Chapter 17

\(\bullet\) A positively charged rubber rod is moved close to a neutral copper ball that is resting on a nonconducting sheet of plastic. (a) Sketch the distribution of charges on the ball. (b) With the rod still close to the ball, a metal wire is briefly connected from the ball to the earth and then removed. After the rubber rod is also removed, sketch the distribution of charges (if any) on the copper ball.

\(\bullet\) Two iron spheres contain excess charge, one positive and the other negative. (a) Show how the charges are arranged on these spheres if they are very far from each other. (b) If the spheres are now brought close to each other, but do not touch, on the copper ball. sketch how the charges will be distributed on their surfaces. (c) In part (b), show how the charges would be distributed if both spheres were negative.

\(\bullet\) Electrical storms. During an electrical storm, clouds can build up very large amounts of charge, and this charge can induce charges on the earth's surface. Sketch the distribution of charges at the earth's surface in the vicinity of a cloud if the cloud is positively charged and the earth behaves like a conductor.

\(\bullet\) Signal propagation in neurons. Neurons are components of the nervous system of the body that transmit signals as elec- trical impulses travel along their length. These impulses propa- gate when charge suddenly rushes into and then out of a part of the neutron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately \(5.6 \times 10^{11} \mathrm{Na}^{+}\) (sodium ions) per meter, each with charge \(+e\) enter the axon. How many coulombs of charge enter a 1.5 \(\mathrm{cm}\) length of the axon during this process?

\(\bullet\) \(\bullet\) 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 car- ries no net charge, how many electrons are in it?

\(\bullet\) At what distance would the repulsive force between two electrons have a magnitude of 2.00 \(\mathrm{N}\) ? Between two protons?

\(\bullet\) A negative charge of \(-0.550 \mu C\) exerts an upward 0.200 \(\mathrm{N}\) force on an unknown charge 0.300 \(\mathrm{m}\) 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\(\mu \mathrm{C}\) charge?

\(\bullet\) (a) What is the total negative charge, in coulombs, of all the electrons in a small 1.00 g sphere of carbon? One mole of C is \(12.0 \mathrm{g},\) and each atom contains 6 protons and 6 electrons. (b) Suppose you could take out all the electrons and hold them in one hand, while in the other hand you hold what is left of the original sphere. If you hold your hands 1.50 \(\mathrm{m}\) apart at arms length, what force will each of them feel? Will it be attractive or repulsive?

\(\bullet\) As you walk across a synthetic-fiber rug on a cold, dry win- ter day, you pick up an excess charge of \(-55 \mu \mathrm{C}\) . (a) How many excess electrons did you pick up? (b) What is the charge on the rug as a result of your walking across it?

\(\bullet$$\bullet\) Two small plastic spheres are given positive electrical charges. When they are 15.0 \(\mathrm{cm}\) apart, the repulsive force between them has magnitude 0.220 \(\mathrm{N} .\) What is the charge on each sphere (a) if the two charges are equal? (b) if one sphere has four times the charge of the other?

\(\bullet$$\bullet\) Two small aluminum spheres, each having mass 0.0250 \(\mathrm{kg}\) , are separated by 80.0 \(\mathrm{cm}\) . (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 \(\mathrm{g} / \mathrm{mol}\) , 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 \(1.00 \times 10^{4} \mathrm{N}\) (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the elec- trons in each sphere does this represent?

\(\bullet$$\bullet\) An average human weighs about 650 \(\mathrm{N}\) . If two such generic humans each carried 1.0 coulomb 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?

\(\bullet$$\bullet\) Three point charges are arranged on a line. Charge \(q_{3}=+5.00 \mathrm{nC}\) and is at the origin. Charge \(q_{2}=-3.00 \mathrm{nC}\) and is at \(x=+4.00 \mathrm{cm} .\) Charge \(q_{1}\) is at \(x=+2.00 \mathrm{cm} .\) What is \(q_{1}\) (magnitude and sign) if the net force on \(q_{3}\) is zero?

