Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for University Physics Volume 2
A charge q is placed in the cavity of a conductor as shown below. Will a charge outside the conductor experience an electric field due to the presence of q?The conductor in the preceding figure has an excess charge of 5.0C . If a 2.0C point charge is placed in the cavity, what is the net charge on the surface of the cavity and on the outer surface of the conductor?A uniform electric field of magnitude 1.1104 N/C is perpendicular to a square sheet with sides 2.0 m long. What is the electric flux through the sheet?Calculate the flux through the sheet of the previous problem if the plane of the sheet is at an angle of to the field. Find the flux for both directions of the normal to the sheet.Find the electric flux through a rectangular area 3 cm x 2 cm between parallel plates where there is a constant electric field of 30 N/C for the following orientations of the area: (a) parallel to the plates, (b) perpendicular to the plates, and (c) the normal to the area making a 300 angle with the direction of the electric field. Note that this angle can also be given as 180o + 30o.The electric flux through a square-shaped area of side 5 cm near a large charged sheet is found to be 3105Nm2/C when the area is parallel to the plate. Find the charge density on the sheet.Two large rectangular aluminum plates of area 150 cm2 face each other with a separation of 3 mm between them. The plates are charged with equal amount of opposite charges, 20C . The charges on the plates face each other. Find the flux through a circle of radius 3 cm between the plates when the normal to the circle makes an angle of 5o with a line perpendicular to the plates. Note that this angle can also be given as 180o + 5o.A square surface of area 2 cm2 is in a space of uniform electric field of magnitude 103 N/C . The amount of flux through it depends on how the square is oriented relative to the direction of the electric field. Find the electric flux through the square, when the normal to it makes the following angles with electric field: (a) 30(, (b) 90(, and (c) 0(. Note that these angles can also be given as 1800 + .A vector field is pointed along the z-axis, v=ax2+y2z . (a) Find the flux of the vector field through a rectangle in the xy-plane between axb and cyd. (b) Do the same through a rectangle in the yz-plane between azb and cyd. (Leave your answer as an integral.)Consider the uniform electric field E=(4.0j+3.0k)103 N/C. What is its electric flux through a circular area of radius 2.0 m that lies in the xy-plane?Repeat the previous problem, given that the circular area is (a) in the yz-plane and (b) 45( above the xy-plane.An infinite charged wire with charge per unit length lies along the central axis of a cylindrical surface of radius r and length l. What is the flux through the surface due to the electric field of the charged wire?Determine the electric flux through each surface whose cross-section is shown below.Find the electric flux through the closed surface whose cross-sections are shown below.A point charge q is located at the center of a cube whose sides are of length a. If there are no other charges in this system, what is the electric flux through one face of the cube?A point charge of 10C is at an unspecified location inside a cube of side 2 cm. Find the net electric flux though the surfaces of the cube.A net flux of 1.0104 N ? m2/C passes inward through the surface of a sphere of radius 5 cm (a) How much charge is inside the sphere? (b) How precisely can we determine the location of the charge from this information?A charge q is placed at one of the comers of a cube of side a, as shown below. Find the magnitude of the electric flux through the shaded face due to q. Assume q0.The electric flux through a cubical box 8.0 cm on aside is 1.2103 N m2/C. What is the total charge enclosed by the box?The electric flux through a spherical surface is 4.0104 N m2/C. What is the net charge enclosed by the surface?A cube whose sides are of length d is placed in a uniform electric field of magnitude E=4.0103 N/C so that the field is perpendicular to two opposite faces of the cube. What is the net flux through the cube?Repeat the previous problem, assuming that the electric field is directed along a body diagonal of the cube.A total charge 5.0106 C is distributed uniformly throughout a cubical volume whose edges are 8.0 cm long. (a) What is the charge density in the cube? (b) What is the electric flux through a cube with 12.0-cm edges that is concentric with the charge distribution? (c) Do the same calculation for cubes whose edges are 10.0 cm long and 5.0 cm long. (d) What is the electric flux through a spherical surface of radius 3.0 cm that is also concentric with the charge distribution?Recall that in the example of a uniform charged sphere, p0=Q/(43R3). Rewrite the answers in terms of the total charge Q on the sphere.Suppose that the charge density of the spherical chargedistribution shown in Figure 6.23 is p(r)=p0r/R for rR and zero for rR . Obtain expressions for theelectric field both inside and outside the distribution.A very long, thin wile has a uniform linear charge density of 50C/m . What is the electric field at a distance 2.0 cm from the wire?A charge of 30C is distributed uniformly a spherical volume of radius 10.0 cm. Determine the electric field due to this charge at a distance of (a) 2.0 cm, (b) 5.0 cm, and (c) 20.0 cm from the center of the sphere.Repeat your calculations for the preceding problem, given that the charge is distributed uniformly over the surface of a spherical conductor of radius 10.0 cm.A total charge Q is distributed uniformly throughout a spherical shell of inner and outer radii r1 and r2 , respectively. Show that the electric field due to the charge is E =Q40r2(r3r13r23r13)r (rr1); (r1rr2); E =Q40r2r (rr1).When a charge is placed on a metal sphere, it ends up in equilibrium at the outer surface. Use this information to determine the electric field of +3.0C charge put on a 5.0-cm aluminum spherical ball at the following two points in space: (a) a point 1.0 cm from the center of the ball (an inside point) and (b) a point 10 cm from the center of the ball (an outside point).A large sheet of charge has a uniform charge density of 10C/m2 . What is the electric field due to this charge at a point just above the surface of the sheet?Determine if approximate cylindrical symmetry holds for the following situations. State why or why not. (a) A 300-cm long copper rod of radius 1 cm is charged with +500 nC of charge and we seek electric field at a point 5 cm from the center of the rod. (b) A 10-cm long copper of radius 1 cm is charged with +500 nC of charge and we seek electric field at a point 5 cm from the center of the rod. (c) A 150-cm wooden rod is glued to a 150-cm plastic rod to make a 300 cm long rod, which is then painted with a charged paint so that one obtains a uniform charge density. The radius of each rod is 1 cm, and we seek