Gauss’s law can tell us how much charge is contained

Referring to Example 19–6 Suppose the magnitude of the electric field 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.

Problem 1P CE Predict/Explain An electrically neutral object is given a positive charge, (a) In principle, does the object’s 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.

Problem 2CQ The fact that the electron has a negative charge and the proton has a positive charge is due to a convention established by Benjamin Franklin. Would there have been any significant consequences if Franklin had chosen the opposite convention? Is there any advantage to naming charges plus and minus as opposed to, say, A and B?

Problem 1CQ When an object that was neutral becomes charged, does the total charge of the universe change? Explain.

Problem 2P CE Predict/Explain An electrically neutral object is given a negative charge, (a) In principle, does the object’s 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 object’s mass; a negative charge decreases its mass. III. Charge is conserved, and therefore the mass of the object will remain the same.

Problem 3CQ Explain why a comb that has been rubbed through your hair attracts small bits of paper, even though the paper is uncharged.

(a) Based on the materials listed in Table 19–1, 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 deflect toward or away from the balloon? Explain.

Problem 4CQ Small bits of paper are attracted to an electrically charged comb, but as soon as they touch the comb they are strongly repelled. Explain this behavior.

This problem refers to the information given in Table 19–1. (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.

Problem 5CQ A charged rod is brought near a suspended object, which is repelled by the rod. Can we conclude that the suspended object is charged? Explain.

Problem 5P Find the net charge of a system consisting of 4.9 × 107 electrons.

Problem 6CQ A charged rod is brought near a suspended object, which is attracted to the rod. Can we conclude that the suspended object is charged? Explain.

Problem 6P Find the net charge of a system consisting of (a) 6.15 × 10 electrons and 7.44 × 106 protons or (b) 212 electrons and 165 protons.

Problem 7CQ Describe some of the similarities and differences between Coulomb’s law and Newton’s law of gravity.

Problem 7P How much negative electric charge is contained in 2 moles of carbon?

Problem 8CQ A point charge +Q is fixed at a height H above the ground. Directly below this charge is a small ball with a charge ?q and a mass m. When the ball is at a height h above the ground, the net force (gravitational plus electrical) acting on it is zero. Is this a stable equilibrium for the object? Explain.

Problem 8P Find the total electric charge of 1.5 kg of (a) electrons and (b) protons.

Problem 9CQ Four identical point charges are placed at the corners of a square. A fifth point charge placed at the center of the square experiences zero net force. Is this a stable equilibrium for the fifth charge? Explain.

Problem 9P 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?

Problem 10CQ A proton moves in a region of constant electric field. Does it follow that the proton’s velocity is parallel to the electric field? Does it follow that the proton’s acceleration is parallel to the electric field? Explain.

Problem 10P 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 0.14 ?C of charge per centimeter, what length of tape must be pulled to transfer 1.8 × 1013 electrons to the remaining tape?

Problem 11CQ Describe some of the differences between charging by induction and charging by contact.

Four pairs of conducting spheres, all with the same radius, are shown in Figure 19–27, 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.

Problem 12P A system of 1525 particles, each of which is either an electron or a proton, has a net charge of ?5.456 × 10?17 C. (a) How many electrons are in this system?(b) What is the mass of this system?

Problem 14CQ Can an electric field exist in a vacuum? Explain.

Problem 12CQ A system consists of two charges of equal magnitude and opposite sign separated by a distance d. Since the total electric charge of this system is zero, can we conclude that the electric field produced by the system is also zero? Does your answer depend on the separation if? Explain.

Problem 13P CE A charge +q and a charge ?q 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.

Problem 13CQ The force experienced by charge 1 at point A is different in direction and magnitude from the force experienced by charge 2 at point B. Can we conclude that the electric fields at points A and B are different? Explain.

Problem 14P CE Repeat the previous question, this time with charges +q and +q at opposite corners of a square.

Problem 15CQ Explain why electric field lines never cross.

Consider the three electric charges, A, B, and C, shown in Figure 19–28. Rank the charges in order of increasing magnitude of the net force they experience. Indicate ties where appropriate.

Problem 16CQ Charge q1 is inside a closed Gaussian surface; charge q2 is just outside the surface. Does the electric flux through the surface depend on q1?Does it depend on q2?Explain.

