Solved: The electric potential a distance r from a point

Problem 112IP IP Referring to Example 20-3 Suppose the charge +q at the origin is replaced with a charge +5q, where q = 4.11 × 10-9 C. The charge -2q is still at x = 1.00 m. (a) Is there a point in the region x<0 where the electric potential passes through zero? (b) Find the location between x = 0 and x = 1.00 m where the electric potential passes through zero, (c) Find the location in the region x > 1.00 m where the electric potential passes through zero.

Problem 1CQ In one region of space the electric potential has a positive constant value. In another region of space the potential has a negative constant value. What can be said about the electric field within each of these two regions of space?

Problem 1P CE An electron is released from rest in a region of space with nonzero electric field. As the electron moves, does it experience an increasing or decreasing electric potential? Explain.

Problem 2P A uniform electric field of magnitude 4.1 × 105 N/C points in the positive x direction. Find the change in electric potential energy of a 4.5-?C charge as it moves from the origin to the points (a) (0, 6.0 m); (b) (6.0 m, 0); and (c) (6.0 m, 6.0 m).

Problem 2CQ Two like charges a distance r apart have a positive electric potential energy. Conversely, two unlike charges a distance r apart have a negative electric potential energy. Explain the physical significance of these observations.

Problem 3P A uniform electric field of magnitude 6.8 × 105 N/C points in the positive x direction. Find the change in electric potential between the origin and the points (a) (0, 6.0 m); (b) (6.0 m, 0); and (c) (6.0 m, 6.0 m).

Problem 3CQ If the electric field is zero in some region of space is the electric potential zero there as well? Explain.

Sketch the equipotential surface that goes through point 1 in Figure 20–19. Repeat for point 2 and for point 3.

Problem 4P BIO Electric Potential Across a Cell Membrane In a typical living cell, the electric potential inside the cell is 0.070 V lower than the electric potential outside the cell. The thickness of the cell membrane is 0.10 ?m. What are the magnitude and direction of the electric field within the cell membrane?

Problem 5CQ How much work is required to move a charge from one location on an equipotential to another point on the same equipo­tential?Explain.

Problem 5P A computer monitor accelerates electrons and directs them to the screen in order to create an image. If the accelerating plates are 1.05 cm apart, and have a potential difference of 25,500 V, what is the magnitude of the uniform electric field between them?

Problem 6CQ It is known that the electric potential is constant on a given two- dimensional surface. What can be said about the electric field on this surface?

Problem 6P Find the change in electric potential energy for an electron that moves from one accelerating plate to the other in the computer monitor described in the previous problem.

Problem 7CQ Explain why equipotenrials are always perpendicular to the electric field.

Problem 7P A parallel-plate capacitor has plates separated by 0.75 mm. If the electric field between the plates has a magnitude of (a) 1.2 × 105 V/m or (b) 2.4 × 104N/C, what is the potential difference between the plates?

Problem 8CQ Two charges are at locations that have the same value of the electric potential. Is the electric potential energy the same for these charges? Explain.

Problem 8P When an ion accelerates through a potential difference of 2140 V, its electric potential energy decreases by 1.37 × 10-15 J. What is the charge on the ion?

Problem 9CQ A capacitor is connected to a battery and fully charged. What becomes of the charge on the capacitor when it is disconnected from the battery? What becomes of the charge when the two terminals of the capacitor are connected to one another?

Problem 9P The Electric Potential of the Earth The Earth has a vertical electric field with a magnitude of approximately 100 V/m near its surface. What is the magnitude of the potential difference between a point on the ground and a point on the same level as the top of the Washington Monument (555 ft high)?

Problem 10CQ It would be unwise to unplug a television set, take off the back, and reach inside. The reason for the danger is that if you happen to touch the terminals of a high-voltage capacitor you could receive a large electrical shock—even though the set is unplugged. Why?

Problem 10P A uniform electric field with a magnitude of 6350 N/C points in the positive x direction. Find the change in electric potential energy when a +12.5-?C charge is moved 5.50 cm in (a) the positive x direction, (b) the negative x direction, and (c) the positive y direction.

Problem 11CQ On which of the following quantities does the capacitance of a capacitor depend: (a) the charge on the plates; (b) the separation of the plates; (c) the voltage difference between the plates; (d) the electric field between the plates; or (e) the area of the plates?

Problem 11P IP A spark plug in a car has electrodes separated by a gap of 0.025 in. To create a spark and ignite the air-fuel mixture in the engine, an electric field of 3.0 × 106 V/m is required in the gap. (a) What potential difference must be applied to the spark plug to initiate a spark? (b) If the separation between electrodes is increased, docs the required potential difference increase, decrease, or stay the same? Explain, (c) Find the potential difference for a separation of 0.050 in.

Problem 12CQ We say that a capacitor stores charge, yet the total charge in a capacitor is zero; that is, Q + (?Q) = 0. In what sense does a capacitor store charge if the net charge within it is zero?

