# Chapter 18

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# Chapter 18 : Electric charge and electric field - all with Video Solutions

### Problem 1

Common static electricity involves charges ranging from nanocoulombs to microcoulombs. (a) How many electrons are needed to form a charge of $-2.00 \textrm{ nC}$ (b) How many electrons must be removed from a neutral object to leave a net charge of $0.500 \textrm{ }\mu\textrm{C}$?

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### Problem 2

If $1.80\times 10^{20}$ electrons move through a pocket calculator during a full day's operation, how many coulombs of charge moved through it?

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### Problem 3

To start a car engine, the car battery moves $3.75 \times 10^{21}$ electrons through the starter motor. How many coulombs of charge were moved?

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### Problem 5

Suppose a speck of dust in an electrostatic precipitator has $1.0000 \times 10^{12}$ protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?

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### Problem 6

An amoeba has $1.00\times 10^{16}$ protons and a net charge of 0.300 pC. (a) How many fewer electrons are there than protons? (b) If you paired them up, what fraction of the protons would have no electrons?

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### Problem 7

A 50.0 g ball of copper has a net charge of $2.00 \textrm{ }\mu\textrm{C}$. What fraction of the copper's electrons has been removed? (Each copper atom has 29 protons, and copper has an atomic mass of 63.5.)

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### Problem 8

What net charge would you place on a 100 g piece of sulfur if you put an extra electron on 1 in $10^{12}$ of its atoms? (Sulfur has an atomic mass of 32.1.)

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### Problem 11

(a) How strong is the attractive force between a glass rod with a $0.700 \textrm{ }\mu\textrm{C}$ charge and a silk cloth with a $-0.600 \textrm{ }\mu\textrm{C}$ charge, which are 12.0 cm apart, using the approximation that they act like point charges? (b) Discuss how the answer to this problem might be affected if the charges are distributed over some area and do not act like point charges.

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### Problem 15

If two equal charges each of 1 C each are separated in air by a distance of 1 km, what is the magnitude of the force acting between them? You will see that even at a distance as large as 1 km, the repulsive force is substantial because 1 C is a very significant amount of charge.

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### Problem 16

A test charge of $+2\textrm{ }\mu\textrm{C}$ is placed halfway between a charge of $+6\textrm{ }\mu\textrm{C}$ and another of $+4 \textrm{ }\mu\textrm{C}$ separated by 10 cm. (a) What is the magnitude of the force on the test charge? (b) What is the direction of this force (away from or toward the $+6\textrm{ }\mu\textrm{C}$ charge)?

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### Problem 17

Bare free charges do not remain stationary when close together. To illustrate this, calculate the acceleration of two isolated protons separated by 2.00 nm (a typical distance between gas atoms). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics.

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### Problem 18

(a) By what factor must you change the distance between two point charges to change the force between them by a factor of 10? (b) Explain how the distance can either increase or decrease by this factor and still cause a factor of 10 change in the force.

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### Problem 19

Suppose you have a total charge $q_{tot}$ that you can split in any manner. Once split, the separation distance is fixed. How do you split the charge to achieve the greatest force?

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### Problem 20

(a) Common transparent tape becomes charged when pulled from a dispenser. If one piece is placed above another, the repulsive force can be great enough to support the top piece's weight. Assuming equal point charges (only an approximation), calculate the magnitude of the charge if electrostatic force is great enough to support the weight of a 10.0 mg piece of tape held 1.00 cm above another. (b) Discuss whether the magnitude of this charge is consistent with what is typical of static electricity.

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### Problem 21

a) Find the ratio of the electrostatic to gravitational force between two electrons. (b) What is this ratio for two protons? (c) Why is the ratio different for electrons and protons?

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### Problem 23

A certain five cent coin contains 5.00 g of nickel. What fraction of the nickel atoms' electrons, removed and placed 1.00 m above it, would support the weight of this coin? The atomic mass of nickel is 58.7, and each nickel atom contains 28 electrons and 28 protons.

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### Problem 24

(a) Two point charges totaling $8.00\textrm{ }\mu\textrm{C}$ exert a repulsive force of 0.150 N on one another when separated by 0.500 m. What is the charge on each? (b) What is the charge on each if the force is attractive?