an electric field at a point that is 4 cm from the center of the rod. (d) Same rod as (c), but we seek electric field at a point that is 500 cm from the center of the rod.A long silver rod of radius 3 cm has a charge of 5C/cm on its surface. (a) Find the electric field at a point 5 cm from the center of the rod (an outside point). (b) Find the electric field at a point 2 cm from the center of the rod (an inside point).ne electric field at 2 cm from the center of long copper rod of radius 1 cm has a magnitude 3 N/C and directed outward from the axis of the rod. (a) How much charge per unit length exists on the rod? (b) What would be the electric flux through a cube of side 5 cm situated such that the rod passes through opposite sides of the cube perpendicularly?A long copper cylindrical shell of inner radius 2 cm and outer radius 3 cm surrounds concentrically a charged long aluminum rod of radius 1 cm with a charge density of 4 pC/m. All charges on the aluminum rod reside at its surface. The inner surface of the copper shell has exactly charge to that of the aluminum rod while the outer surface of the copper shell has the same charge as the aluminum rod. Find the magnitude and direction of the electric field at points that are at the following distances from the center of the aluminum rod: (a) 0.5 cm, (b) 1.5 cm, (c) 2.5 cm, (d) 3.5 cm, and (e) 7 cm.Charge is distributed uniformly with a density p throughout an infinitely long cylindrical volume of radius R. Show that the field of this charge distribution is directed radially with respect to the cylinder and that E=pr20 (rR); E=pR220r (rR).Charge is distributed throughout a very long cylindrical volume of radius R such that the charge density increases with the distance r from the central axis of the cylinder according to p=ar , where a is a constant. Show that the field of this charge distribution is directed radially with respect to the cylinder and that E=ar230 (rR); E=aR330r (rR).The electric field 10.0 cm from the surface of a copper ball of radius 5.0 cm is directed toward the ball's center and has magnitude 4.0102 N/C. How much charge is on the surface of the ball?Charge is distributed throughout a spherical shell of inner radius r1 and outer radius r2 with a volume density given by p=p0r1/r , where p0 is a constant. Determine the electric field due to this charge as a function of r, the distance from the center of the shell.Charge is distributed throughout a spherical volume of radius R with a density p=ar2 , where a is a constant. Determine the electric field due to the charge at pints both inside and outside the sphere.Consider a uranium nucleus to be sphere of radius R=7.41015 m with a charge of 92e distributed uniformly throughout its volume. (a) is the electric force exerted on an electron when it is 3.01015 m from the center of the nucleus? (b) What is the acceleration of the electron at this point?The volume charge density of a spherical charge distribution is given by p(r)=p0ear , where p0 , and a are constants. What is the electric field produced by this charge distribution?An uncharged conductor with an internal cavity is shown in the following figure. Use the closed surface S along with Gauss' law to show that when a charge q is placed in the cavity a total charge q is induced on the inner surface of the conductor. What is the charge on the outer surface of the conductor? Figure 6.46 A charge inside a cavity of a metal. Charges at the outer surface do not depend on how the charges are distributed at the inner surface since E field inside the body of the metal is zero.An uncharged spherical conductor S of radius R has two spherical cavities A and B of radii a and b, respectively as shown below. Two point charges +qa and +qb are placed at the center of the two cavities by using non-conducting supports. In addition, a point charge +q0 is placed outside at a distance r from the center of the sphere. (a) Draw approximate charge distributions in the metal although metal sphere has no net charge. (b) Draw electric field lines. Draw enough lines to represent all distinctly different places.A positive point charge is placed at the angle bisector of two uncharged plane conductors that make an angle of 45o. See below. Draw the electric field lines.A long cylinder of copper of radius 3 cm is charged so that it has a uniform charge per unit length on its surface of 3 C/m. (a) Find the electric field inside and outside the cylinder. (b) Draw electric field lines in a plane perpendicular to the rod.An aluminum spherical ball of radius 4 cm is charged with 5C of charge. A copper spherical shell of inner radius 6 cm and outer radius 8 cm surrounds it. A total charge of 8C is put on the copper shell. (a) Find the electric field at all points in space, including points inside the aluminum and copper shell when copper shell and aluminum sphere are concentric. (b) Find the electric field at all points in space, including points inside the aluminum and copper shell when the centers of copper shell and aluminum sphere are 1 cm apart.A long cylinder of aluminum of radius R meters is charged so that it has a uniform charge per unit length on its surface of . (a) Find the electric field inside and outside the cylinder. (b) Plot electric field as a function of distance from the center of the rod.At the surface of any conductor in electrostatic equilibrium, E=/0. . Show that this equation is consistent with the fact that E=kq/r2 at the surface of a spherical conductor.Two parallel plates 10 cm on a side are given equal and opposite charges of magnitude 5109 C. The plates are 1.5 mm apart. What is the electric field at the center of the region between the plates?Two parallel conducting plates, each of cross-sectional area 400 cm2, are 2.0 cm apart and uncharged. If 1.01012 electrons are transferred from one plate to the other, what are (a) the charge density on each plate? (b) The electric field between the plates?The surface charge density on a long straight metallic pipe is . What is the electric field outside and inside the pipe? Assume the pipe has a diameter of 2a.A point charge q=5.01012 C is placed at the center of a spherical conducting shell of inner radius 3.5 cm and outer radius 4.0 cm. The electric field just above the surface of the conductor is directed radially outward and has magnitude 8.0 N/C. (a) What is the charge density on the inner surface of the shell? (b) What is the charge density on the outer surface of the shell? (c) What is the net charge on the conductor?A solid cylindrical conductor of radius a is surrounded by a concentric cylindrical shell of inner radius b. The solid cylinder and the shell carry charges +Q and Q , respectively. Assuming that the length L of both conductors is much greater than a or b, determine the electric field as a function of r, the distance from the common central axis of the cylinders, for (a) ra; (b) arb; and (c) rb.A vector field E (not necessarily an electric field; note units) is given by E =3x2k. Calculate s E nda, where S is the area shown below. Assume that n=k.Repeat the preceding problem, with E=2xi+3x2k.A circular area S is concentric with the origin, has radius a, and lies in the yz-plane. Calculate s E ndA for E=2z2i.