A uniform electric eld of magnitude points upward. An

Predict/Explain An electrically neutral object is given a positive charge. (a) In principle, does the objects mass increase, decrease, or stay the same as a result of being charged? (b) Choose the best explanation from among the following: I. To give the object a positive charge we must remove some of its electrons; this will reduce its mass. II. Since electric charges have mass, giving the object a positive charge will increase its mass. III. Charge is conserved, and therefore the mass of the object will remain the same.

Predict/Explain An electrically neutral object is given a negative charge. (a) In principle, does the objects mass increase, decrease, or stay the same as a result of being charged? (b) Choose the best explanation from among the following: I. To give the object a negative charge we must give it more electrons, and this will increase its mass. II. A positive charge increases an objects mass; a negative charge decreases its mass. III. Charge is conserved, and therefore the mass of the object will remain the same.

(a) Based on the materials listed in Table 191, is the charge of the rubber balloon shown on page 655 more likely to be positive or negative? Explain. (b)If the charge on the balloon is reversed, will the stream of water deect toward or away from the balloon? Explain.

This problem refers to the information given in Table 191. (a) If rabbit fur is rubbed against glass, what is the sign of the charge each acquires? Explain. (b)Repeat part (a) for the case of glass and rubber. (c) Comparing the situations described in parts (a) and (b), in which case is the magnitude of the triboelectric charge greater? Explain.

Find the net charge of a system consisting of electrons.

Find the net charge of a system consisting of (a) electrons and protons or (b) 212 electrons and 165 protons.

How much negative electric charge is contained in 2 moles of carbon?

Find the total electric charge of 1.5 kg of (a) electrons and (b) protons.

A container holds a gas consisting of 1.85 moles of oxygen molecules. One in a million of these molecules has lost a single electron. What is the net charge of the gas?

The Charge on Adhesive Tape When adhesive tape is pulled from a dispenser, the detached tape acquires a positive charge and the remaining tape in the dispenser acquires a negative charge. If the tape pulled from the dispenser has of charge per centimeter, what length of tape must be pulled to transfer electrons to the remaining tape?

Four pairs of conducting spheres, all with the same radius, are shown in Figure 1927, along with the net charge placed on them initially. The spheres in each pair are now brought into contact, allowing charge to transfer between them. Rank the pairs of spheres in order of increasing magnitude of the charge transferred. Indicate ties where appropriate.

Asystem of 1525 particles, each of which is either an electron or a proton, has a net charge of . (a)How many electrons are in this system? (b) What is the mass of this system?

A charge and a charge are placed at opposite corners of a square. Will a third point charge experience a greater force if it is placed at one of the empty corners of the square, or at the center of the square? Explain.

Repeat the previous question, this time with charges and at opposite corners of a square

Consider the three electric charges, A, B, and C, shown in Figure 1928. Rank the charges in order of increasing magnitude of the net force they experience. Indicate ties where appropriate.

Predict/ExplainSuppose the charged sphere in Active Example 192 is made from a conductor, rather than an insulator. (a) Do you expect the magnitude of the force between the point charge and the conducting sphere to be greater than, less than, or equal to the force between the point charge and an insulating sphere? (b)Choose the bestexplanationfrom among the following: I. The conducting sphere will allow the charges to move, resulting in a greater force. II. The charge of the sphere is the same whether it is conducting or insulating, and therefore the force is the same. III. The charge on

At what separation is the electrostatic force between a point charge and a point charge equal in magnitude to 1.57 N?

The attractive electrostatic force between the point charges and Q has a magnitude of 0.975 N when the separation between the charges is 1.31 m. Find the sign and magnitude of the charge

If the speed of the electron in Example 191 were , what would be the corresponding orbital radius?

Two point charges, the rst with a charge of and the second with a charge of , are separated by 25.5 cm. (a) Find the magnitude of the electrostatic force experienced by the positive charge. (b) Is the magnitude of the force experienced by the negative charge greater than, less than, or the same as that experienced by the positive charge? Explain.

When two identical ions are separated by a distance of , the electrostatic force each exerts on the other is . How many electrons are missing from each ion?

A sphere of radius 4.22 cm and uniform surface charge density exerts an electrostatic force of magnitude on a point charge of . Find the separation between the point charge and the center of the sphere.

