Four identical charges Q are placed at the corners of a

Problem 103P An infinite sheet with positive charge area ? lies in the xy-plane. A second infinite sheet with negative charge per unit area ?? lies in the yz-plane. Find the net electric field at all points that do not lie in either of these planes. Express your answer in terms of the unit vectors and

Problem 1E Excess electrons are placed on a small lead sphere with mass 8.00 g so that its net charge is - 3.20 X 10-9 C. (a) Find the number of excess electrons on the sphere. (b) How many excess electrons are there per lead atom? The atomic number of lead is 82, and its atomic mass is 207 g/mol.

Problem 1DQ If you peel two strips of transparent tape off the same roll and immediately let them hang near each other, they will repel each other. If you then stick the sticky side of one to the shiny side of the other and rip them apart, they will attract each other. Give a plausible explanation, involving transfer of electrons between the strips of tape, for this sequence of events.

Lightning occurs when there is a flow of electric charge (principally electrons) between the ground and a thundercloud. The maximum rate of charge flow in a lightning bolt is about 20,000 C/s; this lasts for 100 \(\mu \mathrm{s}\) or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

Problem 2DQ Two metal spheres are hanging from nylon threads. When you bring the spheres close to each other, they tend to attract. Based on this information alone, discuss all the possible ways that the spheres could be charged. Is it possible that after the spheres touch, they will cling together? Explain.

Problem 3DQ The electric force between two charged particles becomes weaker with increasing distance. Suppose instead that the electric force were ?independent ?of distance. In this case, would a charged comb still cause a neutral insulator to become polarized as in Fig. 21.8? Why or why not? Would the neutral insulator still be attracted to the comb? Again, why or why not?

Problem 3E Estimate how many electrons there are in your body. Make any assumptions you feel are necessary, but cleady state what they are. (?Hint?: Most of the atoms in your body have equal numbers of electrons, protons, and neutrons.) What is the combined charge of all these electrons?

Problem 4DQ Your clothing tends to cling together after going through the dryer. Why? Would you expect more or less clinging if all your clothing were made of the same material (say, cotton) than if you dried different kinds of clothing together? Again, why? (You may want to experiment with your next load of laundry.)

Problem 4E Particles in a Gold Ring. You have a pure (24-karat gold ring with mass 17.7 g. Gold has an atomic mass of 197 g/mol and an atomic number of 79. (a) How many protons are in the ring, and what is their total positive charge? (b) If the ring carries no net charge, how many electrons are in it?

Problem 5DQ An uncharged metal sphere hangs from a nylon thread. When a positively charged glass rod is brought close to the metal sphere, the sphere is drawn toward the rod. But if the sphere touches the rod, it suddenly flies away from the rod. Explain why the sphere is first attracted and then repelled.

Problem 5E BIO Signal Propagation in Neurons. ?Neurons are components of the nervous system of the body that transmit signals as electric impulses travel along their length. These impulses propagate when charge suddenly rushes into and then out of a part of the neuron called an ?axon?. Measurements have shown that, during the inflow part of this cycle, approximately 5.6 X 1011 Na+ (sodium ions) per meter, each with charge + e, enter the axon. How many coulombs of charge enter a 1.5-cm length of the axon during this process?

Problem 6DQ The free electrons in a metal are gravitationally attracted toward the earth. Why, then, don’t they all settle to the bottom of the conductor, like sediment settling to the bottom of a river?

Problem 6E Two small spheres spaced 20.0 cm apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is 4.57 × 10?21 N?

Problem 8DQ Good conductors of electricity, such as metals, are typically good conductors of heat; insulators, such as wood, are typically poor conductors of heat. Explain why there is a relationship between conduction of electricity and conduction of heat in these materials.

Problem 7E An average human weighs about 650 N. If each of two average humans could carry 1.0 C of excess charge, one positive and one negative, how far apart would they have to be for the electric attraction between them to equal their 650-N weight?

Problem 7DQ Figure Q21.7 shows some of the electric field lines due to three point charges arranged along the vertical axis. All three charges have the same magnitude. (a) What are the signs of the three charges? Explain your reasoning. (b) At what point(s) is the magnitude of the electric field the smallest? Explain your reasoning. Explain how the fields produced by each individual point charge combine to give a small net field at this point or points.

Problem 8E Two small aluminum spheres, each having mass 0.0250 kg, are separated by 80.0 cm. (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 g/mol, and its atomic number is 13.) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude 1.00 X 104 N (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the electrons in each sphere does this represent?

Problem 9E Two small plastic spheres are given positive electric charges. When they are 15.0 cm apart, the repulsive force between them has magnitude 0.220 N. What is the charge on each sphere (a) if the two charges are equal and (b) if one sphere has four times the charge of the other?

Problem 9DQ Suppose the charge shown in Fig. 21.28a is fixed in position. A small, positively charged particle is then placed at some point in the figure and released. Will the trajectory of the particle follow an electric field line? Why or why not? Suppose instead that the particle is placed at some point in Fig. 21.28b and released (the positive and negative charges shown in the figure are fixed in position). Will its trajectory follow an electric field line? Again, why or why not? Explain any differences between your answers for the two different situations.

