College Physics

Hugh D. Young

Chapter 18

Electric Potential and Capacitanc - all with Video Answers

Educators

Chapter Questions

Problem 1

A charge of 28.0 $\mathrm{nC}$ is placed in a uniform electric field that is directed vertically upward and that has a magnitude of $4.00 \times 10^{4} \mathrm{N} / \mathrm{C}$ . What work is done by the electric force when the charge moves (a) 0.450 $\mathrm{m}$ to the right; (b) 0.670 $\mathrm{m}$
upward; (c) 2.60 $\mathrm{m}$ at an angle of $45.0^{\circ}$ downward from the
horizontal?

Katie Mcalpine

Katie Mcalpine

Numerade Educator

Problem 2

Two very large charged parallel metal plates are 10.0 $\mathrm{cm}$ apart and produce a uniform electric field of $2.80 \times 10^{6} \mathrm{N} / \mathrm{C}$ between them. A proton is fired perpendicular to these plates with an initial speed of 5.20 $\mathrm{km} / \mathrm{s}$ , starting at the middle of the negative plate and going toward the positive plate. How much work has the electric field done on this proton by the time it reaches the positive plate?

Kai Chen

Princeton University

Problem 3

How far from a $-7.20 \mu \mathrm{C}$ point charge must a $+2.30 \mu \mathrm{C}$ point charge be placed in order for the electric potential energy of the pair of charges to be $-0.400 \mathrm{J} ?$ (Take the energy to be zero when the charges are infinitely far apart.)

Katie Mcalpine

Numerade Educator

Problem 4

A point charge $q_{1}=+2.40 \mu C$ is held stationary at the origin. A second point charge $q_{2}=-4.30 \mu C$ moves from the point $x=0.150 \mathrm{m}, y=0,$ to the point $x=0.250 \mathrm{m},$ $y=0.250 \mathrm{m} .$ How much work is done by the electric forceon $q_{2} ?$

Kai Chen

Princeton University

Problem 5

Two stationary point charges of $+3.00 \mathrm{nC}$ and $+2.00 \mathrm{nC}$ are separated by a distance of 50.0 $\mathrm{cm} .$ An electron is released from rest at a point midway between the charges and moves along the line connecting them. What is the electric potential energy for the electron when it is (a) at the midpoint and (b) 10.0 $\mathrm{cm}$ from the $+3.00 \mathrm{nC}$ charge?

Katie Mcalpine

Numerade Educator

Problem 6

Energy of DNA base pairing, I. (See Problem 24 in Chapter $17 ;$ see also Figure $17.43 .$ ) (a) Calculate the electric potential energy of the adenine thymine bond, using the same combinations of molecules $(\mathrm{O}-\mathrm{H}-\mathrm{N}$ and $\mathrm{N}-\mathrm{H}-\mathrm{N})$ as in Problem 17.24 . (b) Compare this energy with the potential energy of the proton-electron pair in the hydrogen atom.

Kai Chen

Princeton University

Problem 7

Energy of DNA base pairing, II. (See Problem 25 in Chapter $17 ;$ see also Figure $17.44 .$ ) Calculate the electric potential energy of the guanine-cytosine bond, using the same combinations of molecules $(0-\mathrm{H}-\mathrm{O}, \mathrm{N}-\mathrm{H}-\mathrm{N},$ and $\mathrm{O}-\mathrm{H}-\mathrm{N} )$ as in Problem $17.25 .$

Kai Chen

Princeton University

Problem 8

(a) A set of point charges is held in place at the vertices of an equilateral triangle of side $10.0 \mathrm{cm},$ as shown in Figure 18.38$($ a). Find the maximum amount of total kinetic energy that will be produced when the charges are duced when the charges are released from rest in the friction less vold of outer space. (b) If the charges at the vertices of the right triangle in Figure 18.38$(\mathrm{b})$ are released, how much total kinetic energy will they gain? When will this maximum kinetic energy be achieved, just following the release of the charges or after a very long time?

Kai Chen

Princeton University

Problem 9

Three equal $1.20-\mu \mathrm{C}$ point charges are placed at the corners of an equilateral triangle whose sides are 0.500 $\mathrm{m}$ long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Katie Mcalpine

Numerade Educator

Problem 10

When two point charges are a distance $R$ apart, their potential energy is $-2.0 \mathrm{J} .$ How far far (in terms of $R )$ should they be from each other so that their potential energy is $-6.0 \mathrm{J} ?$

Kai Chen

Princeton University

Problem 11

Two large metal parallel plates carry opposite charges of equal magnitude. They are separated by $45.0 \mathrm{mm},$ and the potential difference between them is 360 $\mathrm{V}$ (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge $+2.40 \mathrm{nC}$ ?

Katie Mcalpine

Numerade Educator

Problem 12

A potential difference of 4.75 $\mathrm{kV}$ is established between parallel plates in air. If the air becomes ionized (and hence electrically conducting) when the electric field exceeds $3.00 \times 10^{6} \mathrm{V} / \mathrm{m},$ what is the minimum separation the plates can have without ionizing the air?

Kai Chen

Princeton University

Problem 13

Oscilloscope. Oscilloscopes are found in most science laboratories. Inside, they contain deflecting plates consisting of more-or-less square parallel metal sheets, typically about 2.5 $\mathrm{cm}$ on each side and 2.0 $\mathrm{mm}$ apart. In many experiments, the maximum potential across these plates is about 25 $\mathrm{V}$ . For this maximum potential, (a) what is the strength of the electric field between the plates, and (b) what magnitude of acceleration would this field produce on an electron midway between the plates?

