9 Applications of Electrostatics

The study of electrostatics—the behavior of electric charges at rest and the electric fields they produce—has led to many important technologies used in everyday life, industry, medicine, and scientific research. In the health sciences, electrostatic principles are used to shield sensitive electronic instruments, manipulate charged particles, improve air quality, and produce high-resolution medical and laboratory equipment. In this section, we will examine several practical applications that illustrate how the concepts developed in the previous chapters are used in the real world.

The Van de Graaff Generator

The Van de Graaff generator is one of the most recognizable demonstrations of electrostatics. Although it is commonly seen in science museums and classrooms producing sparks and making people's hair stand on end, it was originally developed as a serious research instrument. The first Van de Graaff generator was built in 1931 by physicist Robert Van de Graaff, building on earlier ideas proposed by Lord Kelvin, and was designed to accelerate charged particles for nuclear physics experiments.

A Van de Graaff generator combines conducting and insulating materials to transport electric charge onto a large hollow metal sphere. An insulating belt carries charge upward, where it is transferred to the conducting dome. Since excess charge on a conductor always moves to its outer surface, the sphere can accumulate an enormous amount of charge, producing extremely high electric potentials (voltages).

There is, however, a practical limit to how much charge can be stored. As the electric field around the sphere becomes stronger, it begins to ionize the surrounding air. Once the air becomes conducting, charge leaks away from the sphere through the ionized air, limiting the maximum achievable voltage. Even with this limitation, large research Van de Graaff generators can reach voltages of approximately 15 million volts.

Figure 9.1 shows the main components of a Van de Graaff generator and how charge is transported to the metal sphere.

Xerography

Most photocopiers and many laser printers produce images using an electrostatic process called xerography, a name derived from Greek words meaning "dry writing." Rather than using liquid ink, xerography relies on electrically charged toner particles that are attracted to selected regions of a photoconducting drum. The basic process is illustrated in Figure 9.2.

The central component is a rotating aluminum drum coated with a thin layer of selenium, a photoconductor. In the dark, selenium behaves as an insulator, but when exposed to light it becomes electrically conducting. Initially, the surface of the drum is given a uniform positive charge by a device called a corotron.

The drum is then exposed to a bright image of the document being copied. Wherever light strikes the selenium, it becomes conducting and the positive surface charge drains away through the grounded aluminum drum. Regions that remain dark retain their positive charge, producing an invisible electrostatic image that matches the original document.

Next, fine black toner particles carrying a negative charge are spread over the drum. Because opposite charges attract, the toner sticks only to the positively charged regions. A sheet of paper is then given a stronger positive charge than the drum, causing the toner to transfer from the drum onto the paper. Finally, heated rollers melt the toner and permanently fuse it to the paper. The drum is then cleaned, recharged, and prepared to produce the next copy.

Figure 9.2 summarizes the main stages of the xerographic printing process.

Laser Printers

Laser printers use the same electrostatic principles as xerography, but instead of projecting an image of an existing document, they use a finely focused laser beam to create the charge pattern directly on a photoconducting drum, as illustrated in Figure 9.3. The laser is controlled by digital information from a computer, allowing it to produce sharp text, graphics, and images with very high precision.

The printing process begins by giving the photoconducting drum a uniform electric charge. The laser then scans across the drum, selectively discharging regions that correspond to the desired image. This creates an invisible electrostatic pattern on the drum. Negatively charged toner particles are attracted to the remaining positively charged regions, transferred to positively charged paper, and permanently fused to the paper by heated rollers.

Although modern laser printers perform sophisticated digital processing—such as rendering fonts, graphics, and page layouts—the underlying physics remains the same. Electrostatic forces control the movement of the toner particles throughout the printing process, making laser printers one of the most widespread everyday applications of electrostatics.

Figure 9.3 shows how a laser printer uses a computer-controlled laser beam and electrostatic attraction to place toner on paper.

Inkjet Printers and Electrostatic Painting

Electrostatic forces are also used in some types of inkjet printers. In these systems, a nozzle produces a continuous stream of tiny ink droplets, which are given a small electric charge as they are formed as shown in Figure 9.4.