A uniform electric field with a magnitude of 1200 N/C points in the negative x direction, as shown in Figure 20–20. (a) What is the difference in electric potential, \(Delta V=V_{B}-V_{A}\), between points A and B? (b) What is the difference in electric potential, \(\Delta V=V_{B}-V_{C}\), between points B and C? (c) What is the difference in electric potential, \(\Delta V=V_{C}-V_{A}\), between points C and A? (d) From the information given in this problem, is it possible to determine the value of the electric potential at point A? If so, determine \(V_{A}\); if not, explain why. Equation Transcription: Text Transcription: \Delta V=V_{B}-V_{A} \Delta V=V_{B}-V_{C} \Delta V=V_{C}-V_{A} V_{A}

Problem 13CQ The plates of a particular parallel-plate capacitor are uncharged. Is the capacitance of this capacitor zero? Explain.

Problem 13P A Charged Battery A typical 12-V car battery can deliver 7.5 × 105 C of charge. If the energy supplied by the battery could be converted entirely to kinetic energy, what speed would it give to a 1400-kg car that is initially at rest?

Problem 14P IP BIO The Sodium Pump Living cells actively "pump" positive sodium ions (Na+) from inside the cell to outside the cell. This process is referred to as pumping because work must be done on the ions to move them from the negatively charged inner surface of the membrane to the positively charged outer surface. Given that the electric potential is 0.070 V higher outside the cell than inside the cell, and that the cell membrane is 0.10 ?m thick, (a) calculate the work that must be done (in joules) to move one sodium ion from inside the cell to outside. (b) If the thickness of the cell membrane is increased, does your answer to part (a) increase, decrease, or stay the same? Explain. (It is estimated that as much as 20% of the energy we consume in a resting state is used in operating this "sodium pump.")

The electric potential of a system as a function of position along the x axis is given in Figure 20–21. (a) In which of the regions, 1, 2, 3, or 4, do you expect \(E_{x}\) to be greatest? In which region does \(E_{x}\) have its greatest magnitude? Explain. (b) Calculate the value of \(E_{x}\) in each of the regions, 1, 2, 3, and 4. Equation Transcription: Text Transcription: Ex Ex Ex

Problem 16P Points A and B have electric potentials of 332 V and 149 V, respectively. When an electron released from rest at point A arrives at point C, its kinetic energy is KA. When the electron is released from rest at point B, however, its kinetic energy when it reaches point C is KB = 2KA. What are (a) the electric potential at point C and (b) the kinetic energy KA?

Problem 17P CE Predict/Explain An electron is released from rest in a region of space with a nonzero electric field, (a) As the electron moves, does the electric potential energy of the system increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. Because the electron has a negative charge its electric potential energy doesn't decrease, as one might expect, but increases instead. II. As the electron begins to move, its kinetic energy increases. The increase in kinetic energy is equal to the decrease in the electric potential energy of the system. III. The electron will move perpendicular to the electric field, and hence its electric potential energy will remain the same.

Problem 18P Calculate the speed of (a) a proton and (b) an electron after each particle accelerates from rest through a potential difference of 275 V.

Problem 19P The electrons in a TV picture tube are accelerated from rest through a potential difference of 25 kV. What is the speed of the electrons after they have been accelerated by this potential difference?

Problem 20P Find the potential difference required to accelerate protons from rest to 10% of the speed of light. (At this point, relativistic effects start to become significant.)

A particle with a mass of 3.8 g and a charge of \(+0.045 \mu C\) is released from rest at point A in Figure 20–20. (a) In which direction will this charge move? (b) What speed will it have after moving through a distance of 5.0 cm? The electric field has a magnitude of 1200 N/C. (c) Suppose the particle continues moving for another 5.0 cm. Will its increase in speed for the second 5.0 cm be greater than, less than, or equal to its increase in speed in the first 5.0 cm? Explain. Equation Transcription: Text Transcription: +0.045 \mu C

A proton has an initial speed of \(4.0\times10^5\mathrm{\ m}/\mathrm{s}\) . (a) What potential difference is required to bring the proton to rest? (b) What potential difference is required to reduce the initial speed of the proton by a factor of 2? (c) What potential difference is required to reduce the initial kinetic energy of the proton by a factor of 2?

In Figure 20–22, it is given that \(q_{1}=+Q\). (a) What value must \(q_{2}\) have if the electric potential at point A is to be zero? (b) With The value for \(q_{2}\) found in part (a), is the electric potential at point B positive, negative, or zero? Explain. Equation Transcription: Text Transcription: q1=+Q q2 q2

Problem 24P CE The charge q1 in Figure 20-22 has the value +Q. (a) What value must q2 have if the electric potential at point B is to be zero? (b) With the value for q2 found in part (a), is the electric potential at point A positive, negative, or zero? Explain.

