Problems

Problems that are not marked with a star are considered the easiest to solve. Problems that are marked with a single star (*) are more difficult, while those marked with a double star (**) are the most difficult.
Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
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Section 18.1 The Origin of Electricity,
Section 18.2 Charged Objects and the Electric Force,
Section 18.3 Conductors and Insulators,
Section 18.4 Charging by Contact and by Induction

| 1 | ssm How many electrons must be removed from an electrically neutral silver dollar to give it a charge of ?

| 2 | A plate carries a charge of , while a rod carries a charge of How many electrons must be transferred from the plate to the rod, so that both objects have the same charge?

| 3 | A metal sphere has a charge of . What is the net charge after electrons have been placed on it?

| 4 | Four identical metallic objects carry the following charges: + 1.6, + 6.2, − 4.8, and . The objects are brought simultaneously into contact, so that each touches the others. Then they are separated, (a) What is the final charge on each object? (b) How many electrons (or protons) make up the final charge on each object?

| 5 | ssm Consider three identical metal spheres, A, B, and C. Sphere A carries a charge of + 5q. Sphere B carries a charge of − q. Sphere C carries no net charge. Spheres A and B are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Last, sphere C is touched to sphere B and separated from it. (a) How much charge ends up on sphere C? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

* | 6 | Water has a mass per mole of 18.0 g/mol, and each water molecule (H2O) has 10 electrons. (a) How many electrons are there in one liter of water? (b) What is the net charge of all these electrons? Section 18.5 Coulomb’s Law

| 7 | ssm Two charges attract each other with a force of 1.5 N. What will be the force if the distance between them is reduced to one-ninth of its original value?

| 8 | Two spherical objects are separated by a distance of . The objects are initially electrically neutral and are very small compared to the distance between them. Each object acquires the same negative charge due to the addition of electrons. As a result, each object experiences an electrostatic force that has a magnitude of . How many electrons did it take to produce the charge on one of the objects?

| 9 | ssm www Two very small spheres are initially neutral and separated by a distance of 0.50 m. Suppose that electrons are removed from one sphere and placed on the other. (a) What is the magnitude of the electrostatic force that acts on each sphere? (b) Is the force attractive or repulsive? Why?

| 10 | A charge + q is located at the origin, while an identical charge is located on the x axis at x = + 0.50 m. A third charge of + 2 q is located on the x axis at such a place that the net electrostatic force on the charge at the origin doubles, its direction remaining unchanged. Where should the third charge be located?

| 11 | ssm Consult Concept Simulation 18.1 at for insight into this problem. Three charges are fixed to an x, y coordinate system. A charge of is on the y axis at y = + 3.0 m. A charge of is at the origin. Last, a charge of is on the x axis at x = + 3.0 m. Determine the magnitude and direction of the net electrostatic force on the charge at x = + 3.0 m. Specify the direction relative to the − x axis.

| 12 | A charge of is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius = 0.100 m). The charges on the circle are at the position due north and at the position due east. What is the magnitude and direction of the net electrostatic force acting on the charge at the center? Specify the direction relative to due east.

| 13 | ssm www Two particles, with identical positive charges and a separation of , are released from rest. Immediately after the release, particle 1 has an acceleration whose magnitude is , while particle 2 has an acceleration whose magnitude is . Particle 1 has a mass of . Find (a) the charge on each particle and (b) the mass of particle 2.

| 14 | Two tiny conducting spheres are identical and carry charges of and . They are separated by a distance of 2.50 cm. (a) What is the magnitude of the force that each sphere experiences, and is the force attractive or repulsive? (b) The spheres are brought into contact and then separated to a distance of 2.50 cm. Determine the magnitude of the force that each sphere now experiences, and state whether the force is attractive or repulsive.

| 15 | Interactive Solution 18.15 at provides a model for solving this type of problem. Two small objects, A and B, are fixed in place and separated by 3.00 cm in a vacuum. Object A has a charge of , and object B has a charge of . How many electrons must be removed from A and put onto B to make the electrostatic force that acts on each object an attractive force whose magnitude is 68.0 N?