Problem 10E What If We Were Not Neutral? A 75-kg person holds out his arms so that his hands are 1.7 m apart. Typically, a person’s hand makes up about 1.0% of his or her body weight. For round numbers, we shall assume that all the weight of each hand is due to the calcium in the bones, and we shall treat the hands as point charges. One mole of Ca contains 40.18 g, and each atom has 20 protons and 20 electrons. Suppose that only 1.0% of the positive in each hand were unbalanced by negative charge. (a) How many Ca atoms does each band contain? (b) How many coulombs of unbalanced charge does each hand contain? (c) What force would the person’s arms have to exert on his hands to prevent them from flying off? Does it seem likely that his arms are capable of exerting such a force?

Problem 11DQ You can use plastic food wrap to cover a container by stretching the material across the top and pressing the overhanging material against the sides. What makes it stick? (Hint: the answer involves the electric force. ) Does the food wrap suck to use if with equal tenacity? Why or why not? Does it work with metallic containers? Again, why or why nor?

Problem 11E Two very small 8.55-g spheres, 15.0 cm apart from center to center, are charged by adding equal numbers of electrons to each of them. Disregarding all other forces, how many electrons would you have to add to each sphere so that the two spheres will accelerate at 25.0g when released? Which way will they accelerate?

Problem 12DQ If you walk across a nylon rug and then touch a large metal object such as a doorknob, you may get a spark and a shock. Why does this tend to happen more on dry days than on humid days? (Hint: See Fig. 21.30.) Why are you less likely to get a shock if you touch a small metal object, such as a paper clip?

Problem 10DQ Two identical metal objects are mounted on insulating stands. Describe how you could place charges of opposite sign but exactly equal magnitude on the two objects.

Problem 12E Just How Strong Is the Electric Force? Suppose you had two small boxes, each containing 1.0 g of protons. (a) If one were placed on the moon by an astronaut and the other were left on the earth, and if they were connected by a very light (and very long!) string, what would be the tension in the string? Express your answer in newtons and in pounds. Do you need to take into account the gravitational forces of the earth and moon on the protons? Why? (b) What gravitational force would each box of protons exert on the other box?

Problem 13DQ You have a negatively charged object. How can you use it to place a net negative charge on an insulated metal sphere? To place a net positive charge on the sphere?

Problem 13E In an experiment in space, one proton is held fixed and another proton is released from rest a distance of 2.50 mm away. (a) What is the initial acceleration of the proton after it is released? (b) Sketch qualitative (no numbers!) acceleration–time and velocity–time graphs of the released proton’s motion.

Problem 14DQ When two point charges of equal mass and charge are released on a frictionless table, each has an initial acceleration (magnitude) a0. If instead you keep one fixed and release the other one, what will be its initial acceleration: a0, 2a0, or a0/2? Explain.

Problem 14E A negative charge of ?0.550 µC exerts an upward 0.200-N force on an unknown charge 0.300 m directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the ?0.550 µC charge?

Problem 15DQ A point charge of mass m and charge Q and another point charge of mass m but charge 2Q are released on a frictionless table. If the charge Q has an initial acceleration a0, what will be the acceleration of 2Q: a0, 2a0, 4a0, a0/2, or a0/4? Explain

Problem 15E Three point charges are arranged on a line. Charge q3 = + 5.00 nC and is at the origin. Charge q2 = -3.00 nC and is at x = + 4.00 cm. Charge q1 is at x = + 2.00 cm. What is q1 (magnitude and sign) if the net force on q3 is zero?

Problem 16DQ A proton is placed in a uniform electric field and then released. Then an electron is placed at this same point and released. Do these two particles experience the same force? The same acceleration? Do they move in the same direction when released?

Problem 16E In Example 21.4, suppose the point charge on the –axis

What similarities do electrical forces have with gravitational forces? What are the most significant differences?

Problem 17E In Example 21.3, calculate the net force on charge q1.

Problem 19DQ Two irregular objects A and B carry charges of opposite sign. Figure Q21.19 shows the electric field lines near each of these objects. (a) Which object is positive, A or B? How do you know? (b) Where is the electric field stronger, close to A or close to B? How do you know?

Problem 20DQ Atomic nuclei are made of protons and neutrons. This shows that there must be another kind of interaction in addition to gravitational and electric forces. Explain.

Repeat Exercise 21.19 for \(q_{3}=+8.00 \mu \mathrm{C}\).

Problem 18E In Example 21.4, what is the net force (magnitude and direction) on charge q1 exerted by the other two charges?

Problem 19E Three point charges are arranged along the x-axis. Charge q1 = + 3.00 µC is at the origin, and charge q2 = - 5.00 µC is at x = 0.200 m. Charge q3 = - 8.00 µC. Where is q3 located if the net force on q1 is 7.00 N in the – x-direction?

Problem 21DQ Sufficiently strong electric fields can cause atoms to become positively ionized—that is, to lose one or more electrons. Explain how this can happen. What determines how strong the field must be to make this happen?

Problem 17DQ In Example 21.1 (Section 21.3) we saw that the electric

Problem 21E Two point charges are located on the y-axis as follows: charge q1 = - 1.50 nC at y = - 0.600 m, and charge q2 = + 3.20 nC at the origin (y = 0). What is the total force (magnitude and direction) exerted by these two charges on a third charge q3 = + 5.00 nC located at y = - 0.400 m?

Problem 23DQ The air temperature and the velocity of the air have different values at different places in the earth’s atmosphere. Is the air velocity a vector field? Why or why not? Is the air temperature a vector field? Again, why or why not?