Katie Mcalpine

Numerade Educator

Problem 14

Axons. Neurons are the basic units of the nervous system. They contain long tubular structures called axons that propagate electrical signals away from the axon contains a solution axon contains a solution of potassium ions $\mathrm{K}^{+}$ and large negative organic ions. The axon membrane prevents the large ions from leaking out, but the smaller $\mathrm{K}^{+}$ ions are able to penetrate the membrane to some degree. (See Figure 18.39 . ) This leaves an excess negative charge on the inner surface of the axon membrane and an excess of positive charge on the outer surface, resulting in a potential difference across the membrane that prevents further $\mathrm{K}^{+}$ ions from leaking out. Measurements show that this potential difference is typically about 70 $\mathrm{mV}$ . The thickness of the axon membrane itself varies from about 5 to $10 \mathrm{nm},$ so we'll use an average of 7.5 $\mathrm{nm}$ . We can model the membrane as a large sheet having equal and opposite charge densities on its faces. (a) Find the electric field inside the axon membrane, assuming (not too realistically) that it is filled with air. Which way does it point, into or out of the axon? (b) Which is at a higher potential, the inside surface or the outside surface of the axon membrane?

Kai Chen

Princeton University

Problem 15

Electrical sensitivity of sharks. Certain sharks can detect an electric field as weak as 1.0$\mu \mathrm{V} / \mathrm{m} .$ To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5 $\mathrm{V}$ A battery across theseplates, how far apart would the plates have to be?

Katie Mcalpine

Numerade Educator

Problem 16

A particle with a charge of $+4.20 \mathrm{nC}$ is in a uniform electric field $\vec{\boldsymbol{E}}$ directed to the left. It is released from rest and moves to the left; after it has moved $6.00 \mathrm{cm},$ its kinetic energy is found to be $+1.50 \times 10^{-6} \mathrm{J}$ (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the endpoint? (c) What is the magnitude of $\vec{E} ?$

Kai Chen

Princeton University

Problem 17

Two very large metal parallel plates are 20.0 $\mathrm{cm}$ apart and carry equal, but opposite, surface charge densities. Figure 18.40 shows a graph of the potential, relative to the negative plate, as a function of $x .$ For this case, $x$ is the distance from the inner surface of the negative plate, measured perpendicular to the plates, and points from the negative plate toward the positive plate. Find the electric field between the plates.

Katie Mcalpine

Numerade Educator

Problem 18

A uniform electric field has magnitude $E$ and is directed in the negative $x$ -direction. The potential difference between point $a$ (at $x=0.60 \mathrm{m} )$ and point $b$ (at $x=0.90 \mathrm{m} )$ is 240 $\mathrm{V}$ . (a) Which point, $a$ or $b$ , is at the higher potential? (b) Calculate the value of $E (\mathrm{c})$ A negative point charge $q=-0.200 \mu \mathrm{C}$ is moved from $b$ to $a$ . Calculate the work done on the point charge by the electric field.

Kai Chen

Princeton University

Problem 19

A point charge has a charge of $2.50 \times 10^{-11} \mathrm{C}$. At what distance from the point charge is the electric potential (a) $90.0 \mathrm{~V} ?$ (b) $30.0 \mathrm{~V}$ ? Take the potential to be zero at an infinite distance from the charge.

Katie Mcalpine

Numerade Educator

Problem 20

(a) An electron is to be accelerated from $3.00 \times 10^{6} \mathrm{m} / \mathrm{s}\right.$ to $8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from
$8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ to a halt?

Kai Chen

Princeton University

Problem 21

A small particle has charge $-5.00 \mu \mathrm{C}$ and mass $2.00 \times$ $10^{-4} \mathrm{kg} .$ It moves from point $A,$ where the electric potential is $V_{A}=+200 \mathrm{V},$ to point $B,$ where the electric potential is $V_{B}=+800 \mathrm{V} .$ The electric force is the only force acting on the particle. The particle has speed 5.00 $\mathrm{m} / \mathrm{s}$ at point $A .$ What is its speed at point $B ?$ Is it moving faster or slower at $B$ than at $A$? Explain.

Katie Mcalpine

Numerade Educator

Problem 22

Two point charges $q_{1}=$ $+2.40 \mathrm{nC}$ and $q_{2}=-6.50 \mathrm{nC}$ are 0.100 $\mathrm{m}$ apart. Point $A$ is midway between them; point $B$ is 0.080 $\mathrm{m}$ from $q_{1}$ and 0.060 m from $q_{2}$ . (See Figure $18.41 .$ ) Take the electric potential to be zero at infinity. Find (a) the potential at point $A ;(b)$ the potental at point $B ;(c)$ the work done by the
electric field on a charge of 2.50 $\mathrm{nC}$ that travels from point $B$ to point $A .$

Kai Chen

Princeton University

Problem 23

A point charge $Q=+4.60 \mu \mathrm{C}$ is held fixed at the origin. A second point charge $q=+1.20 \mu \mathrm{C}$ with mass of $2.80 \times$ $10^{-4} \mathrm{kg}$ is placed on the $x$ axis, 0.250 $\mathrm{m}$ from the origin. (a) What is the electric potential energy $U$ of the pair of charges? (Take $U$ to be zero when the charges have infinite separation.) (b) The second point charge is released from rest. What is its speed when its distance from the origin is (i) $0.500 \mathrm{m} ;$ (ii) 5.00 $\mathrm{m}$ ; (iii) 50.0 $\mathrm{m} ?$

Katie Mcalpine

Numerade Educator

Problem 24

Two protons are released from rest when they are 0.750 $\mathrm{nm}$ apart. (a) What is the maximum speed they will reach? When does this speed occur? (b) What is the maximum acceleration they will achieve? When does this acceleration occur?