After leaving the nozzle, the charged droplets pass between pairs of electrically charged plates. The electric field between the plates deflects each droplet by a controlled amount, allowing it to land at a precise location on the paper. By accurately controlling millions of droplets, the printer produces high-quality text and images. Color printing is achieved by combining droplets of different colored inks, typically cyan, magenta, yellow, and black (CMYK).

Figure 9.4 shows how charged ink droplets are deflected by electric fields between charged plates.

The same electrostatic principles are widely used in electrostatic painting. Paint droplets are given an electric charge as they leave the spray nozzle, while the object being painted is given the opposite charge or is electrically grounded. The resulting electric attraction pulls the paint toward the object, producing a more uniform coating with less wasted paint.
Electrostatic painting is particularly effective for coating objects with complex shapes, such as automobile bodies, bicycles, and household appliances. Because the charged paint droplets repel one another, they spread out evenly in the spray, while the electric field guides them toward the surface. This combination improves coverage, reduces overspray, and increases the efficiency of the painting process.

Smoke Precipitators and Electrostatic Air Cleaning

Electrostatic forces are widely used to remove tiny particles from the air. Devices known as electrostatic precipitators and electrostatic air cleaners use electric fields to capture smoke, dust, pollen, and other airborne particles before they can be released into the environment or recirculated indoors.
As shown in Figure 9.5, air first passes through a set of electrodes that give suspended particles an electric charge. The charged particles then flow between collecting plates carrying the opposite charge. Because opposite charges attract, the particles are pulled out of the air and remain attached to the collecting plates, while the cleaned air continues through the system.
Large electrostatic precipitators are commonly installed in power plants and industrial facilities, where they can remove more than 99% of particulate matter from exhaust gases before they are released into the atmosphere. Smaller electrostatic air cleaners are used in homes, hospitals, and commercial buildings to reduce airborne dust, smoke, pollen, mold spores, and other allergens.
Health Connection: Fine airborne particles can irritate the respiratory system and contribute to asthma, allergies, and cardiovascular disease. By removing these particles from the air, electrostatic filtration systems help improve indoor air quality and reduce exposure to harmful pollutants, particularly for individuals with respiratory conditions.

Figure 9.5 shows how an electrostatic precipitator charges particles and collects them on oppositely charged plates.

Problem-Solving Strategy for Electrostatics

When solving electrostatics problems, the following steps can help organize your work:

1. Identify the situation. Determine whether the problem involves electric charges, electric forces, electric fields, or a combination of these concepts.
2. Sketch the problem. Draw the charges, their locations, and the directions of any known forces or electric fields. A simple diagram often makes the solution much clearer.
3. List the known and unknown quantities. Write down all given values with their units and identify the quantity you need to calculate.
4. Select the appropriate equation. Choose the relationship that applies to the problem, such as Coulomb's law or the definition of the electric field. Whenever possible, solve algebraically before substituting numerical values.
5. Calculate the answer. Substitute the known values, carry the units throughout the calculation, and report the final result with the correct number of significant figures.
6. Check your answer. Verify that the units are correct and that the magnitude and direction of your answer make physical sense.

Integrated Concepts

Integrated Concepts problems combine electrostatics with ideas from earlier chapters. Solving these problems often requires applying several areas of physics rather than treating each topic independently.

Depending on the problem, you may need concepts from:

• Kinematics
• Two-Dimensional Motion
• Newton's Laws of Motion
• Uniform Circular Motion and Gravitation
• Statics and Torque
• Fluid Statics

As you work through these exercises, begin by identifying which physical principles apply before selecting equations. Many real-world problems require combining ideas from multiple areas of physics, making these exercises excellent practice for scientific and engineering problem solving.

The following example illustrates how concepts from electrostatics and mechanics can be combined to solve a real-world problem.

Example 9.1: Acceleration of a Charged Gasoline Droplet

If a gasoline pump is not properly grounded, static electricity can build up as fuel flows through the nozzle. Suppose a tiny gasoline droplet has a mass of [latex]4.00\times10^{-15}\ \text{kg}[/latex] and carries a positive charge of [latex]3.20\times10^{-19}\ \text{C}[/latex]. If an upward electric field of magnitude [latex]3.00\times10^{5}\ \text{N/C}[/latex] exists nearby, determine:

1. the weight of the droplet,
2. the electric force acting on it, and
3. its acceleration.