Given that and , nd the direction and magnitude of the net electrostatic force exerted on the point charge in Figure 1929.

Five point charges, , and , are placed in the vicinity of an insulating spherical shell with a charge , distributed uniformly over its surface, as indicated in Figure 1930. Rank the point charges in order of increasing magnitude of the force exerted on them by the sphere. Indicate ties where appropriate.

Three charges, , and , are at the vertices of an equilateral triangle, as shown in Figure 1931. (a) Rank the three charges in order of increasing magnitude of the electric force they experience. Indicate ties where appropriate.(b)Give the direction angle, , of the net electric force experienced by charge 1. Note that is measured counterclockwise from the positive x axis. (c) Repeat part (b) for charge 2. (d) Repeat part (b) for charge 3.

Given that and , (a)nd the direction and magnitude of the net electrostatic force exerted on the point charge in Figure 1929. (b) How would your answers to part (a) change if the distance d were tripled?

Suppose the charge in Figure 1929 can be moved left or right along the line connecting the charges and . Given that , nd the distance from where experiences a net electrostatic force of zero. (The charges and are separated by a xed distance of 32 cm.)

Find the orbital radius for which the kinetic energy of the electron in Example 191 is 1.51 eV. (Note:

A point charge is xed at the origin. Where must a proton be placed in order for the electric force acting on it to be exactly opposite to its weight? (Let the y axis be vertical and the x axis be horizontal.)

A point charge is xed at the origin. Where must an electron be placed in order for the electric force acting on it to be exactly opposite to its weight? (Let the y axis be vertical and the x axis be horizontal.)

Find the direction and magnitude of the net electrostatic force exerted on the point charge in Figure 1932. Let and .

(a) Find the direction and magnitude of the net electrostatic force exerted on the point charge in Figure 1932. Let (b) How would your answers to part (a) change if the distance d were doubled?

Two point charges lie on the xaxis. Acharge of is at the origin, and a charge of is at . (a) At what position x would a third charge be in equilibrium? (b) Does your answer to part (a) depend on whether is positive or negative? Explain.

A system consists of two positive point charges, and . The total charge of the system is , and each charge experiences an electrostatic force of magnitude 85.0 N when the separation between them is 0.270 m. Find and

The point charges in Figure 1933 have the following values: . (a)Given that the distance d in Figure 1933 is 4.35 cm, nd the direction and magnitude of the net electrostatic force exerted on the point charge . (b)How would your answers to part (a) change if the distance d were doubled? Explain.

Referring to Problem 35, suppose that the magnitude of the net electrostatic force exerted on the point charge in Figure 1933is 0.65 N. (a)Find the distance d.(b)What is the direction of the net force exerted on ?

(a) If the nucleus in Example 191 had a charge of (as would be the case for a nucleus of helium), would the speed of the electron be greater than, less than, or the same as that found in the Example? Explain. (Assume the radius of the electrons orbit is the same.) (b) Find the speed of the electron for a nucleus of charge .

Four point charges are located at the corners of a square with sides of length a. Two of the charges are , and two are . Find the magnitude and direction of the net electric force exerted on a charge , located at the center of the square, for each of the following two arrangements of charge: (a) The charges alternate in sign as you go around the square; (b) the two positive charges are on the top corners, and the two negative charges are on the bottom corners.

Two identical point charges in free space are connected by a string 7.6 cm long. The tension in the string is 0.21 N. (a)Find the magnitude of the charge on each of the point charges. (b) Using the information given in the problem statement, is it possible to determine the sign of the charges? Explain. (c)Find the tension in the string if of charge is transferred from one point charge to the other. Compare with your result from part (a).

Two spheres with uniform surface charge density, one with a radius of 7.2 cm and the other with a radius of 4.7 cm, are separated by a center-to-center distance of 33 cm. The spheres have a combined charge of and repel one another with a force of 0.75 N. What is the surface charge density on each sphere?

Point charges, and , are placed on the x axis, with at and at . A third point charge, , is placed at If the net electrostatic force experienced by the charge is zero, how are and related?

Two electric charges are separated by a nite distance. Somewhere between the charges, on the line connecting them, the net electric eld they produce is zero. (a) Do the charges have the same or opposite signs? Explain. (b) If the point of zero eld is closer to charge 1, is the magnitude of charge 1 greater than or less than the magnitude of charge 2? Explain.