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### Problem 25

Point charges of $5.00 \textrm{ }\mu\textrm{C}$ and $-3.00 \textrm{ }\mu\textrm{C}$ are placed 0.250 m apart. (a) Where can a third charge be placed so that the net force on it is zero? (b) What if both charges are positive?

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### Problem 26

Two point charges $q_1$ and $q_2$ are 3.00 m apart, and their total charge is $20\textrm{ }\mu\textrm{C}$ . (a) If the force of repulsion between them is 0.075N, what are magnitudes of the two charges? (b) If one charge attracts the other with a force of 0.525N, what are the magnitudes of the two charges? Note that you may need to solve a quadratic equation to reach your answer.

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### Problem 27

What is the magnitude and direction of an electric field that exerts a $2.00 \times 10^{-5} \textrm{ N}$ upward force on a $-1.75 \textrm{ }\mu\textrm{C}$ charge?

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### Problem 28

What is the magnitude and direction of the force exerted on a $3.50\textrm{ }\mu\textrm{C}$ charge by a 250 N/C electric field that points due east?

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### Problem 31

Calculate the initial (from rest) acceleration of a proton in a $5.00 \times 10^6 \textrm{ N/C}$ electric field (such as created by a research Van de Graaff). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics.

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### Problem 32

(a) Find the direction and magnitude of an electric field that exerts a $4.80\times 10^{-17}\textrm{ N}$ westward force on an electron. (b) What magnitude and direction force does this field exert on a proton?

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### Problem 33

(a) Sketch the electric field lines near a point charge $+q$. (b) Do the same for a point charge $-3.00q$.

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### Problem 35

Figure 18.47 shows the electric field lines near two charges $q_1$ and $q_2$. What is the ratio of their magnitudes? (b) Sketch the electric field lines a long distance from the charges shown in the figure.

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### Problem 36

Sketch the electric field lines in the vicinity of two opposite charges, where the negative charge is three times greater in magnitude than the positive. (See Figure 18.54 for a similar situation).

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### Problem 37

Sketch the electric field lines in the vicinity of the conductor in Figure 18.47 given the field was originally uniform and parallel to the object's long axis. Is the resulting field small near the long side of the object?

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### Problem 38

Sketch the electric field lines in the vicinity of the conductor in Figure 18.49 given the field was originally uniform and parallel to the object's long axis. Is the resulting field small near the long side of the object?

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### Problem 39

Sketch the electric field between the two conducting plates shown in Figure 18.49, given the top plate is positive and an equal amount of negative charge is on the bottom plate. Be certain to indicate the distribution of charge on the plates.

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### Problem 41

What is the force on the charge located at $x = 8.00\textrm{ cm}$ in Figure 18.51(a) given that $q = 1.00\textrm{ }\mu\textrm{C}$?

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### Problem 42

(a) Find the total electric field at $x = 1.00 \textrm{ cm}$ in Figure 18.52(b) given that $q = 5.00 \textrm{ nC}$. (b) Find the total electric field at $x = 11.00 \textrm{ cm}$ in Figure 18.52(b). (c) If the charges are allowed to move and eventually be brought to rest by friction, what will the final charge configuration be? (That is, will there be a single charge, double charge, etc., and what will its value(s) be?)

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### Problem 43

(a) Find the electric field at $x = 5.00\textrm{ cm}$ in Figure 18.51(a), given that $q = 1.00\textrm{ }\mu\textrm{C}$. (b) At what position between 3.00 and 8.00 cm is the total electric field the same as that for $-2q$ alone? (c) Can the electric field be zero anywhere between 0.00 and 8.00 cm? (d) At very large positive or negative values of x, the electric field approaches zero in both (a) and (b). In which does it most rapidly approach zero and why? (e) At what position to the right of 11.0 cm is the total electric field zero, other than at infinity? (Hint: A graphing calculator can yield considerable insight in this problem.)

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### Problem 44

(a) Find the total Coulomb force on a charge of 2.00 nC located at $x = 4.00 \textrm{ cm}$ in Figure 18.52 (b), given that $q = 1.00 \textrm{ }\mu\textrm{C}$. (b) Find the x-position at which the electric field is zero in Figure 18.52(b).

