(a) A sphere has a surface uniformly charged with 1.00 C.

Problem 3CQ What is the relationship between voltage and energy? More precisely, what is the relationship between potential difference and electric potential energy?

Problem 2PE An evacuated tube uses an accelerating voltage of 40 kV to accelerate electrons to hit a copper plate and produce x rays. Non-relativistically, what would be the maximum speed of these electrons?

Problem 1PE Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be 1.67×10 – 27 kg.

Problem 70PE Unreasonable Results (a) On a particular day, it takes 9.60×103 J of electric energy to start a truck’s engine. Calculate the capacitance of a capacitor that could store that amount of energy at 12.0 V. (b) What is unreasonable about this result? (c) Which assumptions are responsible?

Problem 1CQ Voltage is the common word for potential difference. Which term is more descriptive, voltage or potential difference?

Problem 2CQ If the voltage between two points is zero, can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.

Problem 3PE A bare helium nucleus has two positive charges and a mass of 6.64 ×10–27 kg. (a) Calculate its kinetic energy in joules at 2.00% of the speed of light. (b) What is this in electron volts? (c) What voltage would be needed to obtain this energy?

Problem 4PE Integrated Concepts Singly charged gas ions are accelerated from rest through a voltage of 13.0 V. At what temperature will the average kinetic energy of gas molecules be the same as that given these ions?

Voltages are always measured between two points. Why?

Problem 5CQ How are units of volts and electron volts related? How do they differ?

Problem 5PE Integrated Concepts The temperature near the center of the Sun is thought to be 15 million degrees Celsius (1.5×107 ºC) . Through what voltage must a singly charged ion be accelerated to have the same energy as the average kinetic energy of ions at this temperature?

Problem 7CQ What is the strength of the electric field in a region where the electric potential is constant?

Problem 6PE Integrated Concepts (a) What is the average power output of a heart defibrillator that dissipates 400 J of energy in 10.0 ms? (b) Considering the high-power output, why doesn’t the defibrillator produce serious burns?

Problem 7PE Integrated Concepts A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of 1.00×102 MV . (a) What energy was dissipated? (b) What mass of water could be raised from 15ºC to the boiling point and then boiled by this energy? (c) Discuss the damage that could be caused to the tree by the expansion of the boiling steam.

Problem 6CQ Discuss how potential difference and electric field strength are related. Give an example.

Problem 8CQ Will a negative charge, initially at rest, move toward higher or lower potential? Explain why.

Problem 8PE Integrated Concepts A 12.0 V battery-operated bottle warmer heats 50.0 g of glass, 2.50×102 g of baby formula, and 2.00×102 g of aluminum from 20.0ºC to 90.0ºC . (a) How much charge is moved by the battery? (b) How many electrons per second flow if it takes 5.00 min to warm the formula? (Hint: Assume that the specific heat of baby formula is about the same as the specific heat of water.)

Problem 9PE Integrated Concepts A battery-operated car utilizes a 12.0 V system. Find the charge the batteries must be able to move in order to accelerate the 750 kg car from rest to 25.0 m/s, make it climb a 2.00×102 m high hill, and then cause it to travel at a constant 25.0 m/s by exerting a 5.00×102 N force for an hour.

Problem 9CQ In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?

Problem 11CQ What is an equipotential line? What is an equipotential surface?

Problem 10CQ Can the potential of a non-uniformly charged sphere be the same as that of a point charge? Explain.

Problem 10PE Integrated Concepts Fusion probability is greatly enhanced when appropriate nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This can be done using the kinetic energy of high-temperature gas ions or by accelerating the nuclei toward one another. (a) Calculate the potential energy of two singly charged nuclei separated by 1.00×10–12 m by finding the voltage of one at that distance and multiplying by the charge of the other. (b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?

Problem 11PE Unreasonable Results (a) Find the voltage near a 10.0 cm diameter metal sphere that has 8.00 C of excess positive charge on it. (b) What is unreasonable about this result? (c) Which assumptions are responsible?

Problem 12PE Construct Your Own Problem Consider a battery used to supply energy to a cellular phone. Construct a problem in which you determine the energy that must be supplied by the battery, and then calculate the amount of charge it must be able to move in order to supply this energy. Among the things to be considered are the energy needs and battery voltage. You may need to look ahead to interpret manufacturer’s battery ratings in amperehours as energy in joules.