Problem 22E Two point charges are placed on the x-axis as follows: Charge q1 = + 4.00 nC is located at x = 0.200 m, and charge q2 = + 5.00 nC is at x = - 0.300 m . What are the magnitude and direction of the total force exerted by these two charges on a negative point charge q3 = - 6.00 nC that is placed at the origin?

Problem 22DQ The electric fields at point P due to the positive charges q1 and q2 are shown in Fig. Q21.22. Does the fact that they cross each other violate the statement in Section 21.6 that electric field lines never cross? Explain.

Problem 23E BIO Base Pairing in DNA, I. The two sides of the DNA double helix are connected by pairs of bases (adenine, thymine, cytosine, and guanine). Because of the geometric shape of these molecules, adenine bonds with thymine and cytosine bonds with guanine. Figure E21.21 shows the bonding of thymine and adenine. Each charge shown is and the H—N distance is 0.110 nm. (a) Calculate the net force that thymine exerts on adenine. Is it attractive or repulsive? To keep the calculations fairly simple, yet reasonable, consider only the forces due to the O—H—N and the N—H—N combinations, assuming that these two combinations are parallel to each other. Remember, however, that in the O—H—N set, the O-exerts a force on both the H+ and the N-, and likewise along the N—H—N set. (b) Calculate the force on the electron in the hydrogen atom, which is 0.0529 nm from the proton. Then compare the strength of the bonding force of the electron in hydrogen with the bonding force of the adenine–thymine molecules.

Problem 25E CP A proton is placed in a uniform electric field of 2.75 X 103 N/C. Calculate (a) the magnitude of the electric force felt by the proton; (b) the proton’s acceleration; (c) the proton’s speed after 1.00 µs in the field, assuming it starts from rest.

Problem 26E A particle has charge ?3.00 nC. (a) Find the magnitude and direction of the electric field due to this particle at a point 0.250 m directly above it. (b) At what distance from this particle does its electric field have a magnitude of 12.0 N/C?

Problem 28E CP An electron is released from rest in a uniform electric field. The electron accelerates vertically upward, traveling 4.50 m in the first 3.00 µs after it is released. (a) What are the magnitude and direction of the electric field? (b) Are we justified in ignoring the effects of gravity? Justify your answer quantitatively.

Problem 24E BIO Base Pairing in DNA, II. Refer to Exercise 21.21. Figure E21.22 shows the bonding of cytosine and guanine. The O—H and H—N distances are each 0.110 nm. In this case, assume that the bonding is due only to the forces along the O—H—O, N—H—N, and O—H—N combinations, and assume also that these three combinations are parallel to each other. Calculate the net force that cytosine exerts on guanine due to the preceding three combinations. Is this force attractive or repulsive?

Problem 30E A point charge is placed at each corner of a square with side length a. All charges have magnitude q. Two of the charges are positive and two are negative (Fig. E21.42). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of q and a ?

Problem 29E (a) What must the charge (sign and magnitude) of a 1.45-g particle be for it to remain stationary when placed in a downward-directed electric field of magnitude 650 N/C? (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitude to its weight?

Problem 31E Two point charges are separated by 25.0 cm (Fig. E21.43). Find the net electric field these charges produce at (a) point A and (b) point B. (c) What would be the magnitude and direction of the electric force this combination of charges would produce on a proton at A?

Problem 32E Electric Field of the Earth. The earth has a net electric charge that causes a field at points near its surface equal to 150 N/C and directed in toward the center of the earth. (a) What magnitude and sign of charge would a 60-kg human have to acquire to overcome his or her weight by the force exerted by the earth’s electric field? (b) What would be the force of repulsion between two people each with the charge calculated in part (a) and separated by a distance of 100 m? Is use of the earth’s electric field a feasible means of flight? Why or why not?

Problem 33E CP An electron is projected with an initial speed v0 = 1.60 X 106 m/s into the uniform field between two parallel plates (Fig. E21.29). Assume that the field between the plates is uniform and directed vertically downward and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. (a) If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field. (b) Suppose that the electron in Fig. E21.29 is replaced by a proton with the same initial speed v0. Would the proton hit one of the plates? If not, what would be the magnitude and direction of its vertical displacement as it exits the region between the plates? (c) Compare the paths traveled by the electron and the proton, and explain the differences. (d) Discuss whether it is reasonable to ignore the effects of gravity for each particle.

(a) Calculate the magnitude and direction (relative to the +x-axis) of the electric field in Example 21.6. (b) A point charge is placed at point P in Fig. 21.19. Find the magnitude and direction of (i) the force that the -8.0-nC charge at the origin exerts on this charge and (ii) the force that this charge exerts on the -8.0-nC charge at the origin.

Problem 37E If two electrons are each 1.50 X 10-10 m from a proton (Fig. E21.45), find the magnitude and direction of the net electric force they will exert on the proton.

Problem 35E CP In Exercise 21.29, what is the speed of the electron as it emerges from the field? 21.29 .. CP An electron is projected with an initial speed v0 = 1.60 X 106 m/s into the uniform field between two parallel plates (Fig. E21.29). Assume that the field between the plates is uniform and directed vertically downward and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. (a) If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field. (b) Suppose that the electron in Fig. E21.29 is replaced by a proton with the same initial speed v0. Would the proton hit one of the plates? If not, what would be the magnitude and direction of its vertical displacement as it exits the region between the plates? (c) Compare the paths traveled by the electron and the proton, and explain the differences. (d) Discuss whether it is reasonable to ignore the effects of gravity for each particle.