Vishal Gupta

Numerade Educator

Problem 25

Cathode-ray tube. A cathode-ray tube (CRT) is an evacuated glass tube. Electrons are produced at one end, usually by the heating of a metal. After being focused electromagnetically into a beam, they are accelerated through a potential difference, called the accelerating potential. The electrons then strike a coated screen, where they transfer their energy to the coating through collisions, causing it to glow. CRTs are found in oscilloscopes and computer monitors, as well as in earlier versions of television screens. (a) If an electron of mass $m$ and charge $-e$ is accelerated from rest through an accelerating potential $V,$ show that the speed it gains is $v=\sqrt{2 e V / m}$ . We are assuming that $V$ is small enough that the final speed is much less than the speed of light.) (b) If the accelerating potential is $95 \mathrm{V},$ how fast will the electrons be moving when they hit the screen?

Katie Mcalpine

Numerade Educator

Problem 26

X-ray tube. An X-ray tube is similar to a cathode-ray tube. (See previous problem.) Electrons are accelerated to high speeds at one end of the tube. If they are moving fast enough when they hit the target at the other end, they give up their energy as X-rays (a form of nonvisible light). (a) Through what potential difference should electrons be accelerated so that their speed is 1.0$\%$ of the speed of light when they hit the target? (b) What potential difference would be needed to give protons the same kinetic energy as the electrons? (c) What speed would this potential difference give to protons? Express your answer in $\mathrm{m} / \mathrm{s}$ and as a percent of the speed of light.

Kai Chen

Princeton University

Problem 27

A gold nucleus has a radius of $7.3 \times 10^{-15} \mathrm{m}$ and a charge of $+79 e .$ Through what voltage must an $\alpha$ -particle, with its charge of $+2 e,$ be accelerated so that it has just enough energy to reach a distance of $2.0 \times 10^{-14} \mathrm{m}$ from the surface of a gold nucleus? (Assume the gold nucleus remains stationary and can be treated as a point charge.)

Katie Mcalpine

Numerade Educator

Problem 28

A parallel-plate capacitor having plates 6.0 $\mathrm{cm}$ apart is connected across the terminals of a 12 $\mathrm{V}$ battery. (a) Being as quantitative as you can, describe the location and shape of the equipotential surface that is at a potential of $+6.0 \mathrm{V}$ relative to the potential of the negative plate. Avoid the edges of the plates. (b) Do the same for the equipotential surface that is at $+2.0 \mathrm{V}$ relative to the negative plate. (c) What is the potential gradient between the plates?

Kai Chen

Princeton University

Problem 29

Two very large metal parallel plates that are 25 $\mathrm{cm}$ apart, oriented perpendicular to a sheet of paper, are connected across the terminals of a 50.0 $\mathrm{V}$ battery. (a) Draw to scale the lines where the equipotential surfaces due to these plates intersect the paper. Limit your drawing to the region between the plates, avoiding their edges, and draw the lines for surfaces that are 10.0 $\mathrm{V}$ apart, starting at the low-potential plate. (b) These surfaces are separated equally in potential. Are they also separated equally in distance? (c) In words, describe the shape and orien- tation of the surfaces you just found.

Katie Mcalpine

Numerade Educator

Problem 30

(a) $A+5.00$ pC charge is located on a sheet of paper. (a) Draw to scale the curves where the equipotential surfaces due to these charges intersect the paper. Show only the surfaces that have a potential (relative to infinity) of $1.00 \mathrm{V}, 2.00 \mathrm{V}$ $3.00 \mathrm{V}, 4.00 \mathrm{V},$ and 5.00 $\mathrm{V} .$ (b) The surfaces are separated equally in potential. Are they also separated equally in distance? (c) In words, describe the shape and orientation of the surfaces you just found.

Kai Chen

Princeton University

Problem 31

A metal sphere carrying an evenly distributed charge will have spherical equipotential surfaces surrounding it. Suppose the sphere's radius is 50.0 $\mathrm{cm}$ and it carries a total charge of $+1.50 \mu \mathrm{C}$ (a) Calculate the potential of the sphere's surface.(b) You want to draw equipotential surfaces at intervals of 500 $\mathrm{V}$ outside the sphere's surface. Calculate the distance between the first and the second equipotential surfaces, and between the 20 $\mathrm{th}$ and 21 $\mathrm{st}$ equipotential surfaces. (c) What does the changing spacing of the surfaces tell you about the electric field?

Katie Mcalpine

Numerade Educator

Problem 32

Figure 18.42 shows a set of electric-field lines for a particular distribution of charges. Use these lines to draw a series of equipotem. Limit yourself to the plane of the paper.