Problem 25P CE It is given that the electric potential is zero at the center of the square in Figure 20-22. (a) If q1 = +Q, what is the value of the charge q2? (b) Is the electric potential at point A positive, negative, or zero? Explain, (c) Is the electric potential at point B positive, negative, or zero? Explain.

Problem 26P The electric potential 1.1 m from a point charge q is 2.8 × 104 V. What is the value of q?

Problem 27P A point charge of -7.2 ?C is at the origin. What is the electric- potential at (a) (3.0 m, 0); (b) (?3.0 m, 0); and (c) (3.0 m,-3.0 m)

Problem 28P The Bohr Atom The hydrogen atom consists of one electron and one proton, hi the Bohr model of the hydrogen atom the electron orbits the proton in a circular orbit of radius 0.529 × 10-10 m. What is the electric potential due to the proton at the electron's orbit?

Problem 29P How far must the point charges q1 = +7.22 ?C and q2 = ?26.1 ?C be separated for the electric potential energy of the system to be ?126 J?

Four different arrangements of point charges are shown in Figure 20–23. In each case the charges are the same distance from the origin. Rank the four arrangements in order of increasing electric potential at the origin, taking the potential at infinity to be zero. Indicate ties where appropriate.

Problem 31P IP Point charges +4.1 ?C and ?2.2?C are placed on the x axis at (11 m, 0) and (?11 m, 0), respectively, (a) Sketch the electric potential on the x axis for this system, (b) Your sketch should show one point on the x axis between the two charges where the potential vanishes. Is this point closer to the +4.1-??C charge or closer to the ?2.2-?C charge? Explain, (c) Find the point referred to in part (b).

Problem 32P IP (a) In the previous problem, find the point to the left of the negative charge where the electric potential vanishes, (b) Is the electric field at the point found in part (a) positive, negative, or zero? Explain.

Problem 33P A dipole is formed by point charges +3.6 ?C and ?3.6 ?C placed on the x axis at (0.25 m, 0) and (-0.25 m, 0), respectively. (a) Sketch the electric potential on the x axis for this system. (b) At what positions on the x axis does the potential have the value 7.5 × 105 V?

Problem 34P A charge of 3.05 ?C is held fixed at the origin. A second charge of 3.05 ?C is released from rest at the position (1.25 m, 0.570 m). (a) Tf the mass of the second charge is 2.16 g, what is its speed when it moves infinitely far from the origin? (b) At what distance from the origin does the second charge attain half the speed it will have at infinity?

A charge of \(20.2 \mu C\) is held fixed at the origin. (a) If a \(-5.25 \mu C\) charge with a mass of 3.20 g is released from rest at the position (0.925 m, 1.17 m), what is its speed when it is halfway to the origin? (b) Suppose the \(-5.25 \mu C\) charge is released from rest at the point \(x=\frac{1}{2}(0.925 \mathrm{~m}) \text { and } y=\frac{1}{2}(1.17 \mathrm{~m})\). When it is halfway to the origin, is its speed greater than, less than, or equal to the speed found in part (a)? Explain. (c) Find the speed of the charge for the situation described in part (b). Equation Transcription: Text Transcription: 20.2 \mu C -5.25 \mu C -5.25 \mu C x=\frac{1}{2}(0.925 m) and y=\frac{1}{2}(1.17 ~m)

Problem 36P A charge of -2.205 ?C is located at (3.055 m, 4.501 in), and a charge of 1.800 ?C is located at (?2.533 m, 0). (a) Find the electric potential at time origin, (b) There is one point on the line connecting these two charges where the potential is zero. Find this point.

Problem 38P How much work must be done to move the three charges in Figure 20-24 infinitely far from one another?

(a) Find the electric potential at point Pin Figure 20–25. (b) Suppose the three charges shown in Figure 20–25 are held in place. A Fourth charge, with a charge of \(+6.11 \mu C\) and a mass of 4.71 g, is released from rest at point P. What is the speed of the fourth charge when it has moved infinitely far away from the other three charges? Equation Transcription: Text Transcription: +6.11 \mu C ________________

Figure 20–24 shows three charges at the corners of a rectangle. (a) How much work must be done to move the \(+2.7-\mu C\) charge to infinity? (b) Suppose, instead, that we move the \(-6.1-\mu C\) charge to infinity. Is the work required in this case greater than, less than, or the same as when we moved the \(+2.7-\mu C\) charge to infinity? Explain. (c) Calculate the work needed to move the \(-6.1-\mu C\) charge to infinity. Equation Transcription: Text Transcription: +2.7-\mu C -6.1-\mu C +2.7-\mu C -6.1-\mu C

Problem 40P A square of side a has a charge +Q at each corner. What is the electric potential energy of this system of charges?