\(\bullet$$\bullet\) Two point charges are located on the \(y\) axis as \(\mathrm{fol}\) lows: charge \(q_{1}=-1.50 \mathrm{nC}\) at \(y=-0.600 \mathrm{m},\) and charge \(q_{2}=+3.20 \mathrm{nC}\) at the origin \((y=0) .\) What is the net force (magnitude and direction) exerted by these two charges on a third charge \(a_{3}=+5,00\) nC located at \(y=-0.400 \mathrm{m} ?\)

\(\bullet$$\bullet\) Two point charges are placed on the \(x\) axis as follows: Charge \(q_{1}=+4.00 \mathrm{nC}\) is located at \(x=0.200 \mathrm{m},\) and charge \(q_{2}=+5.00 \mathrm{nC}\) is at \(x=-0.300 \mathrm{m} .\) What are the magnitude and direction of the net force exerted by these two charges on a negative point charge \(q_{3}=-0.600 \mathrm{nC}\) placed at the origin?

\(\bullet$$\bullet\) Two unequal charges repel each other with a force \(F .\) If both charges are doubled in magnitude, what will be the new force in terms of \(F ?\)

\(\bullet$$\bullet$$\bullet\) A charge \(+Q\) is located at the origin and a second charge, \(+4 Q,\) is at distance \(d\) on the \(x\) -axis. Where should a third charge, \(q,\) be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

\(\bullet\) A small object carrying a charge of \(-8.00 \mathrm{nC}\) is acted upon by a downward force of 20.0 \(\mathrm{nN}\) when placed at a certain point in an electric field. (a) What are the magnitude and direction of the electric field at the point in question? (b) What would be the magnitude and direction of the force acting on a proton placed at this same point in the electric field'?

\(\bullet\) (a) What must the charge (sign and magnitude) of a 1.45 \(\mathrm{g}\) particle be for it to remain balanced against gravity when placed in a downward-directed electric field of magnitude 650 \(\mathrm{N} / \mathrm{C}^{?}\) (b) What is the magnitude of an electric field in which the electric force it exerts on a proton is equal in magni- tude to the proton's weight?

\(\bullet\) A uniform electric field exists in the region between two oppositely charged plane parallel plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, 3.20 \(\mathrm{cm}\) distant from the first, in a time interval of \(1.5 \times 10^{-8} \mathrm{s}\) (a) Find the magnitude of this electric field. (b) Find the speed of the elec- tron when it strikes the second plate.

\(\bullet\) A particle has a charge of \(-3.00 \mathrm{nC}\) . (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 the parti- cle does its electric field have a magnitude of 12.0 \(\mathrm{N} / \mathrm{C}\) ?

\(\bullet\) The electric field caused by a certain point charge has a mag- nitude of \(6.50 \times 10^{3} \mathrm{N} / \mathrm{C}\) at a distance of 0.100 \(\mathrm{m}\) from the charge. What is the magnitude of the charge?

\(\bullet\) Electric fields in the atom. (a) Within the nucleus. What strength of electric field does a proton produce at the distance of another proton, about \(5.0 \times 10^{-15} \mathrm{m}\) away? (b) At the elec- trons. What strength of electric field does this proton produce at the distance of the electrons, approximately \(5.0 \times 10^{-10} \mathrm{m}\) away?

\(\bullet\) A proton is traveling horizontally to the right at \(4.50 \times 10^{6} \mathrm{m} / \mathrm{s}\) (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of 3.20 \(\mathrm{cm} .\) (b) How much time does it take the proton to stop after entering the field? (c) What mini- mum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

\(\bullet\) . Electric field of axons. A nerve signal is transmitted through a neuron when an excess of \(\mathrm{Na}^{+}\) ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately 10.0\(\mu \mathrm{m}\) in diameter, and meas- urements show that about \(5.6 \times 10^{11} \mathrm{Na}^{+}\) ions per meter (each of charge \(+e )\) enter during this process. Although the axon is a long cylinder, the charge does not all enter every- where at the same time. A plausible model would be a a series of nearly point charges moving along the axon. Let us look at 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 \(\mathrm{mm}\) 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 \(\mathrm{cm}\) below the skin? (c) Certain sharks can respond to electric fields as weak as 1.0\(\mu \mathrm{N} / \mathrm{C}\) . How far from this segment of axon could a shark be and still detect its electric field?