Problem 39E A point charge is at the origin. With this point charge as the source point, what is the unit vector in the direction of the field point (a) at x = 0, y = - 1.35 m; (b) at x = 12.0 cm, y = 12.0 cm; (c) at x = - 1.10 m, y = 2.60 m ? Express your results in terms of the unit vectors

A uniform electric field exists in the region between two oppositely charged plane parallel plates. A proton is released from rest at the surface of the positively charged plate and strikes the surface of the opposite plate, 1.60 cm distant from the first, in a time interval of \(1.50 \times 10^{-6}\) s. (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.

Point charge \(q_{1}=-5.00\) nC is at the origin and point charge \(q_{2}=+3.00\) nC is on x -axis at x = 3.00 cm. Point P is on the y - axis at y = 4.00 cm. (a) Calculate the electric fields \(\overrightarrow{\boldsymbol{E}}_{1}\) and \(\overrightarrow{\boldsymbol{E}}_{2}\) at point P due to the charges \(q_{1}\) and \(q_{2}\). Express your results in terms of unit vectors (see Example 21.6). (b) Use the results of part (a) to obtain the resultant field at P, expressed in unit vector form.

Problem 41E (a) An electron is moving east in a uniform electric field of 1.50 N/C directed to the west. At point A, the velocity of the electron is 4.50 X 105 m/s toward the east. What is the speed of the electron when it reaches point B, 0.375 m east of point A? (b) A proton is moving in the uniform electric field of part (a). At point A, the velocity of the proton is 1.90 X 104 m/s, east. What is the speed of the proton at point B?

Two point charges Q and +q (where q is positive) produce the net electric field shown at point P in Fig. E21.42. The field points parallel to the line connecting the two charges. (a) What can you conclude about the sign and magnitude of Q? Explain your reasoning. (b) If the lower charge were negative instead, would it be possible for the field to have the direction shown in the figure? Explain your reasoning.

Problem 43E Two positive point charges q are placed on the x-axis, one at x = a and one at x = -a. (a) Find the magnitude and direction of the electric field at x = 0. (b) Derive an expression for the electric field at points on the x -axis. Use your result to graph the x-component of the electric field as a function of x, for values of x between -4a and +4a.

Problem 40E A +8.75-µC point charge is glued down on a horizontal frictionless table. It is tied to a -6.50-µC point charge by a light, nonconducting 2.50-m wire. A uniform electric field of magnitude 1.85 X 10 8 N/C is directed parallel to the wire, as shown in Fig. E21.34. (a) Find the tension in the wire. (b) What would the tension be if both charges were negative?

The two charges \(q_1 \) and \(q_2 \) shown in Fig. E21.44 have equal magnitudes. What is the direction of the net electric field due to these two charges at points A (midway between the charges), B, and C if (a) both charges are negative, (b) both charges are positive, (c) \(q_1 \) is positive and \(q_2 \) is negative.

Problem 45E A + 2.00-nC point charge is at the origin, and a second - 5.00-nC point charge is on the x-axis at x = 0.800 m. (a) Find the electric field (magnitude and direction) at each of the following points on the x-axis: (i) x = 0.200 m; (ii) x = 1.20 m; (iii) x = - 0.200 m. (b) Find the net electric force that the two charges would exert on an electron placed at each point in part (a).

Problem 48E BIO Electric Field of Axons. A nerve signal is transmitted through a neuron when an excess of Na + ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately 10.0 µm in diameter, and measurements show that about 5.6 X 1011 Na+ ions per meter (each of charge +e) enter during this process. Although the axon is a long cylinder, the charge does not all enter everywhere at the same time. A plausible model would be a series of point charges moving along the axon. Consider a 0.10-mm length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many coulombs of charge enter a 0.10-mm length of the axon? (b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is 5.00 cm below the skin? (c) Certain sharks can respond to electric fields as weak as 1.0 µN/C. How far from this segment of axon could a shark be and still detect its electric field?

Problem 46E Repeat Exercise, but now let q1 = ?4.00 nC. Exercise: The two charges q1 and q2 shown in Fig. have equal magnitudes. What is the direction of the net electric field due to these two charges at points A (midway between the charges), B, and C if (a) both charges are negative, (b) both charges are positive, (c) q1 is positive and q2 is negative. Figure:

Problem 47E Three negative point charges lie along a line as shown in Fig. E21.41. Find the magnitude and direction of the electric field this combination of charges produces at point P, which lies 6.00 cm from the - 2.00-µC charge measured perpendicular to the line connecting the three charges.

Problem 49E In a rectangular coordinate system a positive point charge q = 6.00 X 10-9 C is placed at the point x = +0.150 m, y = 0, and an identical point charge is placed at x = -0.150 m, y = 0. Find the x- and y-components, the magnitude, and the direction of the electric field at the following points: (a) the origin; (b) x = 0.300 m, y = 0; (c) x = 0.150 m, y = - 0.400 m; (d) x = 0, y = 0.200 m.

Problem 50E A point charge q1 = -4.00 nC is at the point x = 0.600 m, y = 0.800 m, and a second point charge q2 = + 6.00 nC is at the point x = 0.600 m, y = 0. Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

Problem 51E Repeal Exercise, for the case where the point charge at x = +0.150 m, y = 0 is positive and the other is negative each with magnitude 6.00 × 10?9 C. Exercise: In a rectangular coordinate system a positive point charge q = 6.00 × 10?9 C is placed at the point x = +0.150 m, y = 0 and an identical point charge is placed at x = ?0.150 m, y = 0. Find the x- and y-components, the magnitude, and the direction of the electric field at the following points: (a) the origin; (b) x = 0.300 m, y = 0: (c) x = 0.150 m, y = ?0.400 m. (d) x = 0, y = 0.200 m.