Strategy

This problem combines concepts from mechanics and electrostatics. We first calculate the droplet's weight using the definition of weight, then determine the electric force produced by the electric field. Finally, Newton's second law is applied using the net force acting on the droplet.

Solution

(a) Weight

The weight of the droplet is found from

[latex]w=mg[/latex]

[latex]w=(4.00\times10^{-15}\ \text{kg})(9.80\ \text{m/s}^2)=3.92\times10^{-14}\ \text{N}[/latex]

(b) Electric force

The electric force is found from

[latex]F=qE[/latex]

[latex]F=(3.20\times10^{-19}\ \text{C})(3.00\times10^{5}\ \text{N/C})=9.60\times10^{-14}\ \text{N}[/latex]

Since the droplet carries a positive charge, the electric force acts upward, in the same direction as the electric field.

(c) Acceleration

The electric force acts upward while the weight acts downward, so the net force is

[latex]F_{\text{net}}=F-w[/latex]

[latex]F_{\text{net}}=9.60\times10^{-14}-3.92\times10^{-14}=5.68\times10^{-14}\ \text{N}[/latex]

Applying Newton's second law,

[latex]a=\frac{F_{\text{net}}}{m}[/latex]

[latex]a=\frac{5.68\times10^{-14}\ \text{N}}{4.00\times10^{-15}\ \text{kg}}=14.2\ \text{m/s}^2[/latex]

The droplet accelerates upward.

Discussion

Although both the weight and the electric force are extremely small, the electric force is about 2.5 times larger than the weight of the droplet. As a result, electrostatic forces can easily dominate the motion of microscopic particles. This is why static electricity can attract dust, influence aerosol droplets, and, under certain conditions, create hazards when flammable liquids such as gasoline are handled without proper grounding.

Unreasonable Results

Some problems in this chapter are labeled Unreasonable Results. In these exercises, the physics is applied correctly, but one or more assumptions in the problem are unrealistic or inconsistent. The goal is not only to perform the calculation but also to recognize when the final answer cannot represent a real physical situation.

When solving these problems, follow these guidelines:

1. Solve the problem normally using the appropriate physical principles and problem-solving strategy.
2. Evaluate the result. Ask whether the magnitude, sign, units, and direction of the answer are physically reasonable.
3. If the result is unreasonable, identify which assumption is responsible. For example, an unrealistically large electric force may result from assuming an impossibly large charge or an unrealistically small separation distance.

Developing the habit of checking whether an answer makes physical sense is an important scientific skill. In engineering, medicine, and research, recognizing unrealistic results is often just as valuable as performing the calculation itself.

Section Summary

• Electrostatics is the study of electric charges at rest, the electric fields they produce, and the forces between them.
• Understanding electrostatic principles has led to many important technologies, including Van de Graaff generators, photocopiers, laser printers, inkjet printers, electrostatic painting, and electrostatic air cleaners.
• Many of these applications rely on the controlled motion of charged particles in electric fields to produce useful effects, from high-quality printing to the removal of airborne pollutants.
• Electrostatic principles are also widely used in scientific research, medicine, and industry, where controlling electric charge and electric fields is essential for measurement, manufacturing, and environmental protection.

Problems & Exercises

51. (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 [latex]2.00\,\mu\text{C}[/latex] charge on the Van de Graaff’s belt?
52. (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.
53. A simple and common technique for accelerating electrons is shown in Figure 9.6, where there is a uniform electric field between two plates, as shown in Figure 9.6. 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 [latex]2.50\times10^{4}\,\text{N/C}[/latex]. (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole.Figure 9.6 shows parallel conducting plates used to accelerate electrons.

    

54. Earth 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?
55. Point charges of [latex]25.0\,\mu\text{C}[/latex] and [latex]45.0\,\mu\text{C}[/latex] 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?
56. What can you say about two charges [latex]{q}_{1}[/latex] and [latex]{q}_{2}[/latex], if the electric field one-fourth of the way from [latex]{q}_{1}[/latex] to [latex]{q}_{2}[/latex] is zero?
57. Integrated Concepts Calculate the angular velocity [latex]\omega[/latex] of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is [latex]0.530\times10^{-10}\,\text{m}[/latex]. You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction.
58. Integrated Concepts An electron has an initial velocity of [latex]5.00\times10^{6}\,\text{m/s}[/latex] in a uniform [latex]2.00\times10^{5}\,\text{N/C}[/latex] 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?
59. Integrated Concepts The practical limit to an electric field in air is about [latex]3.00\times10^{6}\,\text{N/C}[/latex]. 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 [latex]3.00%[/latex] of the speed of light, starting from rest. (b) Is this practical in air, or must it occur in a vacuum?
60. Integrated 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 9.7. Given the charge on the ball is [latex]1.00\,\mu\text{C}[/latex], find the strength of the field.Figure 9.7 shows the charged ball hanging at an angle in a horizontal electric field.