Problem 12CQ Explain in your own words why equipotential lines and surfaces must be perpendicular to electric field lines.

Problem 14PE What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of 1.50×104 V ?

Problem 15CQ Use the characteristics of the Coulomb force to explain why capacitance should be proportional to the plate area of a capacitor. Similarly, explain why capacitance should be inversely proportional to the separation between plates.

Problem 14CQ Does the capacitance of a device depend on the applied voltage? What about the charge stored in it?

Problem 15PE The electric field strength between two parallel conducting plates separated by 4.00 cm is 7.50×104 V/m . (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be at zero volts. What is the potential 1.00 cm from that plate (and 3.00 cm from the other)?

Problem 13PE Show that units of V/m and N/C for electric field strength are indeed equivalent.

Problem 13CQ Can different equipotential lines cross? Explain.

Problem 16CQ Give the reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. What is the independent reason that a dielectric material also allows a greater voltage to be applied to a capacitor? (The dielectric thus increases C and permits a greater V .)

Problem 16PE How far apart are two conducting plates that have an electric field strength of 4.50×103 V/m between them, if their potential difference is 15.0 kV?

Problem 18CQ Sparks will occur between the plates of an air-filled capacitor at lower voltage when the air is humid than when dry. Explain why, considering the polar character of water molecules.

Problem 17PE (a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength for air ( 3.0 × 106 V/m ) if the plates are separated by 2.00 mm and a potential difference of 5.0×103 V is applied? (b) How close together can the plates be with this applied voltage?

Problem 19PE Membrane walls of living cells have surprisingly large electric fields across them due to separation of ions.

Problem 18PE The voltage across a membrane forming a cell wall is 80.0 mV and the membrane is 9.00 nm thick. What is the electric field strength? (The value is surprisingly large, but correct. Membranes are discussed in Capacitors and Dielectrics and Nerve Conduction—Electrocardiograms.) You may assume a uniform electric field.

Problem 19CQ Water has a large dielectric constant, but it is rarely used in capacitors. Explain why.

Membranes in living cells, including those in humans, are characterized by a separation of charge across the membrane. Effectively, the membranes are thus charged capacitors with important functions related to the potential difference across the membrane. Is energy required to separate these charges in living membranes and, if so, is its source the metabolization of food energy or some other source? Figure \(19.26\) The semipermeable membrane of a cell has different concentrations of ions inside and out. Diffusion moves the \(K^{+}\) (potassium) and \(\mathrm{Cl}^{-}\) (chloride) ions in the directions shown, until the Coulomb force halts further transfer. This results in a layer of positive charge on the outside, a layer of negative charge on the inside, and thus a voltage across the cell membrane. The membrane is normally impermeable to \(\mathrm{Na}^{+}\) (sodium ions). Equation Transcription: Text Transcription: 19.26 K^+ Cl^- Na^+

Problem 20PE Two parallel conducting plates are separated by 10.0 cm, and one of them is taken to be at zero volts. (a) What is the electric field strength between them, if the potential 8.00 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?

Problem 21CQ If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain.

Problem 21PE Find the maximum potential difference between two parallel conducting plates separated by 0.500 cm of air, given the maximum sustainable electric field strength in air to be 3.0×106 V/m .

Problem 22CQ How does the energy contained in a charged capacitor change when a dielectric is inserted, assuming the capacitor is isolated and its charge is constant? Does this imply that work was done?

Problem 23CQ What happens to the energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work done in the process?

Problem 22PE A doubly charged ion is accelerated to an energy of 32.0 keV by the electric field between two parallel conducting plates separated by 2.00 cm. What is the electric field strength between the plates?

Problem 23PE An electron is to be accelerated in a uniform electric field having a strength of 2.00×106 V/m . (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?

Problem 28PE What are the sign and magnitude of a point charge that produces a potential of –2.00 V at a distance of 1.00 mm?

Problem 29PE If the potential due to a point charge is 5.00×102 V at a distance of 15.0 m, what are the sign and magnitude of the charge?

Problem 24PE A 0.500 cm diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0 pC charge on its surface. What is the potential near its surface?