A square of side \(\alpha\) has charges \(+Q \text { and }-Q\) alternating from one corner to the next, as shown in Figure 20–26. Find the electric potential energy for this system of charges. Equation Transcription: Text Transcription: \alpha +Q and -Q

(a) Is the electric potential at point 1 in Figure 20–19 greater than, less than, or equal to the electric potential at point 3? (b) Choose the best explanation from among the following: 1. The electric field lines point to the right, indicating that the electric potential is greater at point 3 than at point 1. 2. The value of the electric potential is large where the electric field lines are close together, and small where they are widely spaced. Therefore, the electric potential is the same at points 1 and 3. 3. The electric potential decreases as we move in the direction of the electric field, as shown in Figure 20–3. Therefore, the electric potential is greater at point 1 than at point 3.

Problem 42P CE Predict/Explain A positive charge is moved from one location on an equipotential to another point on the same equipotential. (a) Is the work done on the charge positive, negative, or zero? (b) Choose the best explanation from among the following: I. The electric field is perpendicular to an equipotential, therefore the work done in moving along an equipotential is zero. I. Because the charge is positive the work done on it is also positive. II. It takes negative work to keep the positive charge from accelerating as it moves along the equipotential.

Imagine sketching a large number of equipotential surfaces in Figure 20–19, with a constant difference in electric potential between adjacent surfaces. (a) Would the equipotentials at point 2 be more closely spaced, be less closely spaced, or have the same spacing as equipotentials at point 1? (b) Choose the best explanation from among the following: 1. When electric field lines are close together, the correspond-ing equipotentials are far apart. 2. Equipotential surfaces, by definition, always have equal spacing between them. 3. The electric field is more intense at point 2 than at point 1,which means the equipotential surfaces are more closely spaced in that region.

Problem 45P Two point charges are on the x axis. Charge 1 is +q and is located at x = ?1.0 m; charge 2 is ?2q and is located at x = 1.0 m. Make sketches of the equipotential surfaces for this system (a) out to a distance of about 2.0 m from the origin and (b) far from the origin, In each case, indicate the direction in· which the potential increases.

Problem 46P Two point charges are on the x axis. Charge 1 is +q and is located at x = ?1.0 m; charge 2 is +2q and is located at x ? 1.0 m. Make sketches of the equipotential surfaces for this system (a) out to a distance of about 2.0 m from the origin and (b) far from the origin. In each case, indicate the direction in which the potential increases.

Figure 20–27 shows a series of equipotentials in a particular region of space, and five different paths along which an electron is moved. (a) Does the electric field in this region point to the right, to the left, up, or down? Explain. (b) For each path, indicate whether the work done on the electron by the electric field is positive, negative, or zero. (c) Rank the paths in order of increasing amount of work done on the electron by the electric field. Indicate ties where appropriate. (d) Is the electric field near path A greater than, less than, or equal to the electric field near path E? Explain.

Problem 48P IP Consider a region in space where a uniform electric field E = 6500 N/C points in the negative x direction, (a) What is the orientation of the equipotential surfaces? Explain, (b) If you move in the positive x direction, does the electric potential increase or decrease? Explain, (c) What is the distance between the +14-V and the +16-V equipotentials?

A given system has the equipotential surfaces shown in Figure 20–28. (a) What are the magnitude and direction of the electric field? (b) What is the shortest distance one can move to undergo a change in potential of 5.00 V?

Problem 50P A 0.40??F capacitor is connected to a 9.0-V battery. How much charge is on each plate of the capacitor?

Problem 51P It is desired that 5.8 ?C of charge be stored on each plate of a 3.2-? F capacitor. What potential difference is required between the plates?

Problem 52P To operate a given flash lamp requires a charge of 32 ?C. What capacitance is needed to store this much charge in a capacitor with a potential difference between its plates of 9.0 V?

Problem 53P A parallel-plate capacitor is made from two aluminum-foil sheets, each 6.3 cm wide and 5.4 m long. Between the sheets is a Teflon strip of the same width and length that is 0.035 mm thick. What is the capacitance of this capacitor? (The dielectric constant of Teflon is 2.1.)

Problem 54P parallel-plate capacitor is constructed with circular plates of radius 0.056 m. The plates are separated by 0.25 mm, and the space between the plates is filled with a dielectric with dielectric constant ?. When the charge on the capacitor is 1.2 ?C the potential difference between the plates is 750 V. Find the value of the dielectric constant, ?.

Problem 56P IP Consider a parallel-plate capacitor constructed from two circular metal plates of radius R. The plates arc separated by a distance of 1.5 mm. (a) What radius must the plates have if the capacitance of this capacitor is to be 1.0 ?F? (b) If the separation between the plates is increased, should the radius of the plates be increased or decreased to maintain a capacitance of 1.0 ?F? Explain, (c) Find the radius of the plates that gives a capacitance of 1.0 ?F for a plate separation of 3.0 mm.