(a) Calculate the electric flux through the open hemispherical surface due to the electric field E =E0k (see below). (b) If the hemisphere is rotated by 90( around the what is the flux through it?Suppose that the electric field of an isolated point charge were proportional to 1/r2+ rather than 1/r2 . Determine the flux that passes through the surface of a sphere of radius R centered at the charge. Would Gauss's law remain valid?The electric field in a region is given by E=al(b+cx)i, where a=200 N ? m/C, b=2.0 m, and c=2.0 . What is the net charge enclosed by the shaded volume shown below?Two equal and opposite charges of magnitude Q are located on the x-axis at the points +a and a , as shown below. What is the net flux due to these charges through a square surface of side 2a that lies in the yz-plane and is centered at the origin? (Hint: Deter-nine the flux due to each charge separately, then use the principle of superposition. You may be able to make a symmetry argument.)A fellow student calculated the flux through the square for the system in the preceding problem and got 0. What went wrong?A 10cm10cm piece of aluminum foil of 0.1 mm thickness has a charge of that spreads on both wide side surfaces evenly. You may ignore the charges on the thin sides of the edges. (a) Find the charge density. (b) Find the electric field 1 cm from the center, assuring approximate planar symmetry.Two 10cm10cm pieces of aluminum foil of thickness 0.1 mm face each other with a separation of 5 mm. One of the foils has a of +30C and the has 30C . (a) Find the charge density at all surfaces, i.e., on those facing each other and those facing away. (b) Find the electric field between the plates near the center assuming planar symmetry.Two large copper plates facing each other have charge densities 4.0C/m2 on the surface facing the other plate, and zero in between the plates. Find the electric flux through a 3cm4cm rectangular area between the plates, as shown below, for the following orientations of the area. (a) If the area is parallel to the plates, and (b) if the area is tilted =30 from the parallel direction. Note, this angle can also be =180+30.The infinite slab between the planes defined by z=a/2 and z=a/2 contains a uniform volume charge density p (see below). What is the electric field produced by this charge distribution, both inside and outside the distribution?A total charge Q is distributed uniformly throughout a spherical volume that is centered at o1 and has a radius R. Without disturbing the charge remaining charge is removed from the spherical volume that is centered at o2 (see below). Show that the electric field everywhere in the empty region is given by E=Qr40R3 where r is the displacement vector directed from o1 to o2 .A non-conducting spherical shell of inner radius a1 and outer radius b1 is uniformly charged with charged density p1 inside another non-conducting spherical shell of inner radius a2 and outer radius b2 that is also uniformly charged with charge density p2 . See below. Find the electric field at space point P at a distance r from the common center such that (a) rb2 (b) a2rb2 , (c) b1ra2 , (d) a1rb1 , and (e) ra1 .Two non-conducting spheres of radii R1 and R2 are uniformly charged with charge densities p1 and p2 , respectively. They are separated at center-to-center distance a (see below). Find the electric field at point P located at a distance r from the center of sphere 1 and is in the direction from the line joining the two spheres assuming their charge densities are not affected by the presence of the other sphere. (Hint: Work one sphere at a time and use the superposition principle.)A disk of radius R is cut in a non-conducting large plate that is uniformly charged with charge density (coulomb per square meter). See below. Find the electric field at a height h above the center of the disk (hR,hlorw) . (Hint: Fill the hole with . )Concentric conducting spherical shells carry charges Q and -Q, respectively (see below). The inner shell has negligible thickness. Determine the electric field for (a) ra; (b) arb; brc; and (d) rc.Shown below ale two concentric conducting spherical shells of radii R1 and R2 , each of finite thickness much less than either radius. The inner and outer shell carry net charges q1 and q2 , respectively, where both q1 and q2 are positive. What is the electric field for (a) rR1 ; (b) R1rR2 , and (c) rR2 ? (d) What is the net charge on the inner surface of the inner shell, the outer surface of the inner shell, the inner surface of the outer shell, and the outer surface of the outer shell?A point charge of q=5.0108 C is placed at the center of an uncharged spherical conducting shell of inner radius 6.0 cm and outer radius 9.0 cm. Find the electric field at (a) r=4.0 cm, (b) r=8.0 cm, and (c) r=12.0 cm. (d) What are charges induced on the inner and outer surfaces of the shell?Re-derive Gauss's law for the gravitational field, with g directed positively outward.An infinite plate sheet of charge of surface charge density is shown below. What is the electric field at a distance x from the sheet? Compare the result of this calculation with that of worked out in the text.A spherical lubber balloon carries a total charge Q distributed uniformly over its surface. At r=0 , the radius of the balloon is R. The balloon is then slowly inflated until its radius reaches 2R at the time t0 . Determine the electric field due to this charge as a function of time (a) at the surface of the balloon, (b) at the surface of radius R, and (c) at the surface of radius 2R. Ignore any effect on the electric field due to the material of the balloon and assume that the radius increases uniformly with time.Find the electric field of a large conducting plate containing a net charge q. Let A be area of one side of the plate and h the thickness of the plate (see below). The charge on the metal plate will distribute mostly on the two planar sides and very little on the edges if the plate is thin.Check Your Understanding If Q has a mass of 4.00 g , what is the speed of Q at r2 ?Check Your Understanding What is the potential energy of Q relative to the zero reference at infinity at r2 in the above example?Check Your Understanding Is the electrical potential energy of two point charges positive or negative if the charges are of the same sign? Opposite signs? How does this relate to the work necessary to bring the charges into proximity from infinity?Check Your Understanding How much energy does a 1.5-V AAA battery have that can move 100C ?Check Your Understanding How many electrons would go through a 24.0-W lamp?Check Your Understanding How would this example change with a positron? A positron is identical to an electron except the charge is positive.Check Your Understanding From the examples, how