Kai Chen

Princeton University

Problem 33

Dipole. A dipole is located on a sheet of paper. (a) In the plane of that paper, carefully sketch the electric field lines for this dipole. (b) Use your field lines in part (a) to sketch the equipotential curves where the equipotential surfaces intersect the paper.

Katie Mcalpine

Numerade Educator

Problem 34

In a particular Millikan oil-drop apparatus, the plates are 2.25 $\mathrm{cm}$ apart. The oil used has a density of $0.820 \mathrm{g} / \mathrm{cm}^{3},$ and the atomizer that sprays the oil drops produces drops of diameter $1.00 \times 10^{-3} \mathrm{mm}$ . (a) What strength of electric field is needed to hold such a drop stationary against gravity if the drop contains five excess electrons? (b) What should be the potential difference across the plates to produce this electric field? (c) If another drop of the same oil requires a plate potential of 73.8 $\mathrm{V}$ to hold it stationary, how many excess electrons did it contain?

Kai Chen

Princeton University

Problem 35

(a) If an electron and a proton each have a kinetic energy of 1.00 eV, how fast is each one moving? (b) What would be their speeds if each had a kinetic energy of 1.00 $\mathrm{keV}$ ? (c) If they were each traveling at 1.00$\%$ the speed of light, what would be their kinetic energies in keV?

Katie Mcalpine

Numerade Educator

Problem 36

(a) You find that if you place charges of $\pm 1.25 \mu \mathrm{C}$ on two separated metal objects, the potential difference between them is 11.3 $\mathrm{V}$ . What is their capacitance? (b) A capacitor has a capacitance of 7.28$\mu \mathrm{F}$ . What amount of excess charge must be placed on each of its plates to make the potential difference between the plates equal to 25.0 $\mathrm{V}$ ?

Supratim Pal

Numerade Educator

Problem 37

$\bullet$ The plates of a parallel-plate capacitor are 3.28 $\mathrm{mm}$ apart, and each has an area of 12.2 $\mathrm{cm}^{2} .$ Each plate carries a charge of magnitude $4.35 \times 10^{-8} \mathrm{C}$ . The plates are in vacuum. (a) What is the capacitance? (b) What is the potential difference between the plates? (c) What is the magnitude of the electric field between the plates?

Katie Mcalpine

Numerade Educator

Problem 38

The plates of a parallel-plate capacitor are 2.50 $\mathrm{mm}$ apart, and each carries a charge of magnitude 80.0 $\mathrm{nC}$ . The plates are in vacuum. The electric field between the plates has a magnitude of $4.00 \times 10^{3} \mathrm{Vm}$ (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the capacitance?

Kai Chen

Princeton University

Problem 39

A parallel-plate air capacitor has a capacitance of 500.0 $\mathrm{pF}$ and a charge of magnitude 0.200$\mu \mathrm{C}$ on each plate. The plates are 0.600 $\mathrm{mm}$ apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electric-field magnitude between the plates? (d) What is the surface charge density on each plate?

Katie Mcalpine

Numerade Educator

Problem 40

Capacitance of an oscilloscope. Oscilloscopes have parallel metal plates inside them to deflect the electron beam. These plates are called the deflecting plates. Typically, they are squares 3.0 $\mathrm{cm}$ on a side and separated by $5.0 \mathrm{mm},$ with vacuum in between. What is the capacitance of these deflecting plates and hence of the oscilloscope? (This capacitance can
sometimes have an effect on the circuit you are trying to study and must be taken into consideration in your calculations.)

Kai Chen

Princeton University

Problem 41

A 10.0$\mu \mathrm{F}$ parallel-plate capacitor with circular plates is connected to a 12.0 $\mathrm{V}$ battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the 12.0 $\mathrm{V}$ battery after the radius of each plate was doubled without changing their separation?

Katie Mcalpine

Numerade Educator

Problem 42

A 10.0$\mu$ F parallel-plate capacitor is connected to a 12.0 $\mathrm{V}$ battery. After the capacitor is fully charged, the battery is disconnected without loss of any of the charge on the plates. (a) A volt-meter is connected across the two plates without discharging them. What does it read? (b) What would the voltmeter read if (i) the plate separation were doubled; (ii) the radius of each plate was doubled, but the separation between the plates was unchanged?

Kai Chen

Princeton University

Problem 43

You make a capacitor by cutting the $15.0-\mathrm{cm}$ -diameter bottoms out of two aluminum pie plates, separating them by 3.50 $\mathrm{mm},$ and connecting them across a $6.00-\mathrm{V}$ battery. (a) What's the capacitance of your capacitor? (b) If you disconnect the battery and separate the plates to a distance of $3.50 \mathrm{~cm}$ without discharging them, what will be the potential difference between them?

Katie Mcalpine

Numerade Educator

Problem 44

A 5.00 pF parallel-plate air-filled capacitor with circular plates is to be used in a circuit in which it will be subjected to potentials of up to $1.00 \times 10^{2} \mathrm{V}$ . The electric field between he plates is to be no greater than $1.00 \times 10^{4} \mathrm{N} / \mathrm{C} .$ As a budding electrical engineer for Live- Wire Electronics, your tasks are to (a) design the capacitor by finding what its physical dimensions and separation must be and (b) find the maximum charge these plates can hold.

Kai Chen

Princeton University

Problem 45

How far apart would parallel pennies have to be to make a 1.00 -pF capacitor? Does your answer suggest that you are justified in treating these pennies as infinite sheets? Explain.

