189. Applications of ElectrostaticsLearning Objectives
The study of The Van de Graaff Generator
A very large excess charge can be deposited on the sphere, because it moves quickly to the outer surface. Practical limits arise because the large electric fields polarize and eventually ionize surrounding materials, creating free charges that neutralize excess charge or allow it to escape. Nevertheless, voltages of 15 million volts are well within practical limits. TakeHome Experiment: Electrostatics and HumidityRub a comb through your hair and use it to lift pieces of paper. It may help to tear the pieces of paper rather than cut them neatly. Repeat the exercise in your bathroom after you have had a long shower and the air in the bathroom is moist. Is it easier to get electrostatic effects in dry or moist air? Why would torn paper be more attractive to the comb than cut paper? Explain your observations. XerographyMost copy machines use an electrostatic process called A seleniumcoated aluminum drum is sprayed with positive charge from points on a device called a corotron. Selenium is a substance with an interesting property—it is a In the first stage of the xerography process, the conducting aluminum drum is The third stage takes a dry black powder, called toner, and sprays it with a negative charge so that it will be attracted to the positive regions of the drum. Next, a blank piece of paper is given a greater positive charge than on the drum so that it will pull the toner from the drum. Finally, the paper and electrostatically held toner are passed through heated pressure rollers, which melt and permanently adhere the toner within the fibers of the paper. Laser Printers
Ink Jet Printers and Electrostatic PaintingThe Once charged, the droplets can be directed, using pairs of charged plates, with great precision to form letters and images on paper. Ink jet printers can produce color images by using a black jet and three other jets with primary colors, usually cyan, magenta, and yellow, much as a color television produces color. (This is more difficult with xerography, requiring multiple drums and toners.) Electrostatic painting employs electrostatic charge to spray paint onto oddshaped surfaces. Mutual repulsion of like charges causes the paint to fly away from its source. Surface tension forms drops, which are then attracted by unlike charges to the surface to be painted. Electrostatic painting can reach those hardtoget at places, applying an even coat in a controlled manner. If the object is a conductor, the electric field is perpendicular to the surface, tending to bring the drops in perpendicularly. Corners and points on conductors will receive extra paint. Felt can similarly be applied. Smoke Precipitators and Electrostatic Air CleaningAnother important application of electrostatics is found in air cleaners, both large and small. The electrostatic part of the process places excess (usually positive) charge on smoke, dust, pollen, and other particles in the air and then passes the air through an oppositely charged grid that attracts and retains the charged particles. (See Figure 5.) Large ProblemSolving Strategies for Electrostatics
Integrated ConceptsThe Integrated Concepts exercises for this module involve concepts such as electric charges, electric fields, and several other topics. Physics is most interesting when applied to general situations involving more than a narrow set of physical principles. The electric field exerts force on charges, for example, and hence the relevance of Dynamics: Force and Newton’s Laws of Motion. The following topics are involved in some or all of the problems labeled “Integrated Concepts”:
The following worked example illustrates how this strategy is applied to an Integrated Concept problem: Example 1: Acceleration of a Charged Drop of GasolineIf steps are not taken to ground a gasoline pump, static electricity can be placed on gasoline when filling your car’s tank. Suppose a tiny drop of gasoline has a mass of $4.00\times {10}^{\mathrm{\u201315}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ and is given a positive charge of $3.20\times {10}^{\mathrm{\u201319}}\phantom{\rule{0.25em}{0ex}}\text{C}$. (a) Find the weight of the drop. (b) Calculate the electric force on the drop if there is an upward electric field of strength $3.00\times {10}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}$ due to other static electricity in the vicinity. (c) Calculate the drop’s acceleration. Strategy To solve an integrated concept problem, we must first identify the physical principles involved and identify the chapters in which they are found. Part (a) of this example asks for weight. This is a topic of dynamics and is defined in Dynamics: Force and Newton’s Laws of Motion. Part (b) deals with electric force on a charge, a topic of Electric Charge and Electric Field. Part (c) asks for acceleration, knowing forces and mass. These are part of Newton’s laws, also found in Dynamics: Force and Newton’s Laws of Motion. The following solutions to each part of the example illustrate how the specific problemsolving strategies are applied. These involve identifying knowns and unknowns, checking to see if the answer is reasonable, and so on. Solution for (a) Weight is mass times the acceleration due to gravity, as first expressed in $$w=\text{mg}.$$Entering the given mass and the average acceleration due to gravity yields $$w=(\text{4.00}\times {\text{10}}^{\text{15}}\phantom{\rule{0.25em}{0ex}}\text{kg})(9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2})=3\text{.