does the energy of a lightning strike vary with the height of the clouds from the ground? Consider the cloud-ground system to be two parallel plates.Check Your Understanding What is the potential inside the metal sphere in Example 7.10?Check Your Understanding What is the potential on the x-axis? The z-axis?Check Your Understanding What is the potential on the axis of a nonuniform ring of charge, where the charge density is ()=cos ?Check Your Understanding Which coordinate system would you use to calculate the electric field of a dipole?Check Your Understanding What are the equipotential surfaces for an infinite line charge?Would electric potential energy be meaningful if the electric field were not conservative?Why do we need to be careful about work done on the system versus work done by the system in calculations?Does the order in which we assemble a system of point charges affect the total work done?Discuss how potential difference and electric field strength are related. Give an example.What is die strength of the electric field in a region where the electric potential is constant?If a proton is released from rest in an electric field, will it move in the direction of increasing or decreasing potential? Also answer this question for an electron and a neutron. Explain why.Voltage is the common word for potential difference. Which term is more descriptive, voltage or potential difference?If the voltage between two points is zero can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.Wliat is the relationship between voltage and energy? More precisely, what is the relationship between potential difference and electric potential energy?Voltages are always measured between two points Why?How are units of volts and electron-volts related? How do they differ?Can a particle move in a direction of increasing electric potential, yet have its electric potential energy decrease? ExplainCompare the electric dipole moments of charges +Q separated by a distance d and charges +Q/2 separated by a distance d/2.Would Gauss’s law be helpful for determining the electric field of a dipole? Why?In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?Can the potential of a nonuniformly charged sphere be the same as that of a point charge? Explain.If the electric field is zero throughout a region, must the electric potential also be zero in that region?Explain why knowledge of E(x, y, z) is not sufficient to determine V(x,y,z). What about the other way around?If two points are at the same potential, are there any electric field lines connecting them?Suppose you have a map of equipotential surfaces spaced 1.0 V apart. What do die distances between the surfaces in a particular region tell you about the strength of the Ein that region?Is the electric potential necessarily constant over the surface of a conductor?Linder electrostatic conditions, the excess charge on a conductor resides on its surface. Does this mean that all of the conduction electrons in a conductor are on the surface?- Can a positively charged conductor be at a negative potential? Explain.Can equipotential surfaces intersect?Why are the metal support rods for satellite network dishes generally grounded?(a) Why are fish reasonably safe in an electrical storm? (b) Why are swimmers nonetheless ordered to get out of the water in the same circumstance?What are the similarities and differences between the processes in a photocopier and an electrostatic precipitator?About what magnitude of potential is used to charge the drum of a photocopy machine? A web search for “xerography” may be of use.Consider a charge Q1(1+5.0C) fixed at a site with another charge Q2 (Charge +3.0C , mass 6.0g ) moving in die neighboring space, (a) Evaluate die potential energy of Q2 when it is 4.0 cm from Q1 (b) If Q2 starts from rest from a point 4.0 cm from Q1what will be its speed when it is 8.0 cm from Q1 ? (Note: Q1 is held fixed in its place.)Two charges Q1(1+2.00C) and Q2(+2.00C are placed symmetrically along the x-axis at x=3.00cm . Consider a charge Q3 of charge +4.00C and mass 10.0 mg moving along the y-axis. If Q3 starts from rest at y= 2.00 cm, what is its speed when it reaches y = 4.00 cm?To form a hydrogen atom, a proton is fixed at a point and an electron is brought from far away to a distance of 0.5291010 in, the average distance between proton and electron in a hydrogen atom. How much work is done?(a) What is the average power output of a heart defibrillator that dissipates 400 J of energy in 10.0 ms? (b) Considering the high-power output, why doesn’t the defibrillator produce serious bums?Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra election) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be 1.671027 kg.An evacuated tube uses an accelerating voltage of 40 kV to accelerate electrons to hit a copper plate and produce X-rays. Non-relativistic ally, what would be the maximum speed of these electrons?Show that units of V/m and N/C for electric field strength are indeed equivalent.What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of 1.50104 V?The electric field strength between two parallel conducting plates separated by 4.00 cm is 7.50104 V/m. (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be zero volts. What is the potential 1.00 cm from that plate and 3.00 cm from the other?The voltage across a membrane forming a cell wall is 80.0 mV and the membrane is 9.00 nm thick. What is the electric field strength? (The value is surprisingly large, but correct) You may assume a uniform electric field.Two parallel conducting plates are separated by 10.0 cm, and one of them is taken to be at zero volts, (a) What is the electric field strength between them, if the potential 8.00 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?Find the maximum potential difference between two parallel conducting plates separated by 0.500 cm of air, given the maximum sustainable electric field strength in air to be 3.0106 V/m.An electron is to be accelerated in a uniform electric field having a strength of 2.00106 V/m. (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?Use die definition of potential difference in terms of electric field to deduce die formula for potential difference between r=ra and r=rb for a point charge located at the origin. Here r is the spherical radial coordinate.The electric field in a region is pointed away from the z-axis and the magnitude depends upon the distance s from the axis. The magnitude of the electric field is given as E=as where a is a constant. Find the potential difference between points P1and P2, explicitly stating the path over which you conduct die integration for the line integral.Singly charged gas ions are accelerated from rest through a voltage of 13.0 V. At what temperature will the average kinetic energy of gas molecules be the same as that given these ions?A 0.500-cm-diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0-pC charge on its surface. What is tire potential near its surface?How far from a 1.00C point charge is the potential 100 V? At what distance is it 2.00102V ?If the potential due to a point charge is 5.00102 V at a distance of 15.0 m, what are the sign and magnitude of the charge?In nuclear fission, a nucleus splits roughly in half, (a) What is the potential 2.001014 in from a fragment that has 46 protons in it? (b) What is the potential energy in MeV of a similarly charged fragment at this distance?A research Vail de Graaff generator has a 2.00-m- diameter metal sphere with a charge of 5.00 mC on it. (a) What is the potential near its surface? (b) At what distance from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in MeV when the atom is at the distance found in part b?An electrostatic paint sprayer has a 0.200-m-diameter metal sphere at a potential of 25.0 kV that repels paint droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a 0.100-mg drop of paint have to arrive at the object with a speed of 10.0 m/s?