\(\bullet$$\bullet\) A point charge of \(-4.00 \mathrm{nC}\) is at the origin, and a second point charge of \(+6.00 \mathrm{nC}\) is on the \(x\) axis at \(x=0.800 \mathrm{m}\) . Find the magnitude and direction of the electric field at each of the following points on the \(x\) axis: (a) \(x=20.0 \mathrm{cm}\) , (b) \(x=1.20 \mathrm{m},(\mathrm{c}) x=-20.0 \mathrm{cm} .\)

\(\bullet\) In a rectangular coordinate system, a positive point charge \(q=6.00 \mathrm{nC}\) is placed at the point \(x=+0.150 \mathrm{m}, y=0,\) and an identical point charge is placed at \(x=-0.150 \mathrm{m}, y=0 .\) Find the \(x\) and \(y\) components and the magnitude and direction of the electric field at the following points: (a) the origin; (b) \(x=0.300 \mathrm{m}, y=0 ;\) (c) \(x=0.150 \mathrm{m}, y=-0.400 \mathrm{m},\) (d) \(x=0, y=0.200 \mathrm{m}\)

\(\bullet\) \(\bullet\) Two particles having charges of \(+0.500 \mathrm{nC}\) and \(+8.00 \mathrm{nC}\) are separated by a distance of 1.20 \(\mathrm{m}\) . (a) At what point along the line connecting the two charges is the net electric field due to the two charges equal to zero? (b) Where would the net electric field be zero if one of the charges were negative?

\(\bullet\) \(\bullet\) (a) An electron is moving east in a uniform electric field of 1.50 \(\mathrm{N} / \mathrm{C}\) directed to the west. At point \(A\) , the velocity of the electron is \(4.50 \times 10^{5} \mathrm{m} / \mathrm{s}\) toward the east. What is the speed of the electron when it reaches point \(B, 0.375 \mathrm{m}\) 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 \(1.90 \times 10^{4} \mathrm{m} / \mathrm{s},\) east. What is the speed of the proton at point \(B\) ?

\(\bullet\) \(\bullet\) The electric field due to a certain point charge has a magni- tude \(E\) at a distance of 1.0 \(\mathrm{cm}\) from the charge. (a) What will be the magnitude of this field (in terms of \(E\) ) if we move 1.0 \(\mathrm{cm}\) farther away from the charge? (b) What will be the magnitude of the field (in terms of \(E )\) if we move an additional 1.0 \(\mathrm{cm}\) far- ther away than in part (a)?

\(\bullet\) \(\bullet\) Sketch electric field lines in the vicinity of two charges, \(Q\) and \(-4 Q,\) located a small distance apart on the \(x\) -axis.

\(\bullet\) (a) A closed surface encloses a net charge of 2.50\(\mu \mathrm{C}\) . What is the net electric flux through the surface? (b) If the electric flux through a closed surface is determined to be \(1.40 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C},\) how much charge is enclosed by the surface?

\(\bullet\) A point charge 8.00 \(\mathrm{nC}\) is at the center of a cube with sides of length 0.200 \(\mathrm{m}\) What is the electric flux through (a) the surface of the cube, (b) one of the six faces of the cube?

\(\bullet\) A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter \(12.0 \mathrm{cm},\) giving it a charge of \(-15.0 \mu \mathrm{C}\) . Find the electric field (a) just inside the paint layer, (b) just outside the paint layer, and (c) 5.00 \(\mathrm{cm}\) out- side the surface of the paint laver.

\(\bullet\)(a) How many excess elec- trons must be distributed uni- formly within the volume of an isolated plastic sphere 30.0 \(\mathrm{cm}\) in diameter to produce an elec- tric field of 1150 \(\mathrm{N} / \mathrm{C}\) just out- side the surface of the sphere? (b) What is the electric field at a point 10.0 cm outside the surface of the sphere?

\(\bullet$$\bullet\) In a certain region of space, the electric field \(E\) is uniform; i.e., neither its direction nor its magnitude changes in the region. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, there must be no charge in this region. (b) Is the converse true? That is, in a region of space where there is no charge, must \(\vec{E}\) be uniform? Explain.

\(\bullet$$\bullet\) A total charge of magnitude \(Q\) is distributed uniformly within a thick spherical shell of inner radius \(a\) and outer radius b. (a) Use Gauss's law to find the electric field within the cavity \((r \leq a)\) . (b) Use Gauss's law to prove that the electric field outside the shell \((r \geq b)\) is exactly the same as if all the charge were concentrated as a point charge \(Q\) at the center of the sphere. (c) Explain why the result in part (a) for a thick shell is the same as that found in Example 17.10 for a thin shell. A thick shell can be viewed as infinitely many thin shells.)

\(\bullet\) During a violent electrical storm, a car is struck by a falling high-voltage wire that puts an excess charge of \(-850 \mu C\) on the metal car. (a) How much of this charge is on the inner sur- face of the car? (b) How much is on the outer surface?