A straight, nonconducting plastic wire 8.50 cm long carries a charge density of + 175 nC/m distributed uniformly along its length. It is lying on a horizontal tabletop. (a) Find the magnitude and direction of the electric field this wire produces at a point 6.00 cm directly above its midpoint. (b) If the wire is now bent into a circle lying flat on the table, find the magnitude and direction of the electric field it produces at a point 6.00 cm directly above its center.

Problem 52E A very long, straight wire has change per unit length 1.50 × 10?10 C/m. At what distance from the wire is the electric field magnitude equal to 2.50 N/C?

Problem 55E A charge of -6.50 nC is spread uniformly over the surface of one face of a nonconducting disk of radius 1.25 cm. (a) Find the magnitude and direction of the electric field this disk produces at a point P on the axis of the disk a distance of 2.00 cm from its center. (b) Suppose that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Find the magnitude and direction of the electric field at point P. (c) If the charge is all brought to the center of the disk, find the magnitude and direction of the electric field at point P. (d) Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields?

A ring-shaped conductor with radius \(a=2.50\) has a total positive charge \(Q=+0.125 \mathrm{nC}\) uniformly distributed around it, as shown in Fig. 21.23. The center of the ring is at the origin of coordinates O. (a) What is the electric field (magnitude and direction) at point P, which is on the -axis at x = 40.0 cm? (b) A point charge \(q=-2.50 \mu \mathrm{C}\) is placed at the point P described in part (a). What are the magnitude and direction of the force exerted by the charge q on the ring?

Problem 56E The ammonia molecule (NH3) has a dipole moment of 5.0 X 10-30 C ? m. Ammonia molecules in the gas phase are placed in a uniform electric field with magnitude 1.6 X 106 N/C. (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to E S from parallel to perpendicular? (b) At what absolute temperature T is the average translational kinetic energy of a molecule equal to the change in potential energy calculated in part (a)? (Note: Above this temperature, thermal agitation prevents the dipoles from aligning with the electric field.)

Problem 57E Point charges q1 = -4.5 nC and q2 = +4.5 nC are separated by 3.1 mm, forming an electric dipole. (a) Find the electric dipole moment (magnitude and direction). (b) The charges are in a uniform electric field whose direction makes an angle of 36.9° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.2 X 10-9 N ? m?

Problem 58E The dipole moment of the water molecule is

Problem 59E Torque on a Dipole. An electric dipole with dipole moment is in a uniform external electric field (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part (b), the dipole’s own electric field tends to oppose the external field.

Problem 61E Three charges are at the corners of an isosceles tri-angle as shown in Fig. E21.57. The charges form a dipole. (a) Find the force (magnitude and direction) the - 10.00-µC charge exerts on the dipole. (b) For an axis perpendicular to the line connecting the charges at the mid-point of this line, find the torque (magnitude and direction) exerted on the dipole by the - 10.00-µC charge.

A dipole consisting of charges \(\pm e\), 220 nm apart, is placed between two very large (essentially infinite) sheets carrying equal but opposite charge densities of \(125 \mu \mathrm{C} / \mathrm{m}^{2}\) (a) What is the maximum potential energy this dipole can have due to the sheets, and how should it be oriented relative to the sheets to attain this value? (b) What is the maximum torque the sheets can exert on the dipole, and how should it be oriented relative to the sheets to attain this value? (c) What net force do the two sheets exert on the dipole?

Problem 63P Four identical charges Q are placed at the corners of a square of side L. (a) In a free-body diagram, show all of the forces that act on one of the charges. (b) Find the magnitude and direction of the total force exerted on one charge by the other three charges.

Problem 64P Two charges are placed on the x-axis: one, of 2.50 µC, at the origin and the other, of - 3.50 µC, at x = 0.600 m (Fig. P21.60). Find the position on the x-axis where the net force on a small charge +q would be zero.

Three point charges are arranged along the x - axis. Charge \(q_1=-4.50 \mathrm{nC}\) is located at \(x=0.200 \mathrm{~m}\), and charge \(q_2=+2.50 \mathrm{nC}\) is at \(x=-0.300 \mathrm{~m}\). A positive point charge \(q_3\) is located at the origin. (a) What must the value of \(q_3\) be for the net force on this point charge to have magnitude \(4.00 \mu \mathrm{N}\)? (b) What is the direction of the net force on \(q_3\)? (c) Where along the x - axis can \(q_3\) be placed and the net force on it be zero, other than the trivial answers of \(x=+\infty\) and \(x=-\infty\)?

Problem 60E Consider the electric dipole of Example 21.14. (a) Derive an expression for the magnitude of the electric field produced by the dipole at a point on the -axis in Fig. 21.33. What is the direction of this electric field? (b) How does the electric field at points on the -axis depend on when is very large?

Problem 66P A charge q1 = +5.00 nC is placed at the origin of an xy -coordinate system, and a charge q2 = -2.00 nC is placed on the positive x-axis at x = 4.00 cm. (a) If a third charge q3 = +6.00 nC is now placed at the point x = 4.00 cm, y = 3.00 cm, find the x- and y-components of the total force exerted on this charge by the other two. (b) Find the magnitude and direction of this force.