What is the magnitude of the electric eld produced by a charge of magnitude at a distance of (a) 1.00 m and (b) 2.00 m?

A charge experiences a 0.44-N force in the positive y direction. If this charge is replaced with a charge, what force will it experience?

Two point charges lie on the x axis. Acharge of is at the origin, and a charge of is at . What is the net electric eld at

The electric eld on the dashed line in Figure 1928vanishes at innity, but also at two different points a nite distance from the charges. Identify the regions in which you can nd at a nite distance from the charges: region 1, to the left of point A; region 2, between points Aand B; region 3, between points B and C; region 4, to the right of point C.

An object with a charge of and a mass of 0.012 kg experiences an upward electric force, due to a uniform electric eld, equal in magnitude to its weight. (a) Find the direction and magnitude of the electric eld. (b) If the electric charge on the object is doubled while its mass remains the same, nd the direction and magnitude of its acceleration.

Figure 1933 shows a system consisting of three charges, , and , at the vertices of an equilateral triangle of side . (a) Find the magnitude of the electric eld at a point halfway between the charges and . (b)Is the magnitude of the electric eld halfway between the charges and greater than, less than, or the same as the electric eld found in part (a)? Explain. (c)Find the magnitude of the electric eld at the point specied in part (b).

Two point charges of equal magnitude are 7.5 cm apart. At the midpoint of the line connecting them, their combined electric eld has a magnitude of 45 N/C. Find the magnitude of the charges.

Apoint charge is placed at each corner of an equilateral triangle with sides 0.21 m in length. (a) What is the magnitude of the electric eld at the midpoint of any of the three sides of the triangle? (b) Is the magnitude of the electric eld at the center of the triangle greater than, less than, or the same as the magnitude at the midpoint of a side? Explain.

Four point charges, each of magnitude q, are located at the corners of a square with sides of length a. Two of the charges are , and two are . The charges are arranged in one of the following two ways: (1) The charges alternate in sign as you go around the square; (2) the top two corners of the square have positive charges , and the bottom two corners have negative charges . (a) In which case will the electric eld at the center of the square have the greatest magnitude? Explain. (b) Calculate the electric eld at the center of the square for each of these two cases.

The electric eld at the point and points in the positive xdirection with a magnitude of 10.0 N/C. At the point and the electric eld points in the positive x direction with a magnitude of 15.0 N/C. Assuming this electric eld is produced by a single point charge, nd (a)its location and (b) the sign and magnitude of its charge.

The electric eld lines surrounding three charges are shown in Figure 1934. The center charge is . (a) What are the signs of and ? (b) Find .

Make a qualitative sketch of the electric eld lines produced by two equal positive charges, , separated by a distance d.

Make a qualitative sketch of the electric eld lines produced by two charges, and , separated by a distance d.

Referring to Figure 1934, suppose is not known. Instead, it is given that . Find , and .

Make a qualitative sketch of the electric eld lines produced by the four charges, , and , arranged clockwise on the four corners of a square with sides of length d.

Sketch the electric eld lines for the system of charges shown in Figure 1929.

Sketch the electric eld lines for the system of charges described in Problem 35.

Suppose the magnitude of the electric eld between the plates in Example 196 is changed, and a new object with a charge of is attached to the string. If the tension in the string is 0.450 N, and the angle it makes with the vertical is 16, what are (a)the mass of the object and (b)the magnitude of the electric eld?

Predict/Explain Gaussian surface 1 has twice the area of Gaussian surface 2. Both surfaces enclose the same charge Q. (a) Is the electric ux through surface 1 greater than, less than, or the same as the electric ux through surface 2? (b) Choose the best explanation from among the following: I. Gaussian surface 2 is closer to the charge, since it has the smaller area. It follows that it has the greater electric ux. II. The two surfaces enclose the same charge, and hence they have the same electric ux. III. Electric ux is proportional to area. As a result, Gaussian surface 1 has the greater electric ux.

Suppose the conducting shell in Figure 1925which has a point charge at its centerhas a nonzero net charge. How much charge is on the inner and outer surface of the shell when the net charge of the shell is (a) , (b) , and (c) ?

Rank the Gaussian surfaces shown in Figure 1935 in order of increasing electric ux, starting with the most negative. Indicate ties where appropriate.