(a) What is the potential between two points situated 10 cm and 20 cm from a 3.0C point charge? (b) To what location should the point at 20 cm be moved to increase this potential difference by a factor of two?Find the potential at points P1,P2,andP4 in the diagram due to the two given charges.Two charges 20Cand+2.0C are separated by 4.0 cm on the z-axis symmetrically about origin, with the positive one uppermost. Two space points of interest and P1andP2 are located 3.0 cm and 30 cm from origin at an angle 30 with respect to the z-axis. Evaluate electric potentials at P1andP2 and in two ways: (a) Using the exact formula for point charges, and (b) rising the approximate dipole potential formula.(a) Plot the potential of a uniformly charged 1-m rod with 1 C/m charge as a function of the perpendicular distance from the center. Draw your graph from s = 0,1 in to s = 1.0m. (b) On the same graph, plot the potential of a point charge with a 1-C charge at the origin, (c) Which potential is stronger near the rod? (d) What happens to the difference as the distance increases? Interpret your result.Throughout a region, equipotential surfaces are given by z = constant. The surfaces are equally spaced with V = 100 V for z = 0.00 m,= 200 V for 0.50 m, V= 300 V for z = 1.00 m. What is the electric field in this region?In a particular region, the electric potential is given by V=xy2z+4xy . What is the electric field in this region?Calculate the electric field of an infinite line charge, throughout space.Two very large metal plates are placed 2.0 cm apart, with a potential difference of 12 V between them. Consider one plate to be at 12 V, and the other at 0 V. (a) Sketch the equipotential surfaces for 0, 4, 8, and 12 V. (b) Next sketch in some electric field lines, and confirm that they are perpendicular to the equipotential lines.A very large sheet of insulating material has had an excess of electrons placed on it to a surface charge density of 3.00nC/m2 . (a) As the distance from the sheet increases, does the potential increase or decrease? Can you explain why without any calculations? Does the location of your reference point matter? (b) What is the shape of the equipotential surfaces? (c) What is the spacing between surfaces that differ by 1.00 V?A metallic sphere of radius 2.0 cm is charged with +5.0C charge, which spreads on the surface of the sphere uniformly. The metallic sphere stands on an insulated stand and is surrounded by a larger metallic spherical shell, of inner radius 5.0 cm and outer radius 6.0 cm. Now, a charge of 5.0C is placed on the inside of the spherical shell, which spreads out uniformly on the inside surface of the shell. If potential is zero at infinity, what is the potential of (a) the spherical shell, (b) the sphere, (c) the space between the two, (d) inside the sphere, and (e) outside the shell?Two large charged plates of charge density 30C/m2 face each other at a separation of 5.0 mm. (a) Find the electric potential everywhere, (b) An electron is released from rest at the negative plate; with what speed will it strike the positive plate?A long cylinder of aluminum of radius R meters is charged so that it has a uniform charge per unit length on its surface of . (a) Find the electric field inside and outside the cylinder, (b) Find the electric potential inside and outside the cylinder, (c) Plot electric field and electric potential as a function of distance from the center of the rod.Two parallel plates 10 cm on a side are given equal and opposite charges of magnitude 5.0109 C. The plates are 1.5 mm apart. What is the potential difference between the plates?The surface charge density on a long straight metallic pipe is . What is the electric potential outside and inside the pipe? Assume the pipe has a diameter of 2a.Concentric conducting spherical shells carry charges Q and -Q, respectively. The inner shell has negligible thickness. What is the potential difference between the shells?Shown below are two concentric spherical shells of negligible thicknesses and radii R1and R2The inner and outer shell carry net charges q1and q2 respectively where both q1 and q2 positive. What is the electric potential in the regions potential in the regions (a) r < R1, (b) R1< r < R2, and (c) r > R2?A solid cylindrical conductor of radius a is surrounded by a concentric cylindrical shell of inner radius b. The solid cylinder and the shell carry charges Q and —Q, respectively. Assuming that the length L of both conductors is much greater than a or b, what is the potential difference between the two conductors?(a) What is the electric field 5.00 m from die center of the terminal of a Van de Graaff with a 3.00-mC charge, noting that the field is equivalent to that of a point charge at the center of the terminal? (b) At this distance, what force does the field exert on a 2.00C charge on the Van de Graaff’s belt?(a) What is the direction and magnitude of an electric field that supports the weight of a free election near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces.A simple and common technique for accelerating electrons is shown in Figure 7.46, where there is a uniform electric field between two plates. Electrons are released, usually from a hot filament, near the negative plate, and there is a small hole in the positive plate that allows the electrons to continue moving, (a) Calculate the acceleration of the electron if the field strength is 2.50104 N/C . (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole. Figure 7.46 Parallel conducting plates with opposite charges on them create a relatively uniform electric field used to accelerate electrons to the right. Those that go through the hole can be used to make a TV or computer screen glow or to produce X- rays.In a Geiger counter, a thin metallic wire at the center of a metallic tube is kept at a high voltage with respect to the metal tube. Ionizing radiation entering the tube knocks elections off gas molecules or sides of the tube that then accelerate towards the center wire, knocking off even more electrons. This process eventually leads