Problem 55P IP A parallel-plate capacitor has plates with an area of 0.012 m2 and a separation of 0.88 mm. The space between the plates is filled with a dielectric whose dielectric constant is 2.0. (a) What is the potential difference between the plates when the charge on the capacitor plates is 4.7 ?C? (b) Will your answer to part (a) increase, decrease, or stay the same if the dielectric constant is increased? Explain, (c) Calculate the potential difference for the case where the dielectric constant is 4.0.

Problem 57P A parallel-plate capacitor has plates of area 3.45 × 10-4 m2. What plate separation is required if the capacitance is to be 1630 pF? Assume that the space between the plates is filled with (a) air or (b) paper.

Problem 58P IP A parallel-plate capacitor filled with air has plates of are 0.0066 m2 and a separation of 0.45 mm. (a) Find the magnitude of the charge on each plate when the capacitor is connected to a 12-V battery, (b) Will your answer to part (a) increase, decrease or stay the same if the separation between the plates is increased? Explain, (c) Calculate the magnitude of the charge on the plates if the separation is 0.90 mm.

Problem 59P Suppose that after walking across a carpeted floor you reach for a doorknob and just before you touch it a spark jumps 0.50 cm from your finger to the knob. Find the minimum voltage needed between your finger and the doorknob to generate this spark.

Problem 60P (a) What plate area is required if an air-filled, parallel-plate capacitor with a plate separation of 2.6 mm is to have a capacitance of 22 pF? (b) What is the maximum voltage that can be applied to this capacitor without causing dielectric breakdown?

Problem 61P Lightning As a crude model for lightning, consider the ground to be one plate of a parallel-plate capacitor and a cloud at an altitude of 550 m to be the other plate. Assume the surface area of the cloud to be the same as the area of a square that is 0.50 km on a side, (a) What is the capacitance of this capacitor? (b) How much charge can the cloud hold before the dielectric strength of the air is exceeded and a spark (lightning) results?

Problem 62P A parallel-plate capacitor is made from two aluminum-foil sheets, each 3.00 cm wide and 10.0 m long. Between the sheets is a mica strip of the same width and length that is 0.0225 mm thick. What is the maximum charge that can be stored in this capacitor? (The dielectric constant of mica is 5.4, and its dielectric strength is 1.00 x 108 V/m.)

Problem 63P Calculate the work done by a 3.0-V battery as it charges a 7.8-?F capacitor in the flash unit of a camera.

Problem 64P BIO Defibrillator An automatic external defibrillator (AED) delivers 125 J of energy at a voltage of 1050 V. What is the capacitance of this device?

Problem 65P IP BIO Cell Membranes The membrane of a living cell can be approximated by a parallel-plate capacitor with plates of area 4.75 × 10-9 m2, a plate separation of 8.5 × 10-9 m, and a dielectric with a dielectric constant of 4.5. (a) What is the energy stored in such a cell membrane if the potential difference across it is 0.0725 V? (b) Would your answer to part (a) increase, de­crease, or stay the same if the thickness of the cell membrane is increased? Explain.

Problem 66P A 0.22-?F capacitor is charged by a 1.5-V battery. After being charged, the capacitor is connected to a small electric motor. Assuming 100% efficiency, (a) to what height can the motor lift a 5.0-g mass? (b) What initial voltage must the capacitor have if it is to lift a 5.0-g mass through a height of 1.0 cm?

Problem 67P Find the electric energy density between the plates of a 225-?F parallel-plate capacitor. The potential difference between the plates is 345 V, and the plate separation is 0.223 mm.

Problem 68P What electric field strength would store 17.5 Jof energy in every 1.00 mm3 of space?

Problem 69P An electronic flash unit for a camera contains a capacitor with a capacitance of 890 ?F. When the unit is fully charged and ready for operation, the potential difference between the capacitor plates is 330 V. (a) What is the magnitude of the charge on each plate of the fully charged capacitor? (b) Find the energy stored in the "charged-up" flash unit.

Problem 70P A parallel-plate capacitor has plates with an area of 405 cm2 and an air-filled gap between the plates that is 2.25 mm thick. The capacitor is charged by a battery to 575 V and then is disconnected from the battery, (a) How much energy is stored in the capacitor? (b) The separation between the plates is now increased to 4.50 mm. How much energy is stored in the capacitor now? (c) How much work is required to increase the separation of the plates from 2.25 mm to 4.50 mm? Explain your reasoning.

Problem 71GP CE A proton is released from rest in a region of space with a nonzero electric field. As the proton moves, does it experience an increasing or decreasing electric potential? Explain.

Problem 72GP CE Predict/Explain A proton is released from rest in a region of space with a nonzero electric field, (a) As the proton moves, does the electric potential energy of the system increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. As the proton begins to move, its kinetic energy increases. The increase in kinetic energy is equal to the decrease in the electric potential energy of the system. II. Because the proton has a positive charge, its electric potential energy will always increase. III. The proton will move perpendicular to the electric field, and hence its electric potential energy will remain the same.