Problem 27PE How far from a 1.00 µC point charge will the potential be 100 V? At what distance will it be 2.00×102 V?

Problem 25PE What is the potential 0.530 × 10–10 m from a proton (the average distance between the proton and electron in a hydrogen atom)?

Problem 26PE (a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential 5.00 MV? (b) What does your answer imply about the practical aspect of isolating such a large charge?

Problem 30PE In nuclear fission, a nucleus splits roughly in half. (a) What is the potential 2.00×10 – 14 m from a fragment that has 46 protons in it? (b) What is the potential energy in MeV of a similarly charged fragment at this distance?

Problem 32PE An electrostatic paint sprayer has a 0.200-m-diameter metal sphere at a potential of 25.0 kV that repels paint droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a 0.100-mg drop of paint have to arrive at the object with a speed of 10.0 m/s?

Problem 33PE In one of the classic nuclear physics experiments at the beginning of the 20th century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of the doubly charged alpha nucleus was 5.00 MeV, how close to the gold nucleus (79 protons) could it come before being deflected?

Problem 31PE A research Van de Graaff generator has a 2.00-mdiameter metal sphere with a charge of 5.00 mC on it. (a) What is the potential near its surface? (b) At what distance from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in MeV at this distance?

Problem 34PE (a) What is the potential between two points situated 10 cm and 20 cm from a 3.0 µC point charge? (b) To what location should the point at 20 cm be moved to increase this potential difference by a factor of two?

Problem 36PE (a) Sketch the equipotential lines near a point charge + q . Indicate the direction of increasing potential. (b) Do the same for a point charge – 3 q

Problem 35PE Unreasonable Results (a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graaff terminal? (b) What is unreasonable about this result? (c) Which assumptions are responsible?

Sketch the equipotential lines for the two equal positive charges shown in Figure \(19.27\). Indicate the direction of increasing potential. Figure \(19.27\) The electric field near two equal positive charges is directed away from each of the charges. Equation Transcription: Text Transcription: 19.27

Sketch the equipotential lines a long distance from the charges shown in Figure \(19.28\). Indicate the direction of increasing potential. Figure \(19.28\) The electric field near two charges. Equation Transcription: Text Transcription: 19.28

Figure \(19.28\) shows the electric field lines near two charges \(q_{1}\) and \(q_{2}\) , the first having a magnitude four times that of the second. Sketch the equipotential lines for these two charges, and indicate the direction of increasing potential. Equation Transcription: Text Transcription: 19.28 q_1 q_2

Sketch the equipotential lines in the vicinity of two opposite charges, where the negative charge is three times as great in magnitude as the positive. See Figure \(19.28\) for a similar situation. Indicate the direction of increasing potential. Equation Transcription: Text Transcription: 19.28

(a) Sketch the electric field lines in the vicinity of the charged insulator in Figure \(19.31\). Note its non-uniform charge distribution. (b) Sketch equipotential lines surrounding the insulator. Indicate the direction of increasing potential. Figure \(19.31\) A charged insulating rod such as might be used in a classroom demonstration. Equation Transcription: Text Transcription: 19.31

Sketch the equipotential lines surrounding the two conducting plates shown in Figure \(19.30\), given the top plate is positive and the bottom plate has an equal amount of negative charge. Be certain to indicate the distribution of charge on the plates. Is the field strongest where the plates are closest? Why should it be? Figure \(19.30\) Equation Transcription: Text Transcription: 19.30

Sketch the equipotential lines in the vicinity of the negatively charged conductor in Figure \(19.29\). How will these equipotentials look a long distance from the object? Figure \(19.29\) A negatively charged conductor. Equation Transcription: Text Transcription: 19.29

Problem 44PE The naturally occurring charge on the ground on a fine day out in the open country is –1.00 nC/m2 . (a) What is the electric field relative to ground at a height of 3.00 m? (b) Calculate the electric potential at this height. (c) Sketch electric field and equipotential lines for this scenario.

Problem 47PE Find the charge stored when 5.50 V is applied to an 8.00 pF capacitor.

Problem 46PE What charge is stored in a 180 µF capacitor when 120 V is applied to it?

What charge is stored in a \(180 \mu \mathrm{F}\) capacitor when \(120 V\) is applied to it? Equation Transcription: 120 V Text Transcription: 180 mu F 120 V

Problem 50PE What voltage must be applied to an 8.00 nF capacitor to store 0.160 mC of charge?