to an avalanche that is detectable as a current. A particular Geiger counter has a tube of radius K and the inner wire of radius a is at a potential of V0 volts with respect to the outer metal tube. Consider a point P at a distance s from tine center wire and far away from the ends, (a) Find a formula for the electric field at a point P inside using tine infinite wire approximation, (b) Find a formula for the electric potential at a point P inside. (c) Use V0 = 900 V, a = 3.00 min. P = 2.00 cm, and find the value of the electric field at a point 1.00 cm from the center.The practical limit to all electric field in air is about 3.00106 N/C. Above this strength, sparking rakes place because air begins to ionize, (a) At this electric field strength, how far would a proton travel before hitting the speed of light (ignore relativistic effects)? (b) Is it practical to leave air in particle accelerators?To form a helium atom, an alpha particle that contains two protons and two neutrons is fixed at one location, and two electrons are brought in from far away, one at a time. The first electron is placed at 0.6001010 m from the alpha particle and held there while the second electron is brought to 0.6001010 m from the alpha particle on the other side from the first electron. See die final configuration below, (a) How much work is done in each step? (b) What is the electrostatic energy of die alpha particle and two electrons in the final configuration?Find the electrostatic energy of eight equal charges (+3C) each fixed at the corners of a cube of side 2 cm.The probability of fusion occurring is greatly enhanced when appropriate nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This can be done using the kinetic energy of high- temperature gas ions or by accelerating the nuclei toward one another. (a) Calculate the potential energy of two singly charged nuclei separated by 1.001012. (b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?A bare helium nucleus has two positive charges and a mass of 6.641027kg . (a) Calculate its kinetic energy in joules at 2.00% of the speed of light, (b) What is this in electron-volts? (c) What voltage would be needed to obtain this energy?An election enters a region between two large parallel plates made of aluminum separated by a distance of 2.0 cm and kept at a potential difference of 200 V. The electron enters through a small hole in the negative plate and moves toward the positive plate. At the time the electron is near the negative plate, its speed is 4.0103 m/s. Assume the electric field between the plates to be uniform, and find the speed of electron at (a) 0.10 cm, (b) 0.50 cm, (c) 1.0 cm, and (d) 1.5 cm from the negative plate, and (e) immediately before it hits the positive plate.How far apart are two conducting plates that have an electric field strength of 4.50103 V/m between them, if their potential difference is 15.0 kV?(a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength of dry air, which is 3.00106 V/m, if the plates are separated by 2.00 mm and a potential difference of 5.010V is applied? (b) How close together can the plates be with this applied voltage?Membrane walls of living cells have surprisingly large electric fields across them due to separation of ions. What is the voltage across an 8.00-nm-thick membrane if the electric field strength across it is 5.50 MV/m? You may assume a uniform electric field.A double charged ion is accelerated to an energy of 32.0 keV by the electric field between two parallel conducting plates separated by 2.00 cm. What is the electric field strength between the plates?The temperature near the center of the Sun is thought to be 15 million degrees Celsius ( 1.5107oC ) (or kelvin). Through what voltage must a singly charged ion be accelerated to have the same energy as the average kinetic energy of ions at this temperature?A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of 1.00102 MV. (a) What energy was dissipated? (b) What mass of water could be raised from 15 °C to the boiling point and then boiled by this energy? (c) Discuss the damage that could be caused to the tree by die expansion of the boiling steam.What is the potential 0.5301010 m from a proton (the average distance between the proton and electron in a hydrogen atom)?(a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential 5.00 MV? (b) What does your answer imply about the practical aspect of isolating such a large charge?What are the sign and magnitude of a point charge that produces a potential of —2.00 V at a distance of 1.00 mm?In one of the classic nuclear physics experiments at the beginning of the twentieth century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of die doubly charged alpha nucleus was 5.00 MeV, how close to the gold nucleus (79 protons) could it come before being deflected?A 12.0-V battery-operated bottle warmer heats 50.0 g of glass, 2.50102 g of baby formula, and 2.00102 g of aluminum from 20.0C to 90.0C . (a) How much charge is moved by the battery? (b) How many electrons per second flow if it takes 5.00 min to warm the formula? (Hint: Assume that the specific heat of baby formula is about the same as the specific heat of water.)A battery-operated car uses a 12.0-V system. Find the charge the batteries must be able to move in order to accelerate the 750 kg car from rest to 25.0 m/s, make it climb a 2.00102 -in high hill, and finally cause it to travel at a constant 25.0 m/s while climbing with 5.00102 -N force for an hour.(a) Find the voltage near a 10.0 cm diameter metal sphere that has 8.00 C of excess positive charge on it. (b) What is unreasonable about this result? (c) Which assumptions are responsible?A uniformly charged ring of radius 10 cm is placed on a nonconducting table. It is found that 3.0 cm above the center of the half-ring the potential is —3.0 V with respect to zero potential at infinity. How much charge is in the half ring?A glass ring of radius 5.0 cm is painted with a charged paint such that the charge density around the ring varies continuously given by the following function of die polar angle ,=(3.0106C/m)cos2 . Find the potential at a point 15 cm above the center.A CD disk of radius (R = 3.0 cm) is sprayed with a charged paint so that the charge varies continually with radial distance r from the center in the following manner =(6.0C/m)r/R ?. Find the potential at a point 4 cm above the center.(a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graff terminal? (b) What is unreasonable about this result? (c) Which assumptions are responsible?A large metal plate is charged uniformly to a density of a=2.0109C/m2 . How far apart are the equipotential surfaces that represent a potential difference of 25 V?Your friend gets really excited by the idea of making a lightning rod or maybe just a sparking toy by connecting two spheres as shown in Figure 7.39, and making R2so small that the electric field is greater than the dielectric strength of air, just from the usual 150 V/m electric field near the surface of the Earth. If R1is 10 cm. how small does R2to be, and does this seem practical? (Hint: recall the calculation for electric field at the surface of a conductor from Gauss's Law.)