Problem 68P CP Two identical spheres with mass m are hung from silk threads of length L (Fig. P21.62). The spheres have the same charge, so q1 = q2 = q. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Show that if the angle ? is small, the equilibrium separation d between the spheres is (Hint: If ? is small, then

Problem 67P Two positive point charges Q are held fixed on the x-axis at x = a and x = ?a. A third positive point charge q, with mass m, is placed on the x-axis away from the origin at a coordinate x such that |x| ? a. The charge q, which is free to move, along the x-axis, is then released. (a) Find the frequency of oscillation of the charge q. (Hint: Review the definition of simple harmonic motion in Section 14.2. Use the binomial expansion (1+ z)n = 1 + nz + n(n ? 1)z2/2 +..., valid for the case |z| < 1.) (b) Suppose instead that the charge q were placed on the y-axis at a coordinate y such that |y| ? a, and then released. If this charge is free to move anywhere in the xy-plane, what will happen to it? Explain your answer.

Problem 69P CP Two small spheres with mass m = 15.0 g are hung by silk threads of length L = 1.20 m from a common point (Fig. P21.62). When the spheres are given equal quantities of negative charge, so that q1 = q2 = q, each thread hangs at ? = 25.0o from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of q. (c) Both threads are now shortened to length L = 0.600 m, while the charges q1 and q2 remain unchanged. What new angle will each thread make with the vertical? (Hint: This part of the problem can be solved numerically by using trial values for ? and adjusting the values of ? until a self-consistent answer is obtained.)

Problem 70P CP Two identical spheres are each attached to silk threads of length L = 0.500 m and hung from a common point (Fig. P21.62). Each sphere has mass m = 8.00 g. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. One sphere is given positive charge q1, and the other a different positive charge q2; this causes the spheres to separate so that when the spheres are in equilibrium, each thread makes an angle ? = 20.0o with the vertical. (a) Draw a free-body diagram for each sphere when in equilibrium, and label all the forces that act on each sphere. (b) Determine the magnitude of the electrostatic force that acts on each sphere, and determine the tension in each thread. (c) Based on the given information, what can you say about the magnitudes of q1 and q2? Explain. (d) A small wire is now connected between the spheres, allowing charge to be transferred from one sphere to the other until the two spheres have equal charges; the wire is then removed. Each thread now makes an angle of 30.0o with the vertical. Determine the original charges. (Hint: The total charge on the pair of spheres is conserved.)

Problem 71P Sodium chloride (NaCl, ordinary table salt) is made up of positive sodium ions (Na+) and negative chloride ions (Cl?). (a) If a point charge with the same charge and mass as all the Na+ ions in 0.100 mol of NaCl is 2.00 cm from a point charge with the same charge and mass as all the Cl? ions, what is the magnitude of the attractive force between these two point Charges? (b) If the positive point charge in part (a) is held in place and the negative point charge is released from rest, what is initial acceleration? (See Appendix D for atomic masses.) (c) Does it seem reasonable that the ions in NaCl could be separated in this way? Why or why not? (In fact, when sodium chloride dissolves in water, it breaks up into Na+ and Cl? ions. However, in this situation there are additional electric forces exerted by the water molecules on the ions.)

Problem 72P A ?5.00-nC point charge is on the x-axis at x = 1.20 m. A second point charge Q is on the x-axis at ?0.600 m. What must be the sign and magnitude of Q for the resultant electric field at the origin to be (a) 45.0 N/C in the +x-direction, (b) 45.0 N/C in the ?x-direction?

Problem 73P CP A small 12.3-g plastic ball is tied to a very light 28.6-cm string that is attached to the vertical wall of a room (Fig. P21.65). A uniform horizontal electric field exists in this room. When the ball has been given an excess charge of -1.11 µC, you observe that it remains suspended, with the string making an angle of 17.4° with the wall. Find the magnitude and direction of the electric field in the room.

Problem 74P At t = 0 a very small object with mass 0.400 mg and charge +9.00 µC is traveling at 125 m/s in the ?x-direction. The charge is moving in a uniform electric field that is in the +y-direction and that has magnitude E = 895 N/C. The gravitational force on the particle can be neglected. How far is the particle from the origin at t = 7.00 ms?

Problem 75P Two particles having charges q1 = 0.500 nC and q2 = 8.00 nC are separated by a distance of 1.20 m. At what point along the line connecting the two charges is the total electric field due to the two charges equal to zero?

Problem 76P Two point charges q1 and q2 are held in place 4.50 cm apart. Another point charge Q = -1.75 µC, of mass 5.00 g, is initially located 3.00 cm from both of these charges (Fig. P21.72) and released from rest. You observe that the initial acceleration of Q is 324 m/s2 upward, parallel to the line connecting the two point charges. Find q1 and q2.

Problem 77P Three identical point charges q are placed at each of three corners of a square of side L. Find the magnitude and direction of the net force on a point charge -3q placed (a) at the center of the square and (b) at the vacant corner of the square. In each case, draw a free-body diagra

Problem 79P CP Strength of the Electric Force. Imagine two 1.0-g bags of protons, one at the earth’s north pole and the other at the south pole. (a) How many protons are in each bag? (b) Calculate the gravitational attraction and the electric repulsion that each bag exerts on the other. (c) Are the forces in part (b) large enough for you to feel if you were holding one of the bags?