}\text{92}\times {\text{10}}^{\text{14}}\phantom{\rule{0.25em}{0ex}}\text{N}.$$Discussion for (a) This is a small weight, consistent with the small mass of the drop. Solution for (b) The force an electric field exerts on a charge is given by rearranging the following equation: $$F=\text{qE}.$$Here we are given the charge ($3.20\times {10}^{\mathrm{\u201319}}\phantom{\rule{0.25em}{0ex}}\text{C}$ is twice the fundamental unit of charge) and the electric field strength, and so the electric force is found to be $$F=(3.20\times {\text{10}}^{\text{19}}\phantom{\rule{0.25em}{0ex}}\text{C})(3\text{.}\text{00}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C})=9\text{.}\text{60}\times {\text{10}}^{\text{14}}\phantom{\rule{0.25em}{0ex}}\text{N}.$$Discussion for (b) While this is a small force, it is greater than the weight of the drop. Solution for (c) The acceleration can be found using Newton’s second law, provided we can identify all of the external forces acting on the drop. We assume only the drop’s weight and the electric force are significant. Since the drop has a positive charge and the electric field is given to be upward, the electric force is upward. We thus have a onedimensional (vertical direction) problem, and we can state Newton’s second law as $$a=\frac{{F}_{\text{net}}}{m}.$$where ${F}_{\text{net}}=Fw$. Entering this and the known values into the expression for Newton’s second law yields $$\begin{array}{lll}a& =& \frac{Fw}{m}\\ & =& \frac{\text{9.60}\times {\text{10}}^{\text{14}}\phantom{\rule{0.25em}{0ex}}\text{N}\text{3.92}\times {\text{10}}^{\text{14}}\phantom{\rule{0.25em}{0ex}}\text{N}}{\text{4.00}\times {\text{10}}^{\text{15}}\phantom{\rule{0.25em}{0ex}}\text{kg}}\\ & =& \text{14}\text{.}2\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}.\end{array}$$Discussion for (c) This is an upward acceleration great enough to carry the drop to places where you might not wish to have gasoline. This worked example illustrates how to apply problemsolving strategies to situations that include topics in different chapters. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknown using familiar problemsolving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life. The following problems will build your skills in the broad application of physical principles. Unreasonable ResultsThe Unreasonable Results exercises for this module have results that are unreasonable because some premise is unreasonable or because certain of the premises are inconsistent with one another. Physical principles applied correctly then produce unreasonable results. The purpose of these problems is to give practice in assessing whether nature is being accurately described, and if it is not to trace the source of difficulty. ProblemSolving StrategyTo determine if an answer is reasonable, and to determine the cause if it is not, do the following.
Section Summary
Problems & ExercisesExercise 1(a) What is the electric field 5.00 m from the center of the terminal of a Van de Graaff with a 3.00 mC charge, noting that the field is equivalent to that of a point charge at the center of the terminal? (b) At this distance, what force does the field exert on a $2.00\phantom{\rule{0.25em}{0ex}}\mu \text{C}$ charge on the Van de Graaff’s belt? Exercise 2(a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces. Show/Hide Solution Solution(a) $5\text{.}\text{58}\times {\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{N/C}$ (b) the coulomb force is extraordinarily stronger than gravity Exercise 3A simple and common technique for accelerating electrons is shown in Figure 6, where there is a uniform electric field between two plates. Electrons are released, usually from a hot filament, near the negative plate, and there is a small hole in the positive plate that allows the electrons to continue moving. (a) Calculate the acceleration of the electron if the field strength is $2.50\times {10}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/C}$. (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole. Exercise 4Earth has a net charge that produces an electric field of approximately 150 N/C downward at its surface. (a) What is the magnitude and sign of the excess charge, noting the electric field of a conducting sphere is equivalent to a point charge at its center? (b) What acceleration will the field produce on a free electron near Earth’s surface? (c) What mass object with a single extra electron will have its weight supported by this field? Show/Hide Solution Solution(a) $6\text{.}\text{76}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{C}$ (b) $2\text{.}\text{63}\times {\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\phantom{\rule{0.25em}{0ex}}(\text{upward})$ (c) $2\text{.