Problem 73GP CE In the Bohr model of the hydrogen atom, a proton and an electron are separated by a constant distance r. (a) Would the electric potential energy of the system increase, decrease, or stay the same if the electron is replaced with a proton? Explain, (b) Suppose, instead, that the proton is replaced with an electron. Would the electric potential energy of the system increase, decrease, or stay the same? Explain.

Problem 74GP CE The plates of a parallel-plate capacitor have constant charges of +Q and-Q. Do the following quantities increase, decrease, or remain the same as the separation of the plates is increased? (a) The electric field between the plates; (b) the potential difference between the plates; (c) the capacitance; (d) the energy stored in the capacitor.

Problem 75GP CE A parallel-plate capacitor is connected to a battery that maintains a constant potential difference V between the plates. If the plates of the capacitor are pulled farther apart, do the following quantities increase, decrease, or remain the same? (a) The electric field between the plates; (b) the charge on the plates; (c) the capacitance; (d) the energy stored in the capacitor.

Problem 76GP CE The plates of a parallel-plate capacitor have constant charges of +Q and ?Q. Do the following quantities increase, decrease, or remain the same as a dielectric is inserted between the plates? (a) The electric field between the plates; (b) the po­tential difference between the plates; (c) the capacitance; (d) the energy stored in the capacitor.

Problem 78GP Find the difference in electric potential, ?V = VB — VA between the points A and B for the following cases: (a) The electric field does 0.052 J of work as you move a +5.7-?C charge from A to B. (b) The electric field does -0.052 J of work as you move a ?5.7-?C charge from A to B. (c) You perform 0.052 J of work as you slowly move a +5.7-?C charge from A to B.

Problem 77GP CE A parallel-plate capacitor is connected to a battery that maintains a constant potential difference V between the plates. If a dielectric is inserted between the plates of the capacitor, do the following quantities increase, decrease, or remain the same? (a) The electric field between,the plates; (b) the charge on the plates; (c) the capacitance; (d) the energy stored in the capacitor.

Problem 79GP The separation between the plates of a parallel-plate capacitor is doubled and the area of the plates is halved. How is the capacitance affected?

Problem 80GP A parallel-plate capacitor is connected to a battery that maintains a constant potential difference between the plates. If the spacing between the plates is doubled, how is the magnitude of charge on the plates affected?

Problem 81GP CE Two point charges arc placed on the x axis. The charge +2q is at x = 1.5 m, and the charge —q is at x = ?1.5 m. (a) There is a point on the x axis between the two charges where the electric potential is zero. Where is this point? (b) The electric potential also vanishes at a point in one of the following regions: region 1, x between 1.5 m and 5.0 m; region 2, x between ?1.5 m and ?3.0 m; region 3, x between ?3.5 m and ?5.0 m. Identify the appropriate region, (c) Find the value of x referred to in part (b).

Problem 82GP A charge of 24.5 ?C is located at (4.40 m, 6.22 m), and a charge of ?11.2 ?C is located at (?4.50 m, 6.75 m). What charge must be located at (2.23 m, ?3.31 m) if the electric potential is to be zero at the origin?

Problem 83GP The Bohr Model In the Bohr model of the hydrogen atom (see Problem 28) what is the smallest amount of work that must be done on the electron to move it from its circular orbit, with a radius of 0.529 × 10?10 m, to an infinite distance from the proton? This value is referred to as the ionization energy of hydrogen.

Problem 85GP Repeat Problem 84 for the case where both charges arc +1.2 ?C.

A \(+1.2 \mu C\) charge and a \(-1.2 \mu C\) charge are placed at (0.50 m, 0) and (-0.50 m, 0), respectively. (a) In Figure 20–29, at which of the points A, B, C, or D is the electric potential smallest in value? At which of these points does it have its greatest value? Explain. (b) Calculate the electric potential at points A,B, C, and D. Equation Transcription: Text Transcription: +1.2 \mu C -1.2 \mu C

Problem 86GP How much work is required to bring three protons, initially infinitely far apart, to a configuration where each proton is 1.5 × 10-15 m from the other two? (This is a typical separation for protons in a nucleus.) ·

Problem 87GP A point charge Q = +87.1 ?C is held fixed at the origin. A second point charge, with mass m = 0.0576 kg and charge q = ?2.87 ?C, is placed at the location (0.323 m, 0). (a) Find the electric potential energy of this system of charges, (b) If the second charge is released from rest, what is its speed when it reaches the point (0.121 m, 0)?

Problem 88GP Electron Escape Speed An electron is at rest just above the surface of a sphere with a radius of 2.7 mm and a uniformly distributed positive charge of 1.8 × 10-15 C. Like a rocket blasting off from the Earth, the electron is given an initial speed vc radially outward from the sphere. If the electron coasts to infinity, where its kinetic energy drops to zero, what is the escape speed, ve?