A uniform electric eld of magnitude 25,000 N/C makes an angle of 37 with a plane surface of area . What is the electric ux through this surface?

A surface encloses the charges , and . Find the electric ux through this surface

A uniform electric eld of magnitude points upward. An empty, closed shoe box has a top and bottom that are 35.0 cm by 25.0 cm, vertical ends that are 25.0 cm by 20.0 cm, and vertical sides that are 20.0 cm by 35.0 cm. (a)Which side of the box has the greatest positive electric ux? Which side has the greatest negative electric ux? Which sides have zero electric ux? (b) Calculate the electric ux through each of the six sides of the box.

Nerve Cells Nerve cells are long, thin cylinders along which electrical disturbances (nerve impulses) travel. The cell membrane of a typical nerve cell consists of an inner and an outer wall separated by a distance of . The electric eld within the cell membrane is . Approximating the cell membrane as a parallel-plate capacitor, determine the magnitude of the charge density on the inner and outer cell walls.

The electric ux through each of the six sides of a rectangular box are as follows:

Consider a spherical Gaussian surface and three charges: , and . Find the electric ux through the Gaussian surface if it completely encloses (a) only charges and , (b) only charges and , and (c) all three charges. (d) Suppose a fourth charge, Q, is added to the situation described in part (c). Find the sign and magnitude of Q required to give zero electric ux through the surface.

Athin wire of innite extent has a charge per unit length of . Using the cylindrical Gaussian surface shown in Figure 1936, show that the electric eld produced by this wire at a radial distance r has a magnitude given by Note that the direction of the electric eld is always radially away from the wire.

Predict/Explain An electron and a proton are released from rest in space, far from any other objects. The particles move toward each other, due to their mutual electrical attraction. (a) When they meet, is the kinetic energy of the electron greater than, less than, or equal to the kinetic energy of the proton? (b) Choose the best explanation from among the following: I. The proton has the greater mass. Since kinetic energy is proportional to mass, it follows that the proton will have the greater kinetic energy. II. The two particles experience the same force, but the light electron moves farther than the massive proton. Therefore, the work done on the electron, and hence its kinetic energy, is greater. III. The same force acts on the two particles. Therefore, they will have the same kinetic energy and energy will be conserved.

Predict/Explain In Conceptual Checkpoint 193, suppose the charge to be placed at either point A or point B is rather than . (a)Is the magnitude of the net force experienced by the movable charge at point A greater than, less than, or equal to the magnitude of the net force at point B? (b) Choose the best explanation from among the following: I. Point B is farther from the two xed charges. As a result, the net force at point B is less than at point A. II. The net force at point A cancels, just as it does in Conceptual Checkpoint 193. Therefore, the nonzero net force at point B is greater in magnitude than the zero net force at point A. III. The net force is greater in magnitude at point Abecause at that location the movable charge experiences a net repulsion from each of the xed charges.

An electron orbits a helium nucleus . Is the magnitude of the force exerted on the helium nucleus by the electron greater than, less than, or the same as the magnitude of the force exerted on the electron by the helium nucleus? Explain.

In the operating room, technicians and doctors must take care not to create an electric spark, since the presence of the oxygen gas used during an operation increases the risk of a deadly re. Should the operating-room personnel wear shoes that are conducting or nonconducting? Explain.

Under normal conditions, the electric eld at the surface of the Earth points downward, into the ground. What is the sign of the electric charge on the ground?

Two identical spheres are made of conducting material. Initially, sphere 1 has a net charge of and sphere 2 has a net charge of . If the spheres are now brought into contact, what is the nal charge on sphere 1? Explain.

AGaussian surface for the charges shown in Figure 1935 has an electric ux equal to . Which charges are contained within this Gaussian surface?

A proton is released from rest in a uniform electric eld of magnitude . Find the speed of the proton after it has traveled (a) 1.00 cm and (b) 10.0 cm.

Ventricular Fibrillation If a charge of 0.30 C passes through a persons chest in 1.0 s, the heart can go into ventricular brillationa nonrhythmic uttering of the ventricles that results in little or no blood being pumped to the body. If this rate of charge transfer persists for 4.5 s, how many electrons pass through the chest?