Suppose the charged sphere in Active Example 19–2 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 best explanation from among the following: 1. The conducting sphere will allow the charges to move, resulting in a greater force. 2. The charge of the sphere is the same whether it is conducting or insulating, and therefore the force is the same. 3. The charge on a conducting sphere will move as far away as possible from the point charge. This results in a reduced force.

Problem 17P At what separation is the electrostatic force between a +11.2-?C point charge and a +29.1-?Cpoint charge equal in magnitude to 1.57 N?

Problem 17CQ In the previous question, does the electric field at a point on the Gaussian surface depend on q1?Does it depend on q2. Explain.

Problem 18CQ Gauss’s law can tell us how much charge is contained within a Gaussian surface. Can it tell us where inside the surface it is located? Explain.

Problem 18P The attractive electrostatic force between the point charges +8.44 × 10?6 C 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 Q.

Problem 19CQ Explain why Gauss’s law is not very useful in calculating the electric field of a charged disk.

If the speed of the electron in Example 19–1 were \(7.3 \times 10^{5} \mathrm{~m} / \mathrm{s}\), what would be the corresponding orbital radius? Equation Transcription: Text Transcription: 7.3 X 10^5 m/s

Problem 20P IP Two point charges, the first with a charge of +3.13 × 10?6C and the second with a charge of ?4.47 × 10?6 C, 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.

Problem 21P When two identical ions are separated by a distance of 6.2 × 10?10 m, the electrostatic force each exerts on the other is 5.4 × 10?9 N. How many electrons are missing from each ion?

Problem 22P A sphere of radius 4.22 cm and uniform surface charge density +12.1 ?C/m2 exerts an electrostatic force of magnitude 46.9 × 10?3 N on a point charge of +1.95 ?C. Find the separation between the point charge and the center of the sphere.

Given that \(q=+12 \mu \mathrm{C} \text { and } d=16 \mathrm{~cm}\), find the direction and magnitude of the net electrostatic force exerted on the point charge \(q_{1}\) in Figure 19–29. Equation Transcription: Text Transcription: q=+12\mu C and d=16 cm q1

Five point charges, \(q_{1}=+q, q_{2}=+2 q, q_{3}=-3 q, q_{4}=-4 q, \text { and } q_{5}=-5 q\), are placed in the vicinity of an insulating spherical shell with a charge +Q, distributed uniformly over its surface, as indicated in Figure 19–30. Rank the point charges in order of increasing magnitude of the force exerted on them by the sphere. Indicate ties where appropriate. Equation Transcription: Text Transcription: q_1=+q, q_2=+2q, q_3=-3q, q_4, and q_5=-5q

Three charges,\(q_{1}=+q, q_{2}=-q, \text { and } q_{3}=+q\), are at the vertices of an equilateral triangle, as shown in Figure 19–31. (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, \(\theta\), of the net electric force experienced by charge 1. Note that \(\theta\) is measured counterclockwise from the positive x axis. (c) Repeat part (b) for charge 2. (d) Repeat part (b) for charge 3. Equation Transcription: Text Transcription: q_{1}=+q, q_{2}=-q, and q_{3}=+q \\ \theta \theta

Suppose the charge \(q_{2}\) in Figure 19–29can be moved left or right along the line connecting the charges \(q_{1} and q_{3}\). Given that \(q=+12 \mu C\), find the distance from \(q_{1} where q_{2}\) experiences a net electrostatic force of zero. (The charges \(q_{1} and q_{3}\) are sepa-rated by a fixed distance of 32 cm.) Equation Transcription: Text Transcription: q2 q1 and q3 q=+12 \mu C q1 where q2 q1 and q3

Find the orbital radius for which the kinetic energy of the electron in Example 19–1 is 1.51 eV. (Note: 1 eV = 1 electron volt = \(1.6 \times 10^{-19} J\).) Equation Transcription: Text Transcription: 1.6 X 10-19 J

Given that \(q=+12 \mu C \text { and } d=19 \mathrm{~cm}\), (a) find the direc-tion and magnitude of the net electrostatic force exerted on the point charge \(q_{2}\) in Figure 19–29. (b) How would your answers to part (a) change if the distance d were tripled? Equation Transcription: Text Transcription: q=+12\mu C and d=19 cm q2

Problem 29P A point charge q = ?0.35 nC is fixed 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.)

