathrm{m}$ . The desired capacitance is 0.250$\mu \mathrm{F}$ , and the capacitor must withstand a maximum potential difference of 4.00 $\mathrm{kV}$ . Find the minimum area of the capacitor plates.

Salamat A.
Numerade Educator

Problem 64

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 $\mathrm{P} 16.64 .$ (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 $P ?$

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Problem 65

Find the equivalent capacitance of the group of capacitors shown in Figure P16.65.

Salamat A.
Numerade Educator

Problem 66

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 G.
Numerade Educator

Problem 67

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 electric shock to the chest over a time of a few milliseconds. The device contains a capacitor of a few microfarads, charged to several thousand volts. Electrodes called paddles, about 8 cm across and coated with conducting paste, are held against the chest on both sides of the heart. Their handles are insulated to prevent injury to the operator, who calls Clear! and pushes a button on one paddle to discharge the capacitor through the patient's chest. Assume an energy of $3.00 \times 10^{2} \mathrm{W} \cdot \mathrm{s}$ is to be delivered from a $30.0-\mu \mathrm{F}$ capacitor. To what potential difference must it be charged?

Salamat A.
Numerade Educator

Problem 68

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 M.
Numerade Educator

Problem 69

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 A.
Numerade Educator

Problem 70

Two positive charges each of charge $q$ are fixed on the $y$ -axis, one at $y=d$ and the other at $y=-d$ as in Figure P16.70. A third positive charge 2$q$ located on the $x$ -axis at $x=2 d$ is released from rest. Find symbolic expressions for $(a)$ the total electric potential due to the first two charges at the location of the charge $2 q,(b)$ the electric potential energy of the charge $2 q,(c)$ the kinetic energy of the charge 2$q$ after it has moved infinitely far from the other charges, and (d) the speed of the charge 2$q$ after it has moved infinitely far from the other charges if its mass is $m .$

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Anthony I.
Numerade Educator

Problem 71

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 A.
Numerade Educator

Problem 72

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 $\mathrm{P} 16.72 .$ 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.

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