Problem 80P Electric Force With in the Nucleus. Typical dimensions of atomic nuclei are of the order of 10?15 m (l fm). (a) If two protons in a nucleus are 2.0 fm apart, find the magnitude of the electric force each one exerts on the other. Express the answer in newtons and in pounds. Would this force be large enough for a person to feel? (b) Since the protons repel each other so strongly, why don’t they shoot out of the nucleus?

Problem 81P If Atoms Were Not Neutral... Because the charges on the electron and proton have the same absolute value, atoms are electrically neutral. Suppose this were not precisely true, and the absolute value of the charge of the electron were less than the charge of the proton by 0.00100%. (a) Estimate what the net charge of this textbook would be under these circumstances. Make any assumptions you feel are justified, but state clearly what they are. (Hint: Most of the atoms in this textbook have equal numbers of electrons, protons, and neutrons (b) What would be the magnitude of the electric force between two textbooks placed 5.0 m apart? Would this force be altractive or repulsive? Estimate wha the acceleration of each book would be if the books were 5.0 m apart and there were no non electric forces on them (c) Discuss how the act that ordinary matter is stable shows that the absolute values of the charges on the electron and proton must be identical to a very high level of accuracy.

Problem 78P Three point charges are placed on the y-axis a charge q at y = a, a charge ?2q at the origin, and a charge q at y = ?a. Such an arrangement is called an electric quadrupole. (a) Find the magnitude and direction of the electric field at points on the positive x-axis. (b) Use the binomial expansion to find an approximate expression for the electric field valid for x ? a. Contrast this behaviour to that of the electric field of a point charge and that of the electric field of a dipole.

Problem 82P CP Two tiny spheres of mass 6.80 mg carry charges of equal magnitude, 72.0 nC, but opposite sign. They are tied to the same ceiling hook by light strings of length 0.530 m. When a horizontal uniform electric field E that is directed to the left is turned on, the spheres hang at rest with the angle between the strings equal to 58.0° (Fig. P21.74). (a) Which ball (the one on the right or the one on the left) has positive charge? (b) What is the magnitude E of the field?

Problem 83P CP Consider a model of a hydrogen atom in which an electron is in a circular orbit of radius r = 5.29 X 10-11 m around a stationary proton. What is the speed of the electron in its orbit?

A small sphere with mass \(9.00 \mu \mathrm{g}\)and charge \(-4.30 \mu \mathrm{C}\) is moving in a circular orbit around a stationary sphere that has charge \(+7.50 \mu \mathrm{C}\) . If the speed of the small sphere is \(5.90 \times 10^{3} \mathrm{m} / \mathrm{s}\), what is the radius of its orbit? Treat the spheres as point charges and ignore gravity.

Two small copper spheres each have radius 1.00 mm (a) How many atoms does each sphere contain? (b) Assume that each copper atom contains 29 protons and 29 electrons. We know that electrons and protons have charges of exactly the same magnitude, but let’s explore the effect of small differences (see also Problem). If the charge of a proton is +e and the magnitude of the charge of an electron is 0.100% smaller, what is the net charge of each sphere and what force would one sphere exert on the other if they were separated by 1.00 m?

Problem 86P Operation of an Inkjet Printer. In an inkjet printer, letter are built up by squirting drops of ink at the paper from a rapidly moving nozzle. The ink drops, which have a mass of 1.4 × 10?8 g each. leave the nozzle and travel toward the paper at 20 m/s, passing through a charging unit that gives each drop a positive charge q by removing some electrons from it. The drops then pass between parallel deflecting plates 2.0 cm long where there is a uniform vertical electric field with magnitude 8.0 × 104 N/C. If a drop is to be deflected 0.30 mm by the time it reaches the end of the deflection plates, what magnitude of charge must be given to the drop?

Problem 88P A negative point charge q1 = ?4.00 nC is on the x-axis at x = 0.60 m. A second point charge q2 is on the x-axis at x = ?1.20 m. What must the sign and magnitude of q2 be for the net electric field at the origin to be (a) 50.0 N/C in the +x-direction and (b) 50.0 N/C in the –x-direction?

CALC Positive charge Q is distributed uniformly along the x-axis from x = 0 to x = a. A positive point charge q is located on the positive x-axis at x = a + r, a distance r to the right of the end of Q (Fig. P21.89). (a) Calculate the x- and y-components of the electric field produced by the charge distribution Q at points on the positive x-axis where x > a. (b) Calculate the force (magnitude and direction) that the charge distribution Q exerts on q. (c) Show that if r >> a, the magnitude of the force in part (b) is approximately \(Q q / 4 \pi \epsilon_{0} r^{2}\). Explain why this result is obtained.

CP A proton is projected into a uniform electric field that points vertically upward and has magnitude . The initial velocity of the proton has a magnitude \(v_{0}\) and is directed at an angle below the horizontal. (a) Find the maximum distance \(h_{\max }\)that the proton descends vertically below its initial elevation. You can ignore gravitational forces. (b) After what horizontal distance d does the proton return to its original elevation? (c) Sketch the trajectory of the proton. (d) Find the numerical values of \(h_{\max }\) and d if \(E=500 \mathrm{N} / \mathrm{C}\), \(v_{0}=4.00 \times 10^{5} \mathrm{~m} / \mathrm{s}\), and \(\alpha=30.0^{\circ}\)

Problem 90P CALC Positive charge Q is distributed uniformly along the positive y -axis between y = 0 and y = a. A negative point charge -q lies on the positive x-axis, a distance x from the origin (Fig. P21.82). (a) Calculate the x- and y-components of the electric field produced by the charge distribution Q at points on the positive x -axis. (b) Calculate the x- and y-components of the force that the charge distribution Q exerts on q. (c) Show that if x >> a, Explain why this result is obtained.