Problem 48PE What charge is stored in the capacitor in Example 19.8? Example 19.8: Capacitance and Charge Stored in a Parallel Plate Capacitor (a) What is the capacitance of a parallel plate capacitor with metal plates, each of area 1.00 m2 , separated by 1.00 mm? (b) What charge is stored in this capacitor if a voltage of 3.00 × 103 V is applied to it?

Problem 49PE Calculate the voltage applied to a 2.00 µF capacitor when it holds 3.10 µC of charge.

Problem 51PE What capacitance is needed to store 3.00 µC of charge at a voltage of 120 V?

Problem 52PE What is the capacitance of a large Van de Graaff generator’s terminal, given that it stores 8.00 mC of charge at a voltage of 12.0 MV?

Problem 53PE Find the capacitance of a parallel plate capacitor having plates of area 5.00 m2 that are separated by 0.100 mm of Teflon.

Problem 56PE Unreasonable Results (a) A certain parallel plate capacitor has plates of area 4.00 m2 , separated by 0.0100 mm of nylon, and stores 0.170 C of charge. What is the applied voltage? (b) What is unreasonable about this result? (c) Which assumptions are responsible or inconsistent?

Problem 55PE Integrated Concepts A prankster applies 450 V to an 80.0 µF capacitor and then tosses it to an unsuspecting victim. The victim’s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change?

Problem 54PE (a)What is the capacitance of a parallel plate capacitor having plates of area 1.50 m2 that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?

Find the total capacitance of the combination of capacitors in Figure \(19.33\). Figure \(19.33\) A combination of series and parallel connections of capacitors. Equation Transcription: Text Transcription: 19.33

Problem 58PE Suppose you want a capacitor bank with a total capacitance of 0.750 F and you possess numerous 1.50 mF capacitors. What is the smallest number you could hook together to achieve your goal, and how would you connect them?

Find the total capacitance of the combination of capacitors shown in Figure \(19.34\). Figure \(19.34\) A combination of series and parallel connections of capacitors. Equation Transcription: Text Transcription: 19.34

Find the total capacitance of the combination of capacitors shown in Figure \(19.35\). Figure \(19.35\) A combination of series and parallel connections of capacitors. Equation Transcription: Text Transcription: 19.35

Problem 59PE What total capacitances can you make by connecting a 5.00 µF and an 8.00 µF capacitor together?

Problem 62PE Unreasonable Results (a) An 8.00 µF capacitor is connected in parallel to another capacitor, producing a total capacitance of 5.00 µF. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Problem 63PE (a) What is the energy stored in the 10.0 ?F capacitor of a heart defibrillator charged to 9.00×103 V ? (b) Find the amount of stored charge.

Problem 64PE In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the 8.00 ?F capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of stored charge.

Problem 65PE A 165 µF capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?

Problem 66PE Suppose you have a 9.00 V battery, a 2.00 ?F capacitor, and a 7.40 ?F capacitor. (a) Find the charge and energy stored if the capacitors are connected to the battery in series. (b) Do the same for a parallel connection.

Problem 67PE A nervous physicist worries that the two metal shelves of his wood frame bookcase might obtain a high voltage if charged by static electricity, perhaps produced by friction. (a) What is the capacitance of the empty shelves if they have area 1.00×102 m 2 and are 0.200 m apart? (b) What is the voltage between them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small hazard, calculate the energy stored.

Problem 69PE Construct Your Own Problem Consider a heart defibrillator similar to that discussed in Example 19.11. Construct a problem in which you examine the charge stored in the capacitor of a defibrillator as a function of stored energy. Among the things to be considered are the applied voltage and whether it should vary with energy to be delivered, the range of energies involved, and the capacitance of the defibrillator. You may also wish to consider the much smaller energy needed for defibrillation during open-heart surgery as a variation on this problem. Example 19.11: Capacitance in a Heart Defibrillator A heart defibrillator delivers 4.00 × 102 J of energy by discharging a capacitor initially at 1.00 × 104 V . What is its capacitance?

Problem 68PE Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric ( Volume = A · d ). Note that the applied voltage is limited by the dielectric strength.