\(\bullet\) A neutral conductor completely encloses a hole inside of it. You observe that the outer surface of this conductor carries a charge of \(-12 \mu \mathrm{C}\) (a) Can you conclude that there is a charge inside the hole? If so, what is this charge? (b) How much charge is on the inner surface of the conductor?

\(\bullet$$\bullet\) An irregular neutral conductor has a hollow cavity inside of it and is insulated from its surroundings. An excess charge of \(+16 \mathrm{nC}\) is sprayed onto this conductor. (a) Find the charge on the inner and outer surfaces of the conductor. (b) Without touching the conductor, a charge of - 11 \(\mathrm{nC}\) is inserted into the cavity through a small hole in the conductor. Find the charge on the inner and outer surfaces of the conductor in this case.

\(\bullet$$\bullet\) 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 \(x=-\infty\) ?

\(\bullet$$\bullet\) An electron is released from rest in a uniform electric field. The electron accelerates vertically upward, traveling 4.50 \(\mathrm{m}\) in the first 3.00\(\mu\) s after it is released. (a) What are the magnitude and direction of the electric field? (b) Are we justified in ignor- ing the effects of gravity? Justify your answer quantitatively.

\(\bullet$$\bullet\) A charge \(q_{1}=+5.00 \mathrm{nC}\) is placed at the origin of an \(x y-\) coordinate system, and a charge \(q_{2}=-2.00 \mathrm{nC}\) is placed on the positive \(x\) axis at \(x=4.00 \mathrm{cm}\) . (a) If a third charge \(q_{3}=+6.00 \mathrm{nC}\) is now placed at the point \(x=4.00 \mathrm{cm},\) \(y=3.00 \mathrm{cm},\) find the \(x\) and \(y\) components of the total force exerted on this charge by the other two charges. Find the magnitude and direction of this force.

\(\bullet$$\bullet\) A charge of \(-3.00 \mathrm{nC}\) is placed at the origin of an \(x y-\)coordi- nate system, and a charge of 2.00 \(\mathrm{nC}\) is placed on the \(y\) axis at \(y=4.00 \mathrm{cm} .\) (a) If a third charge, of \(5.00 \mathrm{nC},\) is now placed at the point \(x=3.00 \mathrm{cm}, y=4.00 \mathrm{cm},\) find the \(x\) and \(y\) com- ponents of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

\(\bullet$$\bullet\) Point charges of 3.00 \(\mathrm{nC}\) are situated at each of three cor- ners of a square whose side is 0.200 \(\mathrm{m} .\) What are the magni- tude and direction of the resultant force on a point charge of \(-1.00 \mu \mathrm{C}\) if it is placed (a) at the center of the square, (b) at the vacant corner of the square?

\(\bullet$$\bullet\) A small 12.3 g plastic ball is tied to a very light 28.6 \(\mathrm{cm}\) string that is attached to the vertical wall of a room. (See Figure \(17.58 . ) \mathrm{A}\) uniform horizontal electric field exists in this room. When the ball has been given an excess charge of \(-1.11 \mu \mathrm{C},\) you observe that it remains suspended, with the string making an angle of \(17.4^{\circ}\) with the wall. Find the magnitude and direction of the electric field in the room.

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

\(\bullet$$\bullet\) A \(9.60-\mu \mathrm{C}\) point charge is at the center of a cube with sides of length 0.500 \(\mathrm{m}\) . (a) What is the electric flux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were 0.250 m long? Explain.

\(\bullet$$\bullet\) Electrophoresis. Electrophoresis is a process used by biologists to separate dif- ferent biological molecules (such as pro- teins) 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 \(F_{\mathrm{D}}\) propor- tional to the size and speed of the molecule. We can express this relationship as \(F_{11}=K R v,\) where \(R\) is the radius of the molecule (modeled as being spherical), \(v\) 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 \(F=q E .\) (a) Show that when the electric field is adjusted so that the two forces (electrical and vis- cous drag) just balance, the ratio of \(q\) to \(R\) is \(K v / E\) . (b) Show that if we leave the electric field on for a time \(T\) , the distance \(x\) that the molecule moves during that time is \(x=(E T / k)(q / R)\) . (c) Sup- pose you have a sample containing three different biological mole- cules 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 mate- rial 1. Show that the distances migrated by these molecules after the same amount of time are \(x_{2}=2 x_{1}\) and \(x_{3}=3 x_{1} .\) 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 sepa- rated 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. (See Figure 17.60 .) The process can be rather slow, requiring several hours for separa- tions of just a centimeter or so.