Problem 30P A point charge q = ?0.35 nC is fixed 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 \(q_{2}\) in Figure 19–32. Let \(q=+2.4 \mu C and d=33 \mathrm{~cm}\). Equation Transcription: Text Transcription: q2 q=+2.4\mu C and d=33 cm

Problem 32P IP(a) Find the direction and magnitude of the net electrostatic force exerted on the point charge q3 in Figure 19?32. Let q = +2.4 ?C and d = 27 cm. (b) How would your answers to part (a) change if the distance d were doubled?

Problem 34P A system consists of two positive point charges, q1 and q2> q1. The total charge of the system is +62.0 ?C, and each charge experiences an electrostatic force of magnitude 85.0 N when the separation between them is 0.270 m. Find q1 and q2

Problem 33P IP Two point charges lie on the x axis. A charge of +9.9 ?C is at the origin, and a charge of ?5.1 ?C is at x = 10.0 cm. (a) At what position x would a third charge q3 be in equilibrium? (b) Does your answer to part (a) depend on whether q3 is positive or negative? Explain.

The point charges in Figure 19–33have the following values: \(q_{1}=+2.4 \mu C, q_{2}=+6.3 \mu C, q_{3}=-0.89 \mu C\). (a) Given that the distance d in Figure 19–33 is 4.35 cm, find the direction and magnitude of the net electrostatic force exerted on the point charge \(q_{1}\). (b) How would your answers to part (a) change if the distance d were doubled? Explain. Equation Transcription: Text Transcription: q1=+2.4 \mu C, q2=+6.3 \mu C, q3=-0.89 \mu C q1

Problem 36P Referring to Problem 35, suppose that the magnitude of the net electrostatic force exerted on the point charge q2 in Figure 19?33 is 0.65 N. (a) Find the distance d. (b) What is the direction of the net force exerted on q2?

(a) If the nucleus in Example 19–1 had a charge of \(+2 e\) (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 electron’s orbit is the same.) (b) Find the speed of the electron for anucleus of charge \(+2 e\). Equation Transcription: Text Transcription: +2e +2e

Problem 38P Four point charges are located at the corners of a square with sides of length a. Two of the charges are +q, and two are ?q, Find the magnitude and direction of the net electric force exerted on a charge +Q, located at the center of the square, for each of the following two arrangements of charge: (a) The charges alternate in sign (+q, ?q, +q, ?q) 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.

Problem 39P IP 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 +1.0 ?C of charge is transferred from one point charge to the other. Compare with your result from part (a).

Problem 41P Point charges, q1 and q2, are placed on the x axis, with q1 at x = 0 and q2 at x = d. A third point charge, +Q, is placed at x = 3d/4. if the net electrostatic force experienced by the charge +Q is zero, how are q1 and q2 related?

Problem 40P 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 +55 ?Cand repel one another with a force of 0.75 N. What is the surface charge density on each sphere?

Problem 43P What is the magnitude of the electric field produced by a charge of magnitude 7.50 ?Cat a distance of (a) 1.00 m and (b) 2.00 m?

Problem 42P CE Two electric charges are separated by a finite distance. Somewhere between the charges, on the line connecting them, the net electric field they produce is zero, (a) Do the charges have the same or opposite signs? Explain, (b) If the point of zero field is closer to charge 1, is the magnitude of charge 1 greater than or less than the magnitude of charge 2? Explain.

Problem 44P A +5.0 ?C charge experiences a 0.44?N force in the positive y direction. If this charge is replaced with a ? 2.7?Ccharge, what force will it experience?

Problem 47P An object with a charge of ?3.6 ?C and a mass of 0.012 kg experiences an upward electric force, due to a uniform electric field, equal in magnitude to its weight, (a) Find the direction and magnitude of the electric field, (b) If the electric charge on the object is doubled while its mass remains the same, find the direction and magnitude of its acceleration.

Problem 45P Two point charges he on the x axis. A charge of +6.2 ?C is at the origin, and a charge of ?9.5 ?Cis at x = 10.0 cm. What is the net electric field at (a) x = ?4.0 cm and at (b) x = +4.0 cm?

The electric field on the dashed line in Figure 19–28van-ishes at infinity, but also at two different points a finite distance from the charges. Identify the regions in which you can find \(E=0\) at a finite distance from the charges: region 1, to the left of point A; region 2, between points A and B; region 3, between points B and C; region 4, to the right of point C. Equation Transcription: Text Transcription: E=0

Problem 48P IP Figure 19?33 shows a system consisting of three charges, q1 = +5.00 ?C, q2 = +5.00 ?C, and q3 = ?5.00 ?C, at the vertices of an equilateral triangle of side d = 2.95 cm. (a) Find the magnitude of the electric field at a point halfway between the charges q1 and q2. (b) Is the magnitude of the electric field halfway between the charges q2 and q3 greater than, less than, or the same as the electric field found in part (a)? Explain, (c) Find the magnitude of the electric field at the point specified in part (b).