Apoint charge at the origin of a coordinate system produces the electric eld on the xaxis at the location . Determine the sign and magnitude of the charge.

Four lightweight, plastic spheres, labeled A, B, C, and D, are suspended from threads in various combinations, as illustrated in Figure 1937. It is given that the net charge on sphere D is , and that the other spheres have net charges of , or 0. From the results of the four experiments shown in Figure 1937, and the fact that the spheres have equal masses, determine the net charge of (a)sphere A, (b)sphere B, and (c)sphere C.

Find (a) the direction and (b) the magnitude of the net electric eld at the center of the equilateral triangle in Figure 1931. Give your answers in terms of the angle , as dened in Figure 1931, and E, the magnitude of the electric eld produced by any one of the charges at the center of the triangle.

At the moment, the number of electrons in your body is essentially the same as the number of protons, giving you a net charge of zero. Suppose, however, that this balance of charges is off by 1% in both you and your friend, who is 1 meter away. Estimate the magnitude of the electrostatic force each of you experiences, and compare it with your weight.

A small object of mass 0.0150 kg and charge hangs from the ceiling by a thread. A second small object, with a charge of , is placed 1.2 m vertically below the rst charge. Find (a) the electric eld at the position of the upper charge due to the lower charge and (b)the tension in the thread.

Consider a system of three point charges on the x axis. Charge 1 is at , charge 2 is at , and charge 3 is at . In addition, the charges have the following values: . (a) The electric eld vanishes at some point on the xaxis between and . Is the point of zero eld (i) at , (ii) to the left of , or (iii) to the right of ? Explain. (b) Find the point where between and .

Consider the system of three point charges described in the previous problem. (a) The electric eld vanishes at two different points on the xaxis. One point is between and . Is the second point located to the left of charge 1 or to the right of charge 3? Explain. (b) Find the value of x at the second point where .

The electric eld at a radial distance of 47.7 cm from the thin charged wire shown in Figure 1936 has a magnitude of 35,400 N/C. (a) Using the result given in Problem 70, what is the magnitude of the charge per length on this wire? (b) At what distance from the wire is the magnitude of the electric eld equal to ?

A system consisting entirely of electrons and protons has a net charge of and a net mass of . How many (a) electrons and (b) protons are in this system?

Three charges are placed at the vertices of an equilateral triangle of side , as shown in Figure 1938. Charges 1 and 3 are ; charge 2 is . (a) Find the magnitude and direction of the net force acting on charge 3. (b) If charge 3 is moved to the origin, will the net force acting on it there be greater than, less than, or equal to the net force found in part (a)? Explain. (c) Find the net force on charge 3 when it is at the origin.

Consider the system of three charges described in the previous problem and shown in Figure 1938. (a) Do you expect the net force acting on charge 1 to have a magnitude greater than, less than, or the same as the magnitude of the net force acting on charge 2? Explain. (b)Find the magnitude of the net force acting on charge 1. (c) Find the magnitude of the net force acting on charge 2.

Cell Membranes The cell membrane in a nerve cell has a thickness of . (a) Approximating the cell membrane as a parallel-plate capacitor with a surface charge density of , nd the electric eld within the membrane.(b)If the thickness of the membrane were doubled, would your answer to part (a) increase, decrease, or stay the same? Explain.

Asquare with sides of length Lhas a point charge at each of its four corners. Two corners that are diagonally opposite have charges equal to ; the other two diagonal corners have charges Q. Find the magnitude and sign of the charges Q such that each of the charges experiences zero net force.

Suppose a charge is placed on the Earth, and another charge is placed on the Moon. (a) Find the value of Q needed to balance the gravitational attraction between the Earth and the Moon. (b) How would your answer to part (a) change if the distance between the Earth and the Moon were doubled? Explain.

Two small plastic balls hang from threads of negligible mass. Each ball has a mass of 0.14 g and a charge of magnitude q. The balls are attracted to each other, and the threads attached to the balls make an angle of 20.0with the vertical, as shown in Figure 1939. Find (a) the magnitude of the electric force acting on each ball, (b) the tension in each of the threads, and (c) the magnitude of the charge on the balls.

Asmall sphere with a charge of is attached to a relaxed horizontal spring whose force constant is 89.2 N/m. The spring extends along the x axis, and the sphere rests on a frictionless surface with its center at the origin. A point charge is now moved slowly from innity to a point on the xaxis. This causes the small sphere to move to the position . Find d.