    

61. Integrated Concepts Figure 9.8 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 [latex]3.00\times10^{6}\,\text{m/s}[/latex], 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 9.8 shows the electron deflecting as it passes between oppositely charged plates.

    

62. Integrated 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. as shown in Figure 9.9. Given the oil drop to be [latex]1.00\,\mu\text{m}[/latex] in radius and have a density of [latex]920 kg/{m}^{3}[/latex]: (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.Figure 9.9 shows the basic arrangement of the Millikan oil drop experiment.

    

63. Integrated Concepts (a) In Figure 9.10, four equal charges [latex]q[/latex] lie on the corners of a square. A fifth charge [latex]Q[/latex] is on a mass [latex]m[/latex] directly above the center of the square, at a height equal to the length [latex]d[/latex] of one side of the square. Determine the magnitude of [latex]q[/latex] in terms of [latex]Q[/latex], [latex]m[/latex], and [latex]d[/latex], if the Coulomb force is to equal the weight of [latex]m[/latex]. (b) Is this equilibrium stable or unstable? Discuss.Figure 9.10 shows four equal charges at the corners of a square supporting a fifth charge above the center.

    

64. Unreasonable 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?
65. Unreasonable 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?
66. Unreasonable 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?
67. Construct 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.
68. Construct 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.
69. Critical Thinking. A frictionless circular track with a diameter of 4.00 m has two spheres of 10.0 kg each anchored very near it. The track is oriented with a diameter along the x-axis. The anchored spheres are just outside the track. One of the anchored spheres is just left of the origin at a distance of [latex]1.00\times10^{-15}\,\text{m}[/latex] from it, and the other is on the x-axis just right of the track at a distance of [latex]1.00\times10^{-15}\,\text{m}[/latex] past [latex](4,0)[/latex]. Each sphere carries a charge of [latex]2.00\times10^{-9}\,\text{C}[/latex]. (Note: The slight distance from the track is so the particles cannot collide, and does not enter into the calculations to three significant figures.)
    1. 1. A third sphere has a mass of 10.0 kg and carries a charge of [latex]-1.00\times10^{-9}\,\text{C}[/latex]. It is free to move on the track. It is initially placed an equal distance from the two anchored spheres at [latex](2.00,\;2.00)[/latex]. What is the total force on the third sphere? (Be certain to include gravity between the spheres.)
       2. Will the third sphere move?
       3. Label the first two spheres A and B, and the movable sphere C. Now start sphere C at [latex](1.00,\;2.00)[/latex]. What are the forces on sphere C?
       4. Will sphere C now move?
       5. If it does, where will it stop moving and reverse course? If it does not move, you may skip this part.
       6. How many equilibrium points are there on the track at which the movable sphere will remain stationary? Where are they?

Glossary

Van de Graaff generator

A machine that uses a moving insulating belt to accumulate a very large electric charge on a conducting sphere, producing extremely high voltages for demonstrations and scientific applications.

electrostatics

The study of electric charges at rest, the electric fields they produce, and the forces between them.

photoconductor

A material that behaves as an electrical insulator in the dark but becomes conducting when exposed to light.

xerography

A dry copying process that uses electrostatic charges to attract toner and transfer images to paper.

grounded

Connected electrically to Earth so that excess electric charge can flow freely to or from an object.

laser printer

A printer that uses a laser to create an electrostatic image on a photoconducting drum, which attracts toner that is then transferred and fused onto paper.

ink-jet printer

A printer that forms tiny electrically charged ink droplets and uses electric fields to accurately direct them onto paper.

electrostatic precipitator

An air-cleaning device that electrically charges airborne particles and then removes them by attracting them to oppositely charged collecting plates.