Katie Mcalpine

Numerade Educator

Problem 46

A parallel-plate capacitor $C$ is charged up to a potential $V_{0}$ with a charge of magnitude $Q_{0}$ on each plate. It is then disconnected from the battery, and the plates are pulled apart to twice their original separation. (a) What is the new capacitance in terms of $C ?$ (b) How much charge is now on the plates in terms of $Q_{0} ?(\mathrm{c})$ What is the potential difference across the plates in terms of $V_{0} ?$

Kai Chen

Princeton University

Problem 47

For the system of capacitors shown in Figure 18.43 , find the equivalent capacitance (a) between $b$ and $c,$ (b) between $a$ and $c .$

Katie Mcalpine

Numerade Educator

Problem 48

Electric eels. Electric eels and electric fish generate large potential differences that are used to stun 9 enemies and prey. These potentials are produced by cells that each can generate 0.10 V. We can plausibly model such cells as charged capacitors. (a) How should these cells be connected $(\mathrm{in}$ series or in parallel) to produce a total potential of more than 0.10 $\mathrm{V} ?$ (b) Using the connection in part (a), how many cells must be connected together to produce the 500 $\mathrm{V}$ surge of the electric eel?

Kai Chen

Princeton University

Problem 49

In Figure $18.44, \quad C_{1}=$ $6.00 \mu \mathrm{F}, \quad C_{2}=3.00 \mu \mathrm{F}, \quad$ and $C_{3}=5.00 \mu \mathrm{F}$ . The capacitor network is connected to an applied potential $V_{a b}$ . After the charges on the capacitors have reached their final values, the charge on $C_{2}$ is 40.0$\mu C$ (a) What are the charges on capacitors $C_{1}$ and $C_{3} ?$ (b) What is the applied voltage $V_{a b} ?$

Katie Mcalpine

Numerade Educator

Problem 50

You are working on an electronics project requiring a variety of capacitors, but have only a large supply of 100 nF capacitors available. Show how you can connect these capacitors to produce each of the following equivalent capacitances: (a) $50 \mathrm{nF},$ (b) $450 \mathrm{nF},(\mathrm{c}) 25 \mathrm{nF},$ (d) 75 $\mathrm{nF.}$

Kai Chen

Princeton University

Problem 51

In Figure $18.44, C_{1}=3.00 \mu \mathrm{F}$ and $V_{a b}=120 \mathrm{V}$ . The charge on capacitor $C_{1}$ is 150$\mu \mathrm{C}$ . Calculate the voltage across the other two capacitors.

Katie Mcalpine

Numerade Educator

Problem 52

A 4.00$\mu \mathrm{F}$ and a 6.00$\mu \mathrm{F}$ capacitor are connected in series, and this combination is connected across a 48.0 $\mathrm{V}$ potential difference. Calculate (a) the charge on each capacitor and (b) the potential difference across each of them.

Kai Chen

Princeton University

Problem 53

In the circuit shown in Figure $18.45,$ the potential difference across $a b$ is $+24.0 \mathrm{V}$ . Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

Katie Mcalpine

Numerade Educator

Problem 54

In Figure $18.46,$ each capacitor has $C=4.00 \mu \mathrm{F}$ and $V_{a b}=+28.0 \mathrm{V} .$ Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

Kai Chen

Princeton University

Problem 55

Figure 18.47 shows a system of four capacitors, where the potential difference across $a b$ is 50.0 $\mathrm{V}$ . (a) Find the equivalent capacitance of this system between $a$ and $b$ . (b) How much charge is stored by this combination of capacitors? (c) How much charge is stored in each of the 10.0$\mu \mathrm{F}$ and the 9.0$\mu$ F capacitors?

Katie Mcalpine

Numerade Educator

Problem 56

For the system of capacitors shown in Figure $18.48,$ a potential difference of 25 $\mathrm{V}$ is maintainedacross ab. (a) What is the equivalent capacitance of this system between $a$ and b? (b) How much charge is stored by this system? (c) How much charge does the 6.5 $\mathrm{nF}$ rapacitor store? (d) What is the potential difference across the 7.5 $\mathrm{nF}$ capacitor?

Kai Chen

Princeton University

Problem 57

How much charge does a 12 $\mathrm{V}$ battery have to supply to fully charge a 2.5$\mu \mathrm{F}$ capacitor and a 5.0$\mu \mathrm{F}$ capacitor when they're (a) in parallel, (b) in series? (c) How much energy does the battery have to supply in each case?

Katie Mcalpine

Numerade Educator

Problem 58

A 5.80$\mu$ F parallel-plate air capacitor has a plate separation of 5.00 mm and is charged to a potential difference of 400 $\mathrm{V}$ . Calculate the energy density in the region between the plates, in units of $\mathrm{J} / \mathrm{m}^{3} .$

Kai Chen

Princeton University

Problem 59

(a) How much charge does a battery have to supply to a 5.0$\mu \mathrm{F}$ capacitor to create a potential difference of 1.5 $\mathrm{V}$ across its plates? How much energy is stored in the capacitor in this case? (b) How much charge would the battery have to supply to store 1.0 $\mathrm{J}$ of energy in the capacitor? What would be the potential across the capacitor in that case?