}\text{45}\times {\text{10}}^{\text{18}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ Exercise 5Point charges of $25.0\phantom{\rule{0.25em}{0ex}}\mu \text{C}$ and $45.0\phantom{\rule{0.25em}{0ex}}\mu \text{C}$ are placed 0.500 m apart. (a) At what point along the line between them is the electric field zero? (b) What is the electric field halfway between them? Exercise 6What can you say about two charges ${q}_{1}$ and ${q}_{2}$, if the electric field onefourth of the way from ${q}_{1}$ to ${q}_{2}$ is zero? Show/Hide Solution SolutionThe charge ${q}_{2}$ is 9 times greater than ${q}_{1}$. Exercise 7Integrated Concepts Calculate the angular velocity $\mathrm{\omega}$ of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is $0.530\times {10}^{\mathrm{\u201310}}\phantom{\rule{0.25em}{0ex}}\text{m}$. You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction. Exercise 8Integrated Concepts An electron has an initial velocity of $5.00\times {10}^{6}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ in a uniform $2.00\times {10}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}$ strength electric field. The field accelerates the electron in the direction opposite to its initial velocity. (a) What is the direction of the electric field? (b) How far does the electron travel before coming to rest? (c) How long does it take the electron to come to rest? (d) What is the electron’s velocity when it returns to its starting point? Exercise 9Integrated Concepts The practical limit to an electric field in air is about $3.00\times {10}^{6}\phantom{\rule{0.25em}{0ex}}\text{N/C}$. Above this strength, sparking takes place because air begins to ionize and charges flow, reducing the field. (a) Calculate the distance a free proton must travel in this field to reach $\mathrm{3.00\%}$ of the speed of light, starting from rest. (b) Is this practical in air, or must it occur in a vacuum? Exercise 10Integrated Concepts A 5.00 g charged insulating ball hangs on a 30.0 cm long string in a uniform horizontal electric field as shown in Figure 7. Given the charge on the ball is $1.00\phantom{\rule{0.25em}{0ex}}\mu \text{C}$, find the strength of the field. Exercise 11Integrated Concepts Figure 8 shows an electron passing between two charged metal plates that create an 100 N/C vertical electric field perpendicular to the electron’s original horizontal velocity. (These can be used to change the electron’s direction, such as in an oscilloscope.) The initial speed of the electron is $3.00\times {10}^{6}\phantom{\rule{0.25em}{0ex}}\text{m/s}$, and the horizontal distance it travels in the uniform field is 4.00 cm. (a) What is its vertical deflection? (b) What is the vertical component of its final velocity? (c) At what angle does it exit? Neglect any edge effects. Figure 8 Exercise 12Integrated Concepts The classic Millikan oil drop experiment was the first to obtain an accurate measurement of the charge on an electron. In it, oil drops were suspended against the gravitational force by a vertical electric field. (See Figure 9.) Given the oil drop to be $1.00\phantom{\rule{0.25em}{0ex}}\mu \text{m}$ in radius and have a density of $\mathrm{920\; kg/}{\mathrm{m}}^{3}$: (a) Find the weight of the drop. (b) If the drop has a single excess electron, find the electric field strength needed to balance its weight. Exercise 13Integrated Concepts (a) In Figure 10, four equal charges $q$ lie on the corners of a square. A fifth charge $Q$ is on a mass $m$ directly above the center of the square, at a height equal to the length $d$ of one side of the square. Determine the magnitude of $q$ in terms of $Q$, $m$, and $d$, if the Coulomb force is to equal the weight of $m$. (b) Is this equilibrium stable or unstable? Discuss. Exercise 14Unreasonable Results (a) Calculate the electric field strength near a 10.0 cm diameter conducting sphere that has 1.00 C of excess charge on it. (b) What is unreasonable about this result? (c) Which assumptions are responsible? Exercise 15Unreasonable Results (a) Two 0.500 g raindrops in a thunderhead are 1.00 cm apart when they each acquire 1.00 mC charges. Find their acceleration. (b) What is unreasonable about this result? (c) Which premise or assumption is responsible? Exercise 16Unreasonable Results A wrecking yard inventor wants to pick up cars by charging a 0.400 m diameter ball and inducing an equal and opposite charge on the car. If a car has a 1000 kg mass and the ball is to be able to lift it from a distance of 1.00 m: (a) What minimum charge must be used? (b) What is the electric field near the surface of the ball? (c) Why are these results unreasonable? (d) Which premise or assumption is responsible? Exercise 17Construct Your Own Problem Consider two insulating balls with evenly distributed equal and opposite charges on their surfaces, held with a certain distance between the centers of the balls. Construct a problem in which you calculate the electric field (magnitude and direction) due to the balls at various points along a line running through the centers of the balls and extending to infinity on either side. Choose interesting points and comment on the meaning of the field at those points. For example, at what points might the field be just that due to one ball and where does the field become negligibly small? Among the things to be considered are the magnitudes of the charges and the distance between the centers of the balls. Your instructor may wish for you to consider the electric field off axis or for a more complex array of charges, such as those in a water molecule. Exercise 18Construct Your Own Problem Consider identical spherical conducting space ships in deep space where gravitational fields from other bodies are negligible compared to the gravitational attraction between the ships. Construct a problem in which you place identical excess charges on the space ships to exactly counter their gravitational attraction. Calculate the amount of excess charge needed. Examine whether that charge depends on the distance between the centers of the ships, the masses of the ships, or any other factors. Discuss whether this would be an easy, difficult, or even impossible thing to do in practice.