The three charges shown in Figure 20–25are held in place as a fourth charge, q, is brought from infinity to the point P. The Charge q starts at rest at infinity and is also at rest when it is placed at the point P. (a) If q is a positive charge, is the work required to bring it to the point P positive, negative, or zero? Explain. (b) Find the value of q if the work needed to bring it to point \(P \text { is }-1.3 \times 10^{-11} J\). Equation Transcription: Text Transcription: P is -1.3 \times 10^{-11} J

Problem 90GP A parallel-plate capacitor is charged to an electric potential of 325 V by moving 3.75 × 1016 electrons from one plate to the other. How much work is done in charging the capacitor?

(a) In Figure 20–28 we see that the electric potential increases by \(10.0 \mathrm{~V}\) as one moves \(4.00 \mathrm{~cm}\) in the positive x direction. Use This information to calculate the x component of the electric field. (Ignore the y direction for the moment.) (b) Apply the same reasoning as in part (a) to calculate the y component of the electric field. (c) Combine the results from parts (a) and (b) to find the magnitude and direction of the electric field for this system. Equation Transcription: Text Transcription: 10.0 V 4.00 cm

Problem 89GP Quark Model of the Neutron According to the quark model of fundamental particles, neutrons—the neutral particles in an atom's nucleus—are composed of three quarks. Two of these quarks are "down" quarks, each with a charge of —e/3; the third quark is an "up" quark, with a charge of +2e/3. This gives the neutron a net charge of zero. What is the electric potential energy of these three quarks, assuming they are equidistant from one another, with a separation distance of 1.3 × 10-15 m? (Quarks are discussed in Chapter 32.)

Problem 93GP IP BIO Electric Catfish The electric catfish (Malapterurs electricus) is an aggressive fish, 1.0 m in length, found today in tropical Africa (and depicted in Egyptian hieroglyphics). The catfish is capable of generating jolts of electricity up to 350 V by producing a positively charged region of muscle near the head and a negatively charged region near the tail, (a) For the same amount of charge, can the catfish generate a higher voltage by separating the charge from one end of its body to the other, as it does, or from one side of the body to the other? Explain, (b) Estimate the charge generated at each end of a catfish as follows: Treat the catfish as a parallel-plate capacitor with plates of area 1.8 × 10-2 m2, separation 1.0 m, and filled with a dielectric with a dielectric constant K = 95.

As a \(+6.2 \mu C\) charge moves along the x axis from \(x=0 \text { to } x=0.70 \mathrm{~m}\), the electric potential it experiences is shown in Figure 20–21. Find the approximate location(s) of the charge when its electric potential energy is (a) \(2.6 \times 10^{-5} J\) and (b) \(4.3 X 10^{-5} J\). Equation Transcription: Text Transcription: +6.2 \mu C x=0 to x=0.70 m 2.6 X 10-5 J 4.3 X 10-5 J ________________

Problem 95GP IP Computer Keyboards Many computer keyboards operate on the principle of capacitance. As shown in Figure 20-16, each key forms a small parallel-plate capacitor whose separation is reduced when the key is depressed, (a) Does depressing a key increase or decrease its capacitance? Explain, (b) Suppose the plates for each key have an area of 47.5 mm2,and an initial separation of 0.550 mm. In addition, let the dielectric have a dielectric constant of 3.75. If the circuitry of the computer can de­tect a change in capacitance of 0.425 pF, what is the minimum distance a key must be depressed to be detected?

Problem 97GP BIO Cell Membranes and Dielectrics Many cells in the body have a cell membrane whose inner and outer surfaces carry opposite charges, just like the plates of a parallel-plate ca­pacitor. Suppose a typical cell membrane has a thickness of 8.1 × 10-9 m, and its inner and outer surfaces carry charge densities of ?0.58 × 10 3 C/m2 and +0.58 × 10-3 C/m2, respectively. In addition, assume that the material in the cell membrane has a dielectric constant of 5.5. (a) Find the direction and magnitude of the electric field within the cell membrane, (b) Calculate the potential difference between the inner and outer walls of the membrane, and indicate which wall of the membrane has the higher potential.

A point charge of mass 0.081 kg and charge \(+6.77 \mu C\) is suspended by a thread between the vertical parallel plates of a parallel-plate capacitor, as shown in Figure 20–30. (a) If the charge deflects to the right of vertical, as indicated in the figure, which of the two plates is at the higher electric potential? (b) If the angle of deflection is \(22^{\circ}\), and the separation between the plates is 0.025 m, what is the potential difference between the plates? Equation Transcription: Text Transcription: +6.77 \mu C 22^{\circ}

Problem 98GP Long, long ago, on a planet far, far away, a physics experiment was carried out. First, a 0.250-kg ball with zero net charge was dropped from rest at a height of 1.00 m. The ball landed 0.552 s later. Next, the ball was given a net charge of 7.75 ?C and dropped in the same way from the same height. This time the ball fell for 0.680 s before landing. What is the electric potential at a height of 1.00 m above the ground on this planet, given that the electric potential at ground level is zero? (Air resistance can be ignored.)