Problem 92P A Parallel Universe. Imagine a parallel universe in which the electric force has the same properties as in our universe but there is no gravity. In this parallel universe, the sun carries-charge Q, the earth curries charge ?Q, and the electric attraction between them keeps the earth in orbit. The earth in the parallel universe has the same mass, the same orbital radius, and the same orbital period as in our universe. Calculate the value of Q. (Consult Appendix F as needed.)

Problem 91P A charged line like that shown in Fig. 21.24 extends

Problem 93P A uniformly charged disk like the disk in Fig. 21.25 has

Problem 96P CP A small sphere with mass m carries a positive charge q and is attached to one end of a silk fiber of length L. The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density ?. Show that when the sphere is in equilibrium, the fiber makes an angle equal to arctan with the vertical sheet.

Problem 97P CALC Negative charge -Q is distributed uniformly around a quarter-circle of radius a that lies in the first quadrant, with the center of curvature at the origin. Find the x- and y-components of the net electric field at the origin.

Problem 95P Positive Charge +Q is distributed uniformly along the +x-axis from x = 0 to x = a. Negative charge ?Q is distributed uniformly along the ?x-axis from x = 0 to x = ?a. (a) A positive point charge q lies on the positive y-axis, a distance y form the origin. Find the force (magnitude and direction) that the positive negative charge distributions together exert on q. Show that this force is proportional to y?3 for y ? a. (b) Suppose instead that the positive point charge q lies on the positive x-axis, a distance x > a from the orgin. Find the force (magnitude and direction) that the charge distribution exerts on q. Show that this force is proportional to x?3 for x ? a.

Problem 99P Two 1.20-m non-conducting rods meet at a right angle. One rod carries + 2.50 µC of charge distributed uniformly along its length, and the other carries -2.50 µC distributed uniformly along it (Fig. P21.87). (a) Find the magnitude and direction of the electric field these rods produce at point P, which is 60.0 cm from each rod. (b) If an electron is released at P, what are the magnitude and direction of the net force that these rods exert on it?

Problem 94P Electrophoresis. Electrophoresis is a process used by biologists to separate different biological molecules (such as proteins) from each other according to their ratio of charge to size. The materials to be separated are in a viscous solution that produces a drag force FD proportional to the size and speed of the molecule. We can express this relationship as FD = KR?, where R is the radius of the molecule (modeled as being spherical), ? is its speed, and K is a constant that depends on the viscosity of the solution. The solution is placed in an external electric field E so that the electric force on a particle of charge q is F = qE. (a) Show the electric field is adjusted so that the two forces (electric and viscous drag) just balance, the ratio of q to R is K?/E. (b). Show that if we leave the electric field on fir a time T, the distance x that the molecule moves during that time is x = (ET/k)(q/R). (c) Suppose you have a sample containing three different biological Molecules for which the molecular ratio q/R for material 2 is twice that of material 1 and the ratio for material 3 is three times that of material 1. Show that the distances migrated by these molecules after the same amount of time are x2 = 2x1 and x3 = 3x1 in other words material 2 travels twice as far as material 1, and material 3 travels three times as, far as material 1. Therefore we have separated these molecules according to their ratio of charge to size. In practice, this process can be carried out in a special gel or paper, along which the. biological molecules migrate. (Fig.). The process can be rather slow, requiring several hours for separations of just a centimeter or so. Figure:

CALC A semicircle of radius a is in the first and second quadrants, with the center of curvature at the origin. Positive charge +Q is distributed uniformly around the left half of the semicircle, and negative charge is distributed uniformly around the right half of the semicircle (Fig. P21.98). What are the magnitude and direction of the net electric field at the origin produced by this distribution of charge?

Problem 100P Two very large parallel sheets are 5.00 cm apart. Sheet A carries a uniform surface charge density of ?9.50µC/m2, and sheet B, which is to the right or A, carries a uniform charge density of ?11.6 µC/m2. Assume the sheets are large enough to be treated as infinite. Find the magnitude and direction of the net electric field these sheets produce at a point (a) 4.00 cm to the right of sheet A; (b) 4.00 cm to the left of sheet A; (c) 4.00 cm to the right of sheet B.

Problem 101P Repeat Problem for the case where sheet B is positive. Problem: Two very large parallel sheets are 5.00 cm apart. Sheet A carries a uniform surface charge density of ?9.50µC/m2, and sheet B, which is to the right or A, carries a uniform charge density of ?11.6 µC/m2. Assume the sheets are large enough to be treated as infinite. Find the magnitude and direction of the net electric field these sheets produce at a point (a) 4.00 cm to the right of sheet A; (b) 4.00 cm to the left of sheet A; (c) 4.00 cm to the right of sheet B.

Problem 102P Two very huge horizontal sheets are 4.25 cm apart and carry equal but opposite uniform surface charge densities of magnitude ?. You want to use this these sheets to hold stationary in the region between them an oil droplet of mass 324 µg that carries an excess of five electrons. Assuming that the drop is in vacuum. (a) which way should the electric field between the plates point, rind (b) what should ? be?