(a) Find x L limit of the potential of a finite uniformly charged rod and show that it coincides with that of a point charge formula, (b) Why would you expect this result?A small spherical pith ball of radius 0.50 cm is painted with a silver paint and then -10 C of charge is placed on it. The charged pith ball is put at the center of a gold spherical shell of inner radius 2.0 cm and outer radius 2.2 cm. (a) Find the electric potential of the gold shell with respect to zero potential at infinity, (b) How much charge should you put on the gold shell if you want to make its potential 100 V?Two parallel conducting plates, each of cross-sectional area 400 cm2, are 2.0 cm apart and uncharged. If 1.01012 electrons are transferred from one plate to the other, (a) what is the potential difference between the plates? (b) What is the potential difference between the positive plate and a point 1.25 cm from it that is between the plates?A point charge of q=50108 C is placed at the center of an uncharged spherical conducting shell of inner radius 6.0 cm and outer radius 9.0 cm. Find the electric potential at (a) r = 4,0cm, (b) r = 8.0 cm, (c) r — 12.0 cm.Earth has a net charge that produces an electric field of approximately 150 N/C downward at its surface, (a) What is the magnitude and sign of the excess charge, noting the electric field of a conducting sphere is equivalent to a point charge at its center? (b) What acceleration will the field produce on a free electron near Earth’s surface? (c) What mass object with a single extra electron will have its weight supported by this field?Point charges of 25.0/ C and 45. C are placed 0.500 m apart. At what point along die line between them is the electric field zero? What is the electric field halfway between diem?What can you say about two charges q1and q2 if the electric field one-fourth of the way from q1to q2 is zero?Calculate the angular velocity of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is 0.5301010 m. You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction.An electron has an initial velocity of 5.00106m/s in a uniform 2.0010m/s electric field. The field accelerates the election in the direction opposite to its initial velocity, (a) What is the direction of the electric field? (b) How far does the electron travel before coming to rest? (c) How long does it take the electron to come to rest? (d) What is the electron's velocity when it returns to its starting point?Three Na+ and three Cl ions are placed alternately and equally spaced around a circle of radius 50 nm. Find the electrostatic energy stored.Look up (presumably online, or by dismantling an old device and making measurements) the magnitude of the potential deflection plates (and the space between them) in an ink jet printer. Then look up the speed with which the ink comes out the nozzle. Can you calculate the typical mass of an ink drop?Use the electric field of a finite sphere with constant volume charge density to calculate the electric potential, throughout space. Then check your results by calculating the electric field from the potential.Calculate the electric field of a dipole throughout space from the potential.Check Your Understanding The capacitance of a parallel-plate capacitor is 2.0 pF. If the area of each plate is 2.4 cm2, what is the plate separation?Check Your Understanding Verify that /V and e0/d have the same physical units.Check Your Understanding The radius of the outer sphere of a spherical capacitor is five times the radius of its inner shell. What are the dimensions of this capacitor if its capacitance is 5.00 pF?Check Your Understanding When a cylindrical capacitor is given a charge of 0.500 nC, a potential difference of 20.0 V is measured between the cylinders, (a) What is the capacitance of this system? (b) If the cylinders are 1.0 m long, what is the ratio of their radii?Check Your Understanding Determine the net capacitance C of each network of capacitors shown below. Assume the C1= 1.0 pF, C2=2.0pF, C3=4.0pF, and C4=5.0 pF. Find the charge on each capacitor, assuming there is a potential difference of 12.0 V across each network.Check Your Understanding The potential difference across a 5.0-pF capacitor is 0.40 V. (a) What is the energy stored in this capacitor? (b) The potential difference is now increased to 1.20 V. By what factor is the stored energy increased?Check Your Understanding When a dielectric is inserted into an isolated and charged capacitor, the stored energy decreases to 33% of its original value, (a) What is the dielectric constant? (b) How does the capacitance change?Check Your Understanding Continuing with Example 8.12, show that when the battery is connected across the plates the energy stored in dielectric-filled capacitor is U=kU0 (larger than the energy U0 of an empty capacitor kept at the same voltage). Compare this result with the result U=U0/K found previously for an isolated, charged capacitor.Check Your Understanding Repeat the calculations of Example 8.10 for the case in which the battery remains connected while the dielectric is placed in the capacitor.Does the capacitance of a device depend on the applied voltage? Does the capacitance of a device depend on the charge residing on it?Would you place the plates of a parallel-plate capacitor closer together or farther apart to increase their capacitance?The value of the capacitance is zero if the plates are not charged. True or false?If the plates of a capacitor have different areas, will they acquire the same charge when the capacitor is connected across a battery?Does the capacitance of a spherical capacitor depend on which sphere is charged positively or negatively?If you wish to store a large amount of charge in a capacitor bank, would you connect capacitors in series or in parallel? Explain.What is the maximum capacitance you can get by connecting three 1.0F capacitors? What is the minimum capacitance?If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain.Discuss what would happen if a conducting slab rather than a dielectric were inserted into the gap between the capacitor plates.Discuss how the energy stored in an empty but charged capacitor changes when a dielectric is inserted if (a) the capacitor is isolated so that its charge does not change; (b) the capacitor remains connected to a battery so that the potential difference between its plates does not change.Distinguish between dielectric strength and dielectric constant.Water is a good solvent because it has a high dielectric constant. Explain.Water has a high dielectric constant. Explain why it is then not used as a dielectric material in capacitors.Elaborate on why molecules in a dielectric material experience net forces oil them in a non-uniform electrical field but not in a uniform field.Explain why the dielectric constant of a substance