Katie Mcalpine

Numerade Educator

Problem 60

In the text, it was shown that the energy stored in a capacitor $C$ charged to a potential $V$ is $U=\frac{1}{2} Q V$ . Show that this energy can also be expressed as (a) $U=Q^{2} / 2 C$ and (b) $U=\frac{1}{2} C V^{2}$

Kai Chen

Princeton University

Problem 61

A parallel-plate vacuum capacitor has 8.38 J of energy stored in it. The separation between the plates is 2.30 $\mathrm{mm}$ . If the separation is decreased to $1.15 \mathrm{mm},$ what is the energy stored (a) if the capacitor is disconnected from the potential source so the charge on the plates remains constant, and (b) if the capacitor remains connected to the potential source so the potential difference between the plates remains constant?

Katie Mcalpine

Numerade Educator

Problem 62

(a) How many excess electrons must be added to one plate and removed from the other to give a 5.00 $\mathrm{nF}$ parallel-plate capacitor 25.0$\mu \mathrm{J}$ of stored energy? (b) How could you modify the geometry of this capacitor to get it to store 50.0$\mu$ J of energy without changing the charge on its plates?

Kai Chen

Princeton University

Problem 63

For the capacitor network shown in Figure 18.49 , the potential difference across ab is 36 V. Find (a) the total charge stored in this network, (b) the charge on each capacitor, (c) the total energy stored in the network, (d) the energy stored in each capacitor, and (e) the potential difference across each capacitor.

Katie Mcalpine

Numerade Educator

Problem 64

For the capacitor network shown in Figure 18.50 , the potential difference across ab is 220 $\mathrm{V}$ . Find (a) the total charge stored in this network, (b) the charge on each capacitor, (c) the total energy stored in the network, (d) the energy stored in each capacitor, and (e) the potential difference across each capacitor.

Kai Chen

Princeton University

Problem 65

A $\mathrm{A} 20.0 \mu \mathrm{F}$ capacitor is charged to a potential difference of 800 $\mathrm{V} .$ The terminals of the charged capacitor are then connected to those of an uncharged 10.0$\mu \mathrm{F}$ capacitor. Compute (a) the onginal charge of the system, (b) the final potential difference across each capacitor, (c) the final energy of the system, and (d) the decrease in energy when the capacitors are connected.

Katie Mcalpine

Numerade Educator

Problem 66

For the capacitor network shown in Figure 18.51 , the potential difference across $a b$ is 12.0 V. Find (a) the total energy stored in this network and (b) the energy stored in the 4.80$\mu$ F capacitor.

Kai Chen

Princeton University

Problem 67

A parallel-plate air capacitor has a capacitance of 920 pF. The charge on each plate is 2.55$\mu \mathrm{C}$ . (a) What is the potential difference between the plates? (b) If the charge is kept constant, what will be the potential difference between the plates if the separation is doubled? (c) How much work is required to double the separation?

Katie Mcalpine

Numerade Educator

Problem 68

A parallel-plate capacitor has capacitance $C_{0}=5.00 \mathrm{pF}$ when there is air between the plates. The separation between the plates is 1.50 $\mathrm{mm}$ (a) What is the maximum magnitude of charge $Q$ that can be placed on each plate if the electric field in the region between the plates is not to exceed $3.00 \times 10^{4} \mathrm{V} / \mathrm{m}$ (b) A dielectric with $K=2.70$ is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed $3.00 \times 10^{4} \mathrm{V} / \mathrm{m} ?$

Prabhu Ramji

Numerade Educator

Problem 69

Cell membranes. Cell membranes (the walled enclosure around a cell) are typically about 7.5 nm thick. They are partially permeable to allow charged material to pass in and out, as needed. Equal but opposite charge densities build up on the inside and outside faces of such a membrane, and these charges prevent additional charges from passing through the cell wall. We can model a cell membrane as a parallel-plate capacitor, with the membrane itself containing proteins embedded in an organic material to give the membrane a dielectric constant of about $10 .$ (See Figure $18.52 . )$ (a) What
is the capacitance per square centimeter of such a cell wall? (b) In its normal resting state, a cell has a potential difference of 85 $\mathrm{mV}$ across its membrane. What is the electric field inside this membrane?

Katie Mcalpine

Numerade Educator

Problem 70

A parallel-plate capacitor is to be constructed by using, as a dielectric, rubber with a dielectric constant of 3.20 and a dielectric strength of 20.0 $\mathrm{MV} / \mathrm{m}$ . The capacitor is to have a capacitance of 1.50 $\mathrm{nF}$ and must be able to withstand a maximum potential difference of 4.00 $\mathrm{kV} .$ What is the minimum area the plates of this capacitor can have?

Kai Chen

Princeton University

Problem 71

A $\mathrm{A} 12.5 \mu \mathrm{F}$ capacitor is connected to a power supply that keeps a constant potential difference of 24.0 $\mathrm{V}$ across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the insertion? Did it increase or decrease?

Katie Mcalpine

Numerade Educator

Problem 72

The paper dielectric in a paper-and-foil capacitor is 0.0800 mm thick. Its dielectric constant is $2.50,$ and its dielectric strength is 50.0 $\mathrm{MV} / \mathrm{m}$ . Assume that the geometry is that of a parallel-plate capacitor, with the metal foil serving as the plates. (a) What area of each plate is required for a 0.200$\mu F$ capacitor? (b) If the electric field in the paper is not to exceed one-half the dielectric strength, what is the maximum potential difference that can be applied across the capacitor?