Problem 49P Two point charges of equal magnitude are 7.5 cm apart. At the midpoint of the line connecting them, their combined electric field has a magnitude of 45 N/C. Find the magnitude of the charges.

Problem 50P IP A point charge q1 = +4.7 ?Cis placed at each corner of an equilateral triangle with sides 0.21 m in length, (a) What is the magnitude of the electric field at the midpoint of any of the three sides of the triangle? (b) Is the magnitude of the electric field at the center of the triangle greater than, less than, or the same as the magnitude at the midpoint of a side? Explain.

Problem 51P IP 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 +q, and two aie ?q. The charges are arranged in one of the following two ways: (1) The charges alternate in sign (+q, ?q, +q, ?q) as you go around the square; (2) the top two comers of the square have positive charges (+q, +q), and the bottom two corners have negative charges (?q, ?q). (a) In which case will the electric field at the center of the square have the greatest magnitude? Explain, (b) Calculate the electric field at the center of the square for each of these two cases.

Problem 52P The electric field at the point x = 5.00 cm and y = 0 points in the positive x direction with a magnitude of 10.0 N/C. At the point x = 10.0 cm and y = 0 the electric field points in the positive x direction with a magnitude of 15.0 N/C. Assuming this electric field is produced by a single point charge, find (a) its location and (b) the sign and magnitude of its charge.

The electric field lines surrounding three charges are shown in Figure 19–34. The center charge is \(q_{2}=-10.0 \mu C\). (a) What are the signs of \(q_{1} \text { and } q_{3}\)? (b) Find \(q_{1}\) (c) Find \(q_{3}\) Equation Transcription: Text Transcription: q2=-10.0 \muC q1 and q3 q1 q3

Problem 54P Make a qualitative sketch of the electric field lines produced by two equal positive charges, +q, separated by a distance d.

Problem 55P Make a qualitative sketch of the electric field lines produced by two charges, +q and ?q, separated by a distance d.

Referring to Figure 19–34, suppose \(q_{2}\) is not known. Instead, it is given that \(q_{1}+q_{2}=-2.5 \mu C\). Find \(q_{1}, q_{2}=\text { and } q_{3}\). Equation Transcription: Text Transcription: q2 q1+q2=-2.5 \muC q1,q2, and q3

Problem 57P Make a qualitative sketch of the electric field lines produced by the four charges, +q, ?q, +q, and ?q, arranged clockwise on the four corners of a square with sides of length d.

Sketch the electric field lines for the system of charges shown in Figure 19–29.

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

Suppose the magnitude of the electric field between the plates in Example 19–6 is changed, and a new object with a charge of \(-2.05 \mu C\) is attached to the string. If the tension in the string is 0.450 N, and the angle it makes with the vertical is \(16^{\circ}\), what are (a) the mass of the object and (b) the magnitude of the electric field? Equation Transcription: Text Transcription: -2.05 \mu C 16^{\circ}

Problem 61P CE Predict/Explain Gaussian surface 1 has twice the area of Gaussian surface 2. Both surfaces enclose the same charge Q. (a) Ts the electric flux through surface 1 greater than, less than, or the same as the electric flux 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 flux. II. The two surfaces enclose the same charge, and hence they have the same electric flux. III. Electric flux is proportional to area. As a result, Gaussian surface 1 has the greater electric flux.

Rank the Gaussian surfaces shown in Figure 19–35 in order of increasing electric flux, starting with the most negative. Indicate ties where appropriate.

Suppose the conducting shell in Figure 19–25—which has a point charge \(+Q\) at its center—has 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) \(-2 Q\) (b) \(-Q\), and (c) \(+Q\)? Equation Transcription: Text Transcription: +Q -2Q -Q +Q

Problem 64P A uniform electric field of magnitude 25,000 N/C makes an angle of 37° with a plane surface of area 0.0153 m2, What is the electric flux through this surface?

Problem 65P A surface encloses the charges q1 = 3.2 ?C, q2 ? 6.9 ?C, and q3 = ?4.1 ?C. Find the electric flux through this surface.