* | 16 | Multiple-Concept Example 3 provides some pertinent background for this problem. Suppose a single electron orbits about a nucleus containing two protons (+ 2e), as would be the case for a helium atom from which one of the two naturally occurring electrons is removed. The radius of the orbit is . Determine the magnitude of the electron’s centripetal acceleration.

* | 17 | Interactive LearningWare 18.2 at provides one approach to solving problems such as this one. The drawing shows three point charges fixed in place. The charge at the coordinate origin has a value of ; the other two charges have identical magnitudes, but opposite signs: and . (a) Determine the net force (magnitude and direction) exerted on q1 by the other two charges. (b) If q1 had a mass of 1.50 g and it were free to move, what would be its acceleration?

* | 18 | The drawing shows an equilateral triangle, each side of which has a length of 2.00 cm. Point charges are fixed to each corner, as shown. The charge experiences a net force due to the charges qA and qB. This net force points vertically downward and has a magnitude of 405 N. Determine the magnitudes and algebraic signs of the charges qA and qB.

* | 19 | ssm Two spheres are mounted on identical horizontal springs and reston a frictionless table, as in the drawing. When the spheres are uncharged, the spacing between them is 0.0500 m, and the springs are unstrained. When each sphere has a charge of , the spacing doubles. Assuming that the spheres have a negligible diameter, determine the spring constant of the springs.

* | 20 | Multiple-Concept Example 3 illustrates several of the concepts used in this problem. A single electron orbits a lithium nucleus that contains three protons (+ 3e). The radius of the orbit is . Determine the kinetic energy of the electron.

* | 21 | An electrically neutral model airplane is flying in a horizontal circle on a 3.0-m guideline, which is nearly parallel to the ground. The line breaks when the kinetic energy of the plane is 50.0 J. Reconsider the same situation, except that now there is a point charge of + q on the plane and a point charge of − q at the other end of the guideline. In this case, the line breaks when the kinetic energy of the plane is 51.8 J. Find the magnitude of the charges.

** | 22 | Two identical small insulating balls are suspended by separate 0.25-m threads that are attached to a common point on the ceiling. Each ball has a mass of 8.0 × 10−4 kg. Initially the balls are uncharged and hang straight down. They are then given identical positive charges and, as a result, spread apart with an angle of 36° between the threads. Determine (a) the charge on each ball and (b) the tension in the threads.

** | 23 | ssm A small spherical insulator of mass and charge is hung by a thin wire of negligible mass. A charge of is held 0.150 m away from the sphere and directly to the right of it, so the wire makes an angle with the vertical (see the drawing). Find (a) the angle and (b) the tension in the wire.

** | 24 | There are four charges, each with a magnitude of . Two are positive and two are negative. The charges are fixed to the corners of a 0.30-m square, one to a corner, in such a way that the net force on any charge is directed toward the center of the square. Find the magnitude of the net electrostatic force experienced by any charge.
Section 18.6 The Electric Field,
Section 18.7 Electric Field Lines,
Section 18.8 The Electric Field Inside a Conductor: Shielding

| 25 | Concept Simulation 18.2 at provides background concerning the electric field lines that are the focus of this problem. Review the important features of electric field lines discussed in Conceptual Example 13. Three point charges (+ q, + 2q, and −3q) are at the corners of an equilateral triangle. Sketch in six electric field lines between the three charges

| 26 | Review Conceptual Example 13 as an aid in working this problem. Charges of −4q are fixed to diagonally opposite corners of a square. A charge of + 5q is fixed to one of the remaining corners, and a charge of + 3q is fixed to the last corner. Assuming that ten electric field lines emerge from the + 5q charge, sketch the field lines in the vicinity of the four charges.

| 27 | ssm www A tiny ball (mass = 0.012 kg) carries a charge of . What electric field (magnitude and direction) is needed to cause the ball to float above the ground?