Problem 99GP Rutherford's Planetary Model of the Atom In 1911, Ernest Rutherford developed a planetary model of the atom, in which a small positively charged nucleus is orbited by electrons. The model was motivated by an experiment carried out by Rutherford and his graduate students, Geiger and Marsden. In tIn s experiment, they fired alpha particles with an initial speed of 1.75 × 107 m/s at a tIn n sheet of gold. (Alpha particles are obtained from certain radioactive decays. They have a charge of +2e and a mass of 6.64 × 10-27 kg.) Flow close can the alpha particles get to a gold nucleus (charge = +79e), assuming the nucleus remains stationary? (This calculation sets an upper limit on the size of the gold nucleus. See Chapter 31 for further details.)

(a) One of the charges in Figure 20–26 is given an outward “kick” that sends it off with an initial speed \(v_{0}\) while the other three charges are held at rest. If the moving charge has a mass m, what is its speed when it is infinitely far from the other charges? (b) Suppose the remaining \(-Q\) charge, which also has a mass m, is now given the same initial speed, \(v_{0}\). When it is infinitely far away from the two \(+Q\) charges, is its speed greater than, less than, or the same as the speed found in part (a)? Explain. Equation Transcription: Text Transcription: v0 -Q v0 +Q

Problem 101GP Figure 20-30 shows a charge q = +6.77 ?C with a mass m = 0.071 kg suspended by a thread of length L = 0.022 m between the plates of a capacitor, (a) Plot the electric potential energy of the system as a function of the angle ? the thread makes with the vertical. (The electric field between the plates has a magnitude E = 4.16 × 104 V/m.) (b) Repeat part (a) for tine case of the gravitational potential energy of the system, (c) Show that the total potential energy of the system (electric plus gravitational) is a minimum when the angle ? satisfies the equilibrium condition for the charge, tan ? = qE/mg. TIn s relation implies that ? = 22°.

Problem 102GP The electric potential a distance r from a point charge q is 2.70 × 104 V. One meter farther away from the charge the potential is 6140 V. Find the charge q and the initial distance r.

Problem 105GP The electric potential a distance r from a point charge q is 155 V, and the magnitude of the electric field is 2240 N/C. Find the values of q and r.

Problem 106PP Electric eels produce an electric field within their body. In which direction does the electric field point? A. toward the head B. toward the tail C. upward D. downward

Problem 107PP As a rough approximation, consider an electric eel to be a parallel-plate capacitor with plates of area 1.8 × 10-2 m2 separated by 2.0 m and filled with a dielectric whose dielectric constant is ? = 95. What is the capacitance of the eel in this model? A. 8.0 × 10-14 F B. 7.6 × 10-12 F C. 1.5 ×10-11 F D. 9.3 ×10-8 F

Problem 103GP Referring to Problem 84, calculate and plot the electric potential on the circle centered at (0.50 m, 0). Give your results in terms of the angle ×, defined as follows: ? is the angle measured counterclockwise from a vertex at the center of the circle, with ? = 0 at point C.

Problem 108PP In terms of the parallel-plate model of the previous problem, how much charge does an electric eel generate at each end of its body when it produces a voltage of 650 V? A. 1.2 ×10-14C B. 5.2 ×10-11 C C. 4.9×10-9 C D. 6.1 ×10-5C

Problem 104GP When the potential difference between the plates of a capacitor is increased by 3.25 V, the magnitude of the charge on each plate increases by 13.5 ?C. What is the capacitance of this capacitor?

Problem 109PP How much energy is stored by an electric eel when it is charged up to 650 V. Use the same parallel-plate model discussed in the previous two problems. A. 1.8 ×10-17J B. 1.7 ×10-8J C. 1.6×10-6J D. 2.0×10-2J

Problem 110IP IP Referring to Example 20-3 Suppose the charge -2q at x = 1.00 m is replaced with a charge -3q, where q = 4.11 ×10-9 C. The charge +q is at the origin, (a) Is the electric potential positive, negative, or zero at the point x = 0.333 m? Explain. (b) Find the point between x = 0 and x = 1.00 m where the electric potential vanishes, (c) Is there a point in the region x < 0 where the electric potential passes through zero?

Problem 111IP Referring to Example 20-3 Suppose we can change the location of the charge ?2q on the x axis. The charge +q (where q = 4.11 × 10-9C) is still at the origin, (a) Where should the charge ?2q be placed to ensure that the electric potential vanishes at x = 0.500 m? (b) With the location of ?2q found in part (a), where does the electric potential pass through zero in the region x < 0?