containing permanent molecular electric dipoles decreases with increasing temperature.Give a reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. How does a dielectric material also allow a greater voltage to be applied to a capacitor? (The dielectric thus increases C and permits a greater V.)Elaborate on the way in which the polar character of water molecules helps to explain water's relatively large dielectric constant.Sparks will occur between the plates of an ail-filled capacitor at a lower voltage when the air is humid than when it is dry. Discuss why, considering the polar character of water molecules.What charge is stored in a 180.0F capacitor when 120.0 V is applied to it?Find the charge stored when 5.50 V is applied to an 8.00-pF capacitor.Calculate the voltage applied to a 2.00F capacitor when it holds 3.10C of charge.What voltage must be applied to an 8.00-nF capacitor to store 0.160 mC of charge?What capacitance is needed to store 3.00C of charge at a voltage of 120 V?What is the capacitance of a large Van de Graaff generator’s terminal, given that it stores 8.00 mC of charge at a voltage of 12.0 MV?The plates of an empty parallel-plate capacitor of capacitance 5.0 pF are 2.0 mm apart. What is the area of each plate?A 60.0-pF vacuum capacitor has a plate area of 0.010 m2. What is the separation between its plates?A set of parallel plates has a capacitance of 5.0F . How much charge must be added to the plates to increase the potential difference between them by 100 V?Consider Earth to be a spherical conductor of radius 6400 km and calculate its capacitance.If the capacitance per unit length of a cylindrical capacitor is 20 pF/m, what is the ratio of tine radii of the two cylinders?, An empty parallel-plate capacitor has a capacitance of 20F . How much charge must leak off its plates before the voltage across them is reduced by 100 V?A 4.00-pF is connected in series with an 8.00-pF capacitor and a 400-V potential difference is applied across the pair, (a) What is the charge on each capacitor? (b) What is the voltage across each capacitor?Three capacitors, with capacitances of C1=2.0F , C2=3.0F, and C3=6.0F , respectively, are connected in parallel. A 500-V potential difference is applied across the combination. Determine the voltage across each capacitor and the charge on each capacitor.Find the total capacitance of this combination of series and parallel capacitors shown below.Suppose you need a capacitor bank with a total capacitance of 0.750 F but you have only 1.50-mF capacitors at your disposal. What is the smallest number of capacitors you could connect together to achieve your goal, and how would you connect them?What total capacitances can yon make by connecting a 5.00F and a 8.00F capacitor?Find the equivalent capacitance of the combination of series and parallel capacitors shown below.Find the net capacitance of the combination of series and parallel capacitors shown below.A 40-pF capacitor is charged to a potential difference of 500 V. Its terminals are then connected to those of an uncharged 10-pF capacitor. Calculate: (a) the original charge on the 40-pF capacitor; (b) the charge on each capacitor after the connection is made; and (c) the potential difference across the plates of each capacitor after the connection.A 2.0F capacitor and a 4.0F capacitor are connected in series across a 1.0-kV potential. The charged capacitors are then disconnected from the source and connected to each other with terminals of like sign together. Find the charge on each capacitor and the voltage across each capacitor.How much energy is stored in an 8.00F capacitor whose plates are at a potential difference of 6.00 V?A capacitor has a charge of 2.5C when connected to a 6.0-V battery. How much energy is stored in this capacitor?How much energy is stored in die electrical field of a metal sphere of radius 2.0 m that is kept at a 10.0-V potential?(a) What is the energy stored in the 10.0F capacitor of a heart defibrillator charged to 9.00103 V ?(b) Find the amount of the stored charge.In open-heart surgery, a much smaller amount of energy will defibrillate the heart, (a) What voltage is applied to the 8.00F capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of the stored charge.A 165F capacitor is used in conjunction with a dc motor. How much energy is stored in it when 119 V is applied?Suppose you have a 9.00-V battery, a 2.00F capacitor, and a 7.40F capacitor, (a) Find the charge and energy stored if the capacitors are connected to the battery in series, (b) Do the same for a parallel connection.An anxious physicist worries that the two metal shelves of a wood frame bookcase might obtain a high voltage if charged by static electricity,, perhaps produced by friction, (a) What is the capacitance of the empty shelves if they have area 1.00102m2 and are 0.200 m apart? (b) What is the voltage between them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small hazard, calculate the energy stored, (d) The actual shelves have an area 100 times smaller than these hypothetical shelves. Are his fears justified?A parallel-plate capacitor is made of two square plates 25 cm on a side and 1.0 mm apart. The capacitor is connected to a 50.0-V battery. With the battery still connected, the plates are pulled apart to a separation of 2.00 mm. What are the energies stored in the capacitor before and after the plates are pulled farther apart? Why does the energy decrease even though work is done in separating the plates?Suppose that the capacitance of a variable capacitor can be manually changed from 100 pF to 800 pF by turning a dial, connected to one set of plates by a shaft from 0° to 180°. With the dial set at 180° (corresponding to C — 800 pF), the capacitor is connected to a 500-V source. After charging, the capacitor is disconnected from the source, and the dial is turned to 0°. If friction is negligible, how much work is required to turn the dial from 180° to 0°?Show that for a given dielectric material, the maximum energy a parallel-plate capacitor can store is directly proportional to the volume of dielectric.An air-filled capacitor is made from two flat parallel plates 1.0 mm apart. The inside area of each plate is 8.0cm2. (a) What is the capacitance of this set of plates? (b) If the region between the plates is filled with a material whose dielectric constant is 6.0, what is the new capacitance?A capacitor is made from two concentric spheres, one with radius 5.00 cm, the other with radius 8.00 cm. (a) What is the capacitance of this set of conductors? (b) If the region between the conductors is filled with a material whose dielectric constant is 6.00, what is the capacitance of the system?A parallel-plate capacitor has charge of magnitude 9.00F on each plate and capacitance 3.00F when there is air between the plates. The plates are separated by 2.00 mm. With the charge on the plates kept constant, a dielectric with =5 . is inserted between the plates, completely filling the volume between the plates, (a) What