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### Problem 45

Using the symmetry of the arrangement, determine the direction of the force on q in the figure below, given that $q_a = q_b = +7.50\textrm{ }\mu\textrm{C}$ and $q_c=q_d = -7.50\textrm{ }\mu\textrm{C}$. (b) Calculate the magnitude of the force on the charge $q$ , given that the square is 10.0 cm on a side and $q = 2.00\textrm{ }\mu\textrm{C}$.

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### Problem 46

(a) Using the symmetry of the arrangement, determine the direction of the electric field at the center of the square in Figure 18.53, given that $q_a = q_b = -1.00 \textrm{ }\mu\textrm{C}$ and $q_c = q_d = +1.00 \textrm{ }\mu\textrm{C}$. (b) Calculate the magnitude of the electric field at the location of $q$, given that the square is 5.00 cm on a side.

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### Problem 47

Find the electric field at the location of qa in Figure 18.52 given that $q_b = q_c = q_d = +2.00\textrm{ nC}$, $q = -1.00\textrm{ nC}$, and the square is 20.0 cm on a side.

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### Problem 48

Find the total Coulomb force on the charge $q$ in Figure 18.53, given that $q = 1.00 \textrm{ }\mu\textrm{C}$, $q_a = 2.00 \textrm{ }\mu\textrm{C}$, $q_b = -3.00 \textrm{ }\mu\textrm{C}$, $q_c = -4.00 \textrm{ }\mu\textrm{C}$, and $q_d = +1.00 \textrm{ }\mu\textrm{C}$.The square is 50.0 cm on a side.

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### Problem 49

(a) Find the electric field at the location of $q_a$ in Figure 18.53, given that $q_b = +10.00\textrm{ }\mu\textrm{C}$ and $q_c = -5.00\textrm{ }\mu\textrm{C}$. (b) What is the force on $q_a$, given that $q_a = +1.50\textrm{ nC}$?

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### Problem 50

(a) Find the electric field at the center of the triangular configuration of charges in Figure 18.54, given that $q_a = +2.50 \textrm{ nC}$, $q_b = -8.00 \textrm{ nC}$, and $q_c = +1.50 \textrm{ nC}$. (b) Is there any combination of charges, other than $q_a = q_b = q_c$, that will produce a zero strength electric field at the center of the triangular configuration? The equilateral triangle has a side length of $25.0 \textrm{ cm}$.

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### Problem 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 $2.00 \textrm{ }\mu\textrm{C}$ charge on the Van de Graaff's belt?

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### Problem 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.

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### Problem 53

A simple and common technique for accelerating electrons is shown in Figure 18.55, 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 $2.50 \times 10^4 \textrm{ N/C}$. (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole.

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### Problem 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?

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### Problem 55

Point charges of $25.0 \textrm{ }\mu\textrm{C}$ and $45.0 \textrm{ }\mu\textrm{C}$ 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?

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### Problem 57

Calculate the angular velocity $\Omega$ of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is $0.530 \times 10^{-10} \textrm{ m}$You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction.

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### Problem 58

An electron has an initial velocity of $5.00\times 10^{6}\textrm{ m/s}$ in a uniform $2.00\times 10^{5}\textrm{ N/C}$ 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?

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### Problem 59

The practical limit to an electric field in air is about $3.00 \times 10^6 \textrm{ N/C}$. 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 3.00% of the speed of light, starting from rest. (b) Is this practical in air, or must it occur in a vacuum?

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### Problem 60

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 18.56. Given the charge on the ball is $1.00\textrm{ }\mu\textrm{C}$ , find the strength of the field.

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### Problem 61

Figure 18.57 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 $3.00 \times 10^6 \textrm{ m/s}$, 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.

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### Problem 62

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 18.58.) Given the oil drop to be $1.00\textrm{ }\mu\textrm{m}$ in radius and have a density of $920\textrm{ kg/m}^3$: (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.

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### Problem 63

(a) In Figure 18.59, 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.

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### Problem 64

(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?

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### Problem 65

(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?

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### Problem 66

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?

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