Problems & Exercises

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?

A bare helium nucleus has two positive charges and a mass of $6\text{.}\text{64}\times {\text{10}}^{\text{–27}}\text{kg}\text{.}$ (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?

Integrated Concepts

The temperature near the center of the Sun is thought to be 15 million degrees Celsius $\left(1.5\times {\text{10}}^7\mathrm{ºC}\right)$. 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?

Integrated Concepts

A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of $1.00\times {\text{10}}^{\text{2}}\text{MV}$. (a) What energy was dissipated? (b) What mass of water could be raised from $\text{15ºC}$ 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.

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 $1\text{.}\text{00}\times {\text{10}}^{\text{–12}}\text{m}$ 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?

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 ampere-hours as energy in joules.

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

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 $1\text{.}\text{50}\times {\text{10}}^4\mathrm{V}$?

How far apart are two conducting plates that have an electric field strength of $4\text{.}\text{50}\times {\text{10}}^3\text{V/m}$ between them, if their potential difference is 15.0 kV?

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.

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?

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?

What is the potential $0\text{.}\text{530}\times {\text{10}}^{\mathrm{–10}}\text{m}$ from a proton (the average distance between the proton and electron in a hydrogen atom)?

How far from a $1\text{.}\text{00 µC}$ point charge will the potential be 100 V? At what distance will it be $\text{2.00}\times {\text{10}}^2\text{V}?$

If the potential due to a point charge is $5\text{.}\text{00}\times {\text{10}}^2\text{V}$ at a distance of 15.0 m, what are the sign and magnitude of the charge?

A research Van de Graaff generator has a 2.00-m-diameter 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?

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?

(a) What is the potential between two points situated 10 cm and 20 cm from a $3\text{.}\mathrm{0\; \mu C}$ point charge? (b) To what location should the point at 20 cm be moved to increase this potential difference by a factor of two?

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

Sketch the equipotential lines for the two equal positive charges shown in Figure 19.32. Indicate the direction of increasing potential.

Figure 19.32 The electric field near two equal positive charges is directed away from each of the charges.

Figure 19.33 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.

Sketch the equipotential lines a long distance from the charges shown in Figure 19.33. Indicate the direction of increasing potential.

Figure 19.33 The electric field near two charges.

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 19.33 for a similar situation. Indicate the direction of increasing potential.

Sketch the equipotential lines in the vicinity of the negatively charged conductor in Figure 19.34. How will these equipotentials look a long distance from the object?

Figure 19.34 A negatively charged conductor.

Sketch the equipotential lines surrounding the two conducting plates shown in Figure 19.35, 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 19.35

(a) Sketch the electric field lines in the vicinity of the charged insulator in Figure 19.36. Note its non-uniform charge distribution. (b) Sketch equipotential lines surrounding the insulator. Indicate the direction of increasing potential.

Figure 19.36 A charged insulating rod such as might be used in a classroom demonstration.

The naturally occurring charge on the ground on a fine day out in the open country is $\text{–1}\text{.}\text{00}{\text{nC/m}}^2$. (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.

The lesser electric ray (Narcine bancroftii) maintains an incredible charge on its head and a charge equal in magnitude but opposite in sign on its tail (Figure 19.37). (a) Sketch the equipotential lines surrounding the ray. (b) Sketch the equipotentials when the ray is near a ship with a conducting surface. (c) How could this charge distribution be of use to the ray?

Figure 19.37 Lesser electric ray (Narcine bancroftii) (credit: National Oceanic and Atmospheric Administration, NOAA's Fisheries Collection).

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

Calculate the voltage applied to a $2\text{.}\text{00 µF}$ capacitor when it holds $3\text{.}\text{10 µC}$ of charge.

What capacitance is needed to store $3\text{.}\text{00 µC}$ of charge at a voltage of 120 V?

Find the capacitance of a parallel plate capacitor having plates of area $5\text{.}\text{00}{\text{m}}^2$ that are separated by 0.100 mm of Teflon.

Integrated Concepts

A prankster applies 450 V to an $\text{80}\text{.}\mathrm{0\; \mu F}$ 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?

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 19.40.

Figure 19.40 A combination of series and parallel connections of capacitors.

A $1\text{65 µF}$ capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?

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

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.