| 28 | Four point charges have the same magnitude of and are fixed to the corners of a square that is 4.0 cm on a side. Three of the charges are positive and one is negative. Determine the magnitude of the net electric field that exists at the center of the square.

| 29 | Two charges, −16 and , are fixed in place and separated by 3.0 m. (a) At what spot along a line through the charges is the net electric field zero? Locate this spot relative to the positive charge. (Hint: The spot does not necessarily lie between the two charges.) (b) What would be the force on a charge of placed at this spot?

| 30 | The membrane surrounding a living cell consists of an inner and an outer wall that are separated by a small space. Assume that the membrane acts like a parallel plate capacitor in which the effective charge density on the inner and outer walls has a magnitude of . (a) What is the magnitude of the electric field within the cell membrane? (b) Find the magnitude of the electric force that would be exerted on a potassium ion (K+; charge = + e) placed inside the membrane.

| 31 | ssm Two charges are placed on the x axis. One of the charges is at x1 = + 3.0 cm and the other is at x1 = + 9.0 cm. Find the net electric field (magnitude and direction) at (a) x = 0 cm and (b) x = + 6.0 cm.

| 32 | A long, thin rod (length = 4.0 m) lies along the x axis, with its midpoint at the origin. In a vacuum, a point charge is fixed to one end of the rod, and a point charge is fixed to the other end. Everywhere in the x, y plane there is a constant external electric field that is perpendicular to the rod. With respect to the z axis, find the magnitude of the net torque applied to the rod.

| 33 | ssm A small drop of water is suspended motionless in air by a uniform electric field that is directed upward and has a magnitude of 8480 N/C. The mass of the water drop is . (a) Is the excess charge on the water drop positive or negative? Why? (b) How many excess electrons or protons reside on the drop?

| 34 | Review Conceptual Example 12 before attempting to work this problem. The magnitude of each of the charges in Figure 18-21 is . The lengths of the sides of the rectangles are 3.00 cm and 5.00 cm. Find the magnitude of the electric field at the center of the rectangle in Figures 18-21a and b.

* | 35 | Two charges are placed between the plates of a parallel plate capacitor. One charge is + q1 and the other is . The charge per unit area on each plate has a magnitude of . The force on q1 due to q2 equals the force on q1 due to the electric field of the parallel plate capacitor. What is the distance r between the two charges?

* | 36 | Two parallel plate capacitors have circular plates. The magnitude of the charge on these plates is the same. However, the electric field between the plates of the first capacitor is , while the field within the second capacitor is . Determine the ratio r2/r1 of the plate radius for the second capacitor to the plate radius for the first capacitor.

* | 37 | Interactive Solution 18.37 at provides a model for problems of this kind. A small object has a mass of and a charge of . It is placed at a certain spot where there is an electric field. When released, the object experiences an acceleration of in the direction of the + x axis. Determine the magnitude and direction of the electric field.

* | 38 | Multiple-Concept Example 9 illustrates some of the concepts pertinent to this problem. An electron is released from rest at the negative plate of a parallel plate capacitor. The charge per unit area on each plate is , and the plates are separated by a distance of . How fast is the electron moving just before it reaches the positive plate?

* | 39 | ssm www A rectangle has a length of 2d and a height of d. Each of the following three charges is located at a corner of the rectangle: + q1 (upper left corner), + q2 (lower right corner), and −q (lower left corner). The net electric field at the (empty) upper right corner is zero. Find the magnitudes of q1 and q2. Express your an swers in terms of q.

* | 40 | In Multiple-Concept Example 9 you can see the concepts that are important in this problem. A particle of charge and mass is released from rest in a region where there is a constant electric field of + 480 N/C. What is the displacement of the particle after a time of ?

* | 41 | Interactive Solution 18.41 at for help with this problem. The drawing shows two positive charges q1 and q2 fixed to a circle. At the center of the circle they produce a net electric field that is directed upward along the vertical axis. Determine the ratio / of the charge magnitudes.