Kai Chen

Princeton University

Problem 73

A constant potential difference of 12 $\mathrm{V}$ is maintained between the terminals of a $0.25-\mu \mathrm{F}$ , parallel-plate, air capacitor. (a) A sheet of Mylar is inserted between the plates of the capacitor, completely filling the space between the plates. When this is done, how much additional charge flows onto the positive plate of the capacitor (see Table 18.1$) ?$ (b) What is the total induced charge on either face of the Mylar sheet? (c) What effect does the Mylar sheet have on the electric field between the plates? Explain how you can reconcile this with the increase in charge on the plates, which acts to increase the electric field.

Katie Mcalpine

Numerade Educator

Problem 74

(a) If a spherical raindrop of radius 0.650 $\mathrm{mm}$ carries a charge of $-1.20 \mathrm{pC}$ uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop. (b) Two identical raindrops, each with radius and charge specified in part (a) collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

Kai Chen

Princeton University

Problem 75

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 V and $12.0 \mathrm{V} / \mathrm{m},$ respectively. (Take the potential to be zero at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the point charge?

Katie Mcalpine

Numerade Educator

Problem 76

Two oppositely charged identical insulating spheres, each 50.0 $\mathrm{cm}$ in diameter and carrying a uniform charge of magnitude $175 \mu \mathrm{C},$ are placed 1.00 $\mathrm{m}$ apart center to center Fig. 18.53 ). (a) If a voltmeter is connected between the nearest points $(a$ and $b)$ on their surfaces, what will it read? (b) Which point, $a$ or $b,$ is at the higher potential? How can you know this without any calculations?

Kai Chen

Princeton University

Problem 77

Potential in human cells. Some cell walls in the human body have a layer of negative charge on the inside surface and a layer of positive charge of equal magnitude on the outside surface. Suppose that the charge density on either surface is $\pm 0.50 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2},$ the cell wall is 5.0 $\mathrm{nm}$ thick, and the cell-wall material is air. (a) Find the magnitude of $\vec{E}$ in the wall between the two layers of charge. (b) Find the potential difference between the inside and the outside of the cell. Which is at the higher potential? (c) A typical cell in the human body has a volume of $10^{-16} \mathrm{m}^{3} .$ Estimate the total electric-field energy stored in the wall of a cell of this size. (Hint: Assume that the cell is spherical, and calculate the volume of the cell wall.) (d) In reality, the cell wall is made up, not of air, but of tissue with a dielectric constant of $5.4 .$ Repeat parts (a) and (b) in this case.

Katie Mcalpine

Numerade Educator

Problem 78

An alpha particle with a kinetic energy of 10.0 MeV makes a head-on collision with a gold nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the gold nucleus remains stationary and that it may be treated as a point charge. The atomic number of gold is $79,$ and an alpha particle is a helium nucleus consisting of two protons and two neutrons.)

Kai Chen

Princeton University

Problem 79

In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius $r .$ Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron's speed. (b) Obtain an expression
for the electron's kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using $r=$ $5.29 \times 10^{-11} \mathrm{m} .$ Give your numerical result in joules and in electron volts.

Katie Mcalpine

Numerade Educator

Problem 80

A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha particle (a helium nucleus) has essentially four times the mass and two times the charge of a proton. Find the maximum speed and maximum acceleration of each of these particles. When do these maxima occur, just following the release of the particles or after a very long time?

Kai Chen

Princeton University

Problem 81

A parallel-plate air capacitor is made from two plates 0.200 m square, spaced 0.800 $\mathrm{cm}$ apart. It is connected to a $120-\mathrm{V}$ battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of $1.60 \mathrm{cm},$ what are the answers to parts (a), (b), $(c),$ and $(d) ?$

Katie Mcalpine

Numerade Educator

Problem 82

In the previous problem, suppose the battery remains connected while the plates are pulled apart. What are the answers then to parts (a), ( b), (c), and (d) after the plates have been pulled apart?

Kai Chen

Princeton University

Problem 83

A capacitor consists of two parallel plates, each with an area of 16.0 $\mathrm{cm}^{2}$ , separated by a distance of 0.200 $\mathrm{cm} .$ The material that fills the volume between the plates has a dielectric constant of $5.00 .$ The plates of the capacitor are connected to a $300-\mathrm{V}$ battery. (a) What is the capacitance of the capacitor? (b) What is the charge on either plate? (c) How much energy is stored in the charged capacitor?

Katie Mcalpine

Numerade Educator

Problem 84

Electronic flash units for cameras contain a capacitor for storing the energy used to produce the flash. In one such unit, the flash lasts for $\frac{1}{675}$ s with an average light power output of $2.70 \times 10^{5} \mathrm{W}$ (a) If the conversion of electrical energy to light is 95$\%$ efficient (the rest of the energy goes to thermal energy), how much energy must be stored in the capacitor for one flash? (b) The capacitor has a potential difference between its plates of 125 $\mathrm{V}$ when the stored energy equals the value calculated in part (a). What is the capacitance?