Problem 67P BIO 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 0.10 ? m. The electric field within the cell membrane is 7.0 × 105 N/C. Approximating the cell membrane as a parallel-plate capacitor, determine the magnitude of the charge density on the inner and outer cell walls.

Problem 66P IPA uniform electric field of magnitude 6.00 × 103 N/C 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 flux? Which side has the greatest negative electric flux? Which sides have zero electric flux? (b) Calculate the electric flux through each of the six sides of the box.

Problem 68P The electric flux through each of the six sides of a rectangular box are as follows: ?1 = +150.0 N · m2/C; ?2 = +250.0 N · m2/C; ?3 = ?350.0 N · m2/C; ?4 = +175.0 N · m2/C; ?5 = ?100.0 N · m2/C; ?6 = +450.0 N · m2/C. How much charge is in this box?

Problem 69P Consider a spherical Gaussian surface and three charges: q1 = 1.61 ?C,q2 = ?2.62 ?C, and q3 = 3.91 ?C. Find the electric flux through the Gaussian surface if it completely encloses (a) only charges q1 and q2, (b) only charges q2 and q3, 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 flux through the surface.

A thin wire of infinite extent has a charge per unit length of \(\lambda\). Using the cylindrical Gaussian surface shown in Figure 19–36, show that the electric field produced by this wire at a radial distance r has a magnitude given by \(E=\frac{\lambda}{2 \pi \varepsilon_{0} r}\) Note that the direction of the electric field is always radially away from the wire. Equation Transcription: Text Transcription: \lambda E=\frac{\lambda}{2 \pi \varepsilon_{0} r}

Problem 71GP CE 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.

In Conceptual Checkpoint 19–3, suppose the charge to be placed at either point A or point B is +q rather than -q. (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: 1. Point B is farther from the two fixed charges. As a result, the net force at point B is less than at point A. 2. The net force at point A cancels, just as it does in Conceptual Checkpoint 19–3. Therefore, the nonzero net force at point B is greater in magnitude than the zero net force at point A. 3. The net force is greater in magnitude at point A because at that location the movable charge experiences a net repulsion from each of the fixed charges.

Problem 73GP CE An electron (charge = ?e) orbits a helium nucleus (charge = +2e). Is the magnitude of the force exerted on the he? Hum 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.

Problem 75GP CE Under normal conditions, the electric field at the surface of the Earth points downward, into the ground. What is the sign of the electric charge on the ground?

Problem 74GP CE 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 fire. Should the operating?room personnel wear shoes that are conducting or non-conducting? Explain.

Problem 76GP CE Two identical spheres are made of conducting material. Initially, sphere 1 has a net charge of +35Q and sphere 2 has a net charge of ?26Q. If the spheres are now brought into contact, what is the final charge on sphere 1? Explain.

A Gaussian surface for the charges shown in Figure 19–35 has an electric flux equal to \(+3 q / \varepsilon_{0}\) Which charges are contained within this Gaussian surface? Equation Transcription: Text Transcription: +3 q / \varepsilon_0

Problem 78GP A proton is released from rest in a uniform electric field of magnitude 1.08 × 105 N/C. Find the speed of the proton after it has traveled (a) 1.00 cm and (b) 10.0 cm.

A point charge at the origin of a coordinate system produces the electric field \(\vec{E}=(36,000 \mathrm{~N} / \mathrm{C})\) \(\hat{x}\) on the x axis at the location \(x=-0.75 \mathrm{~m}\). Determine the sign and magnitude of the charge. Equation Transcription: Text Transcription: \ve E=(36,000 N/C) \hat{x} x=-0.75 m

Problem 79GP BIO Ventricular Fibrillation If a charge of 0.30 C passes through a person's chest in 1.0 s, the heart can go into ventricular fibrillation?a nonrhythmic "fluttering" 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?

Four lightweight, plastic spheres, labeled A, B, C, and D, are suspended from threads in various combinations, as illustrated in Figure 19–37. It is given that the net charge on sphere D is \(+Q\), and that the other spheres have net charges of \(+Q,-Q, \text { or } 0\). From the results of the four experiments shown in Figure 19–37, and the fact that the spheres have equal masses, determine the net charge of (a) sphere A, (b) sphere B, and (c) sphere C. Equation Transcription: Text Transcription: +Q +Q, -Q, or 0

Find (a) the direction and (b) the magnitude of the net electric field at the center of the equilateral triangle in Figure 19–31. Give your answers in terms of the angle \(\theta\), as defined in Figure 19–31, and E, the magnitude of the electric field produced by any one of the charges at the center of the triangle. Equation Transcription: Text Transcription: \theta

Problem 83GP 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.