** | 42 | The drawing shows an electron entering the lower left side of a parallel plate capacitor and exiting at the upper right side. The initial speed of the electron is . The capacitor is 2.00 cm long, and its plates are separated by 0.150 cm. Assume that the electric field between the plates is uniform everywhere and find its magnitude.

** | 43 | A small plastic ball with a mass of and with a charge of is suspended from an insulating thread and hangs between the plates of a capacitor (see the drawing). The ball is in equilibrium, with the thread making an angle of 30.0° with respect to the vertical. The area of each plate is 0.0150 m2. What is the magnitude of the charge on each plate?

** | 44 | Two particles are in a uniform electric field whose value is + 2500 N/C. The mass and charge of particle 1 are and , while the corresponding values for particle 2 are and . Initially the particles are at rest. The particles are both located on the same electric field line, but are separated from each other by a distance d. When released, they accelerate, but always remain at this same distance from each other. Find d.

** | 45 | ssm Two point charges of the same magnitude but opposite signs are fixed to either end of the base of an isosceles triangle, as the drawing shows. The electric field at the midpoint M between the charges has a magnitude EM. The field directly above the midpoint at point P has a magnitude EP. The ratio of these two field magnitudes is . Find the angle in the drawing.

Section 18.9 Gauss’ Law

| 46 | A spherical surface completely surrounds a collection of charges. Find the electric flux through the surface if the collection consists of (a) a single charge, (b) a single charge, and (c) both of the charges in (a) and (b).

| 47 | ssm The drawing shows an edge-on view of two planar surfaces that intersect and are mutually perpendicular. Surface 1 has an area of 1.7 m2, while surface 2 has an area of 3.2 m2. The electric field in the drawing is uniform and has a magnitude of 250 N/C. Find the electric flux through (a) surface 1 and (b) surface 2.

| 48 | A rectangular surface is oriented in a uniform electric field of 580 N/C. What is the maximum possible electric flux through the surface?

| 49 | A vertical wall in a house faces due east. A uniform electric field has a magnitude of 150 N/C. This field is parallel to the ground and points 35° north of east. What is the electric flux through the wall?

| 50 | A charge Q is located inside a rectangular box. The electric flux through each of the six surfaces of the box is: . What is Q?

* | 51 | Interactive Solution 18.51 at offers help with this problem in an interactive environment. A solid nonconducting sphere has a positive charge q spread uniformly throughout its volume. The charge density or charge per unit volume, therefore, is . Use Gauss’ law to show that the electric field at a point within the sphere at a radius r has a magnitude of . (Hint: For a Gaussian surface, use a sphere of radius r centered within the solid sphere. Note that the net charge within any volume is the charge density times the volume.)

* | 52 | Refer to Concept Simulation 18.3 at for a perspective that is useful in solving this problem. Two spherical shells have a common center. A charge is spread uniformly over the inner shell, which has a radius of 0.050 m. A charge is spread uniformly over the outer shell, which has a radius of 0.15 m. Find the magnitude and direction of the electric field at a distance (measured from the common center) of (a) 0.20 m, (b) 0.10 m, and (c) 0.025 m.

* | 53 | ssm A cube is located with one corner at the origin of an x, y, z coordinate system. One of the cube’s faces lies in the x, y plane, another in the y, z plane, and another in the x, z plane. In other words, the cube is in the first octant of the coordinate system. The edges of the cube are 0.20 m long. A uniform electric field is parallel to the x, y plane and points in the direction of the + y axis. The magnitude of the field is 1500 N/C. (a) Find the electric flux through each of the six faces of the cube. (b) Add the six values obtained in part (a) to show that the electric flux through the cubical surface is zero, as Gauss’ law predicts, since there is no net charge within the cube.

** | 54 | A long, thin, straight wire of length L has a positive charge Q distributed uniformly along it. Use Gauss’ law to show that the electric field created by this wire at a radial distance r has a magnitude of , where . (Hint: For a Gaussian surface, use a cylinder aligned with its axis along the wire and note that the cylinder has a flat surface at either end, as well as a curved surface.)

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