Kai Chen

Princeton University

Problem 85

In Figure $18.54,$ each capacitance $C_{1}$ is 6.9$\mu \mathrm{F}$ and each capacitance $C_{2}$ is 4.6$\mu \mathrm{F}$ , (a) Compute the equivalent capacitance of the network between points $a$ and $b .$ (b) Compute the charge on each of the three capacitors nearest $a$ and $b$ when $V_{a b}=420 \mathrm{V}$

Vishal Gupta

Numerade Educator

Problem 86

A parallel-plate capacitor is made from two plates 12.0 $\mathrm{cm}$ on each side and 4.50 mm apart. Half of the space between these plates contains only air, but the other half is filled with Plexiglas\oplus of dielectric constant 3.40 . (See Figure $18.55 .$ An 18.0 V battery is connected across the plates. (a) What is the capacitance of this combination? (Hint: Can you think of this capacitor as equivalent to two capacitors in parallel? (b) How much energy is stored in the capacitor? (c) If we remove the Plexiglas@, but change nothing else, how much energy will be stored in the capacitor?

Kai Chen

Princeton University

Problem 87

A parallel-plate capacitor with plate separation $d$ has the space between the plates filled with two slabs of dielectric, one with constant $K_{1}$ and the other with constant $K_{2, \text { and each }}$ having thickness $d / 2$ . (a) Show that the capacitance is given by $C=\frac{2 \epsilon_{o} A}{d}\left(\frac{K_{1} K_{2}}{K_{1}+K_{2}}\right) \cdot($Hint: Can you think of this combination as two capacitors in series? (b) To see if your answer is reasonable, check it in the following cases: (i) There is only one dielectric, with constant $K,$ and it completely fills the space between the plates. (ii) The plates have nothing but air, which we can treat as vacuum, between them.

Katie Mcalpine

Numerade Educator

Problem 88

BIO The electric egg. The eggs of many species undergo a rapid change in the electrical potential difference across the outer membrane when they are fertilized. This change in potential difference affects the physiological development of the eggs. The poterntial difference across the membrane is called the membrane potential, $V_{m},$ defined as the inside potential minus the outside potential. The membrane potential $V_{m}$ arises when protein enzymes use the energy available in ATP to actively expel sodium ions (Na') and accumulate potassium ions $\left(\mathrm{K}^{+}\right) .$ Because the membrane of the unfertilized egg is selectively permeable to $\mathrm{K}^{+},$ the $V_{m}$ of the resting sea urchin egg is about $-70 \mathrm{mV}$ ; that is, the inside has a potential of 70 $\mathrm{mV}$ less than that of the outside. The egg membrane behaves as a capacitor with a specific capacitance of about 1$\mu \mathrm{F} / \mathrm{cm}^{2} .$ When a sea urchin egg is fertilized, Na' channels in the membrane are opened, $\mathrm{Na}^{+}$ enters the egg, and $V_{m}$ rapidly changes to $+30 \mathrm{mV},$ where it remains for several minutes. The concentration of $\mathrm{Na}^{+}$ in the egg's interior is about 30 mmoles/liter (30 $\mathrm{mM} )$ and 450 $\mathrm{mM}$ in the surrounding sea water. The inside $\mathrm{K}^{*}$ concentration is about 200 $\mathrm{mM}$ and the outside $\mathrm{K}^{+}$ is 10 $\mathrm{mM} .$ A useful constant that connects electrical and chemical units is the Faraday number, which has a value of approximately $10^{5}$ coulomb/mole. That is, an Avogadro number (a mole) of monovalent ions such as Na^ + or $\mathrm{K}^{+}$ carries a charge of $10^{5} \mathrm{C}$ .

How many moles of $\mathrm{Na}^{+}$ must move per unit area of membrane to change $V_{m}$ from $-70 \mathrm{mV}$ to $+30 \mathrm{mV},$ making the assumption that the membrane behaves purely as a capacitor?

A. $10^{-4}$ mole $/ \mathrm{cm}^{2}$
B. $10^{-9}$ mole/cm $^{2}$
C. $10^{-12} \mathrm{mole} / \mathrm{cm}^{2}$
D. $10^{-14} \mathrm{mole} / \mathrm{cm}^{2}$

Kai Chen

Princeton University

Problem 89

Suppose the egg has a diameter of 200$\mu \mathrm{m} .$ What fractional change in internal $\mathrm{Na}^{+}$ concentration results from the fertilization-induced change in $V_{m} ?$ Assume that the $\mathrm{Na}^{+}$ ions are distributed throughout the cell volume.

A. Increases by 1 part in $10^{4}$
B. Increases by 1 part in $10^{5}$
C. Increases by 1 part in $10^{6}$
D. Increases by 1 part in $10^{7}$

Vidhi Bhatt

Numerade Educator

Problem 90

Suppose the change in $V_{m}$ was caused by the entry of $\mathrm{Ca}^{2+}$ instead of $\mathrm{Na}^{+} .$ How many $\mathrm{Ca}^{2+}$ ions would have to enter the cell per unit membrane to produce the change?

A. Half as many as for $\mathrm{Na}^{+}$
B. The same as for $\mathrm{Na}^{+}$
C. Twice as many as for Na^{+}
D. Cannot say without knowing the inside and outside concentrations of $C a^{2+}$

Kai Chen

Princeton University

Problem 91

What is the minimum amount of work that must be done by the cell to restore $V_{m}$ to its original value?

A. 3 $\mathrm{mJ}$
B. 3$\mu \mathrm{J}$
C. 3 $\mathrm{nJ}$
D. 3 $\mathrm{pJ}$

Vidhi Bhatt

Numerade Educator