Problem 84GP A small object of mass 0.0150 kg and charge 3.1 ?Changs from the ceiling by a thread. A second small object, with a charge of 4.2 ?C, is placed 1.2 m vertically below the first charge. Find (a) the electric field at the position of the upper charge due to the lower charge and (b) the tension in the thread.

Problem 86GP IP Consider the system of three point charges described in the previous problem, (a) The electric field vanishes at two different points on the.x axis. One point is between x = 0.20 m and x = 0.40 m. 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 E = 0.

Problem 85GP IP Consider a system of three point charges on the x axis. Charge 1 is at x = 0, charge 2 is at x = 0.20 m, and charge 3 is at x = 0.40 m. In addition, the charges have the following values: q1 = ?19 ?C, q2 = q3 = +19 ?C(a)The electric field vanishes at some point on the x axis between x = 0.20 m and x = 0.40 m. Ts the point of zero field (i) at x = 0.30 m, (ii) to the left of x = 0.30 m, or (iii) to the right of x = 0.30 m? Explain, (b) find the point where E = 0 between x = 0.20 m and x = 0.40 m.

The electric field at a radial distance of 47.7 cm from the thin charged wire shown in Figure 19–36 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 field equal to \(\frac{1}{2}(35,400 \mathrm{~N} / \mathrm{C})\)? Equation Transcription: Text Transcription: 1 over 2(35, 400 N/C)

Problem 88GP A system consisting entirely of electrons and protons has a net charge of 1.84 × 10?15 C and a net mass of 4.56 × 10?23 kg. How many (a) electrons and (b) protons are in this system?

Consider the system of three charges described in the previous problem and shown in Figure 19–38. (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.

Three charges are placed at the vertices of an equilateral triangle of side \(a=0.93 \mathrm{~m} \text { and } 3 \text { are }+7.3 \mu C \text {; charge } 2 \text { is }-7.3 \mu C\) (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. Equation Transcription: Text Transcription: a=0.93 m and 3 are +7.3 \mu C; charge 2 is -7.3 \mu C

Problem 91GP IP BIO Cell Membranes The cell membrane in a nerve cell has a thickness of 0.12? m. (a) Approximating the cell membrane as a parallel?plate capacitor with a surface charge density of 5.9 × 10?6 C/m2, find the electric field 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.

Problem 92GP A square with sides of length L has a point charge at each of its four corners. Two corners that are diagonally opposite have charges equal to +2.25 ?C; the other two diagonal comers have charges Q. Find the magnitude and sign of the charges Q such that each of the +2.25?C charges experiences zero net force.

Problem 93GP IP Suppose a charge +Q is placed on the Earth, and another charge +Q 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 \(20.0^{\circ}\) with the vertical, as shown in Figure 19–39. 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. Equation Transcription: Text Transcription: 20.0^\circ

Problem 95GP A small sphere with a charge of +2.44 ?Cis 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 Q = ?8.55 ?Cis now moved slowly from infinity to a point x = d > 0 on the x axis. This causes the small sphere to move to the position x = 0.124 m. Find d.

Problem 96GP Twelve identical point charges q are equally spaced around the circumference of a circle of radius R. The circle is centered al 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.

Problem 97GP BIO Nerve Impulses When a nerve impulse propagates along a nerve cell, the electric field within the cell membrane changes from 7.0 × 105 N/C inone direction to 3.0 × 105 N/C in the other direction. Approximating the cell membrane as a parallel?plate capacitor, find the magnitude of the change in charge density on the walls of the cell membrane.

Problem 98GP IP The Electric Field of the Earth The Earth produces an approximately uniform electric field at ground level. This electric field 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 6.38 × 106 m, find 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 field at the surface be greater than, less than, or equal to 110 N/C? Explain.

An object of mass \(m=3.1 \mathrm{~g}\) and charge \(Q=+48 \mu C\) is attached to a string and placed in a uniform electric field that is inclined at an angle of \(30.0^{\circ}\) with the horizontal (Figure 19–40). The object is in static equilibrium when the string is horizontal. Find (a) the magnitude of the electric field and (b) the tension in the string. Equation Transcription: Text Transcription: m=3.1 g Q=+48 \mu C 30.0°

Problem 100GP Four identical charges, +Q, occupy the corners of a square with sides of length a. A fifth charge, q, can be placed at any desired location. Find the location of the fifth charge, and the value of q, such that the net electric force acting on each of the original four charges, +Q, is zero.