Twelve identical point charges q are equally spaced around the circumference of a circle of radius R. The circle is centered at the origin. One of the twelve charges, which happens to be on the positive x axis, is now moved to the center of the circle. Find (a) the direction and (b) the magnitude of the net electric force exerted on this charge.

Nerve Impulses When a nerve impulse propagates along a nerve cell, the electric eld within the cell membrane changes from in one direction to in the other direction. Approximating the cell membrane as a parallel-plate capacitor, nd the magnitude of the change in charge density on the walls of the cell membrane.

The Electric Field of the Earth The Earth produces an approximately uniform electric eld at ground level. This electric eld has a magnitude of 110 N/C and points radially inward, toward the center of the Earth. (a) Find the surface charge density (sign and magnitude) on the surface of the Earth. (b) Given that the radius of the Earth is , nd the total electric charge on the Earth. (c) If the Moon had the same amount of electric charge distributed uniformly over its surface, would its electric eld at the surface be greater than, less than, or equal to 110 N/C? Explain.

An object of mass and charge is attached to a string and placed in a uniform electric eld that is inclined at an angle of 30.0 with the horizontal (Figure 1940). The object is in static equilibrium when the string is horizontal. Find(a)the magnitude of the electric eld and (b)the tension in the string.

Four identical charges, , occupy the corners of a square with sides of length a. Afth charge, q, can be placed at any desired location. Find the location of the fth charge, and the value of q, such that the net electric force acting on each of the original four charges, , is zero

Figure 1941 shows an electron entering a parallel-plate capacitor with a speed of . The electric eld of the capacitor has deected the electron downward by a distance of 0.618 cm at the point where the electron exits the capacitor. Find(a)the magnitude of the electric eld in the capacitor and (b) the speed of the electron when it exits the capacitor.

Two identical conducting spheres are separated by a xed center-to-center distance of 45 cm and have different charges. Initially, the spheres attract each other with a force of 0.095 N. The spheres are now connected by a thin conducting wire. After the wire is removed, the spheres are positively charged and repel one another with a force of 0.032 N. Find (a)the nal and (b) the initial charges on the spheres.

How many electrons must be transferred away from a bee to produce a charge of pC? A. B. C. D

Suppose two bees, each with a charge of 93.0 pC, are separated by a distance of 1.20 cm. Treating the bees as point charges, what is the magnitude of the electrostatic force experienced by the bees? (In comparison, the weight of a 0.140-g bee is N.) A. N B. N C. N D.

The force required to detach a grain of pollen from an avocado stigma is approximately . What is the maximum distance at which the electrostatic force between a bee and a grain of pollen is sufcient to detach the pollen? Treat the bee and pollen as point charges, and assume the pollen has a charge opposite in sign and equal in magnitude to the bee. A. m B. 1.9 mm C. 4.4 cm D.

The Earth produces an electric eld of magnitude 110 N/C. What force does this electric eld exert on a bee carrying a charge of 93.0 pC? (Again, for comparison, the weight of a bee is approximately N.) A. N B. N C. N D.

Referring to Example 195 Suppose is no longer at the origin, but is now on the y axis between . The charge is at and , and point 3 is at . (a) Is the magnitude of the net electric eld at point 3, which we call , greater than, less than, or equal to its previous value? Explain. (b) Is the angle that makes with the x axis greater than, less than, or equal to its previous value? Explain. Find the new values of (c) and (d) if is at .

Referring to Example 195 In this system, the charge is at the origin, the charge is at and , and point 3 is at . Suppose that , but that is increased to a value greater than . As a result, do (a) and (b) increase, decrease, or stay the same? Explain. If , nd (c) and (d) .

Referring to Example 196 The magnitude of the charge is changed until the angle the thread makes with the vertical is . The electric eld is and the mass of the object is 0.0250 kg. (a)Is the new magnitude of qgreater than or less than its previous value? Explain. (b)Find the new value of q.

Referring to Example 196 Suppose the magnitude of the electric eld is adjusted to give a tension of 0.253 N in the thread. This will also change the angle the thread makes with the vertical. (a)Find the new value of E.(b)Find the new angle between the thread and the vertical.