Figure 19–41 shows an electron entering a parallel-plate capacitor with a speed of \(5.45 \times 10^{6} \mathrm{~m} / \mathrm{s}\). The electric field of the capacitor has deflected the electron downward by a distanceof 0.618 cm at the point where the electron exits the capacitor. Find (a) the magnitude of the electric field in the capacitor and (b) the speed of the electron when it exits the capacitor. Equation Transcription: Text Transcription: 5.45 X 106 m/s

Problem 102GP Two identical conducting spheres are separated by a fixed 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 final and (b) the initial charges on the spheres.

Problem 103PP How many electrons must be transferred away from a bee to produce a charge of +93.0 pC? A. 1.72 × 10?9 B. 5.81 × 108 C. 1.02 × 1020 D. 1.49 × 1029

Problem 104PP 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 0.140g bee is 1.37 × 10?3 N.) A. 6.01 × 10?17 N B. 6.48 × 10?9 N C. 5.40 × 10?7N D. 5.81 × 10?3N

Problem 105PP The force required to detach a grain of pollen from an avocado stigma is approximately 4.0 × 10?5 N. What is the maximum distance at which the electrostatic force between a bee and a grain of pollen is sufficient 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. 4.7 × 10?7 m B. 1.9 mm C. 4.4 cm D. 220 m

Referring to Example 19–5 Suppose \(q_{1}=+2.90 \mu C\) is no longer at the origin, but is now on the y axis between \(y=0 \text { and } y=0.500 \mathrm{~m}\). The charge \(q_{2}=+2.90 \mu C \text { is at } x=0 \text { and } y=0.500 \mathrm{~m}\), and point 3 is at \(x=y=0.500 \mathrm{~m}\) (a) Is the magnitude of the net electric field at point 3, which we call \(E_{n e t}\), greater than, less than, or equal to its previous value? Explain. (b) Is the angle \(\theta\) that \(E_{n t}\) makes with the x axis greater than,less than, or equal to its previous value? Explain. Find the newvalues of (c) \(E_{n t}\) and (d) \(\theta \text { if } q_{1} \text { is at } y=0.250 \mathrm{~m}\) Equation Transcription: Text Transcription: q_1=+2.90 \mu C y=0 and y=0.500 m q_{2}=+2.90 \mu C is at x=0 and y=0.500 m x=y=0.500 m E_n e t \theta E_n t E_n e t \theta if q_1 is at y=0.250 ~m

Problem 106PP The Earth produces an electric field of magnitude 110 N/C. What force does this electric field exert on a bee carrying a charge of 93.0 pC? (Again, for comparison, the weight of a bee is approximately 1.37 × 10?3 N.) A. 1.76 × 10?17N B. S.45 × 10?13 N C. 1.02 × 10?8 N D. 1.13 × 10?6 N

Referring to Example 19–5 In this system, the charge \(q_{1}\) is at the origin, the charge \(q_{2}\) is at \(x=0 \text { and } y=0.500 \mathrm{~m}\), and point 3 is at \(x=y=0.500 \mathrm{~m}\). Suppose that \(q_{1}=+2.90 \mu \mathrm{C}\), but that \(q_{2}\) is increased to a value greater than \(+2.90 \mu \mathrm{C}\). As a result, do (a) \(E_{n e t}\) and (b) \(\theta\) increase, decrease,or stay the same? Explain. If \(E_{n e t}=1.66 \mathrm{X} 10^{5} \mathrm{~N} / \mathrm{C}\), find (c) \(q_{2}\) and (d) \(\theta\). Equation Transcription: Text Transcription: q1 q2 x=0 and y=0.500 m x=y=0.500m q1=+2.90 \muC q2 +2.90 \mu C Enet \theta Enet=1.66 X 105 N/C q2 \theta

Referring to Example 19–6 The magnitude of the charge is changed until the angle the thread makes with the vertical is \(\theta=15^{\circ}\). The electric field is \(1.46 X 10^{4} \mathrm{~N} / \mathrm{C}\) and the mass of the object is 0.0250 kg. (a) Is the new magnitude of q greater than or less than its previous value? Explain. (b) Find the new value of q. Equation Transcription: Text Transcription: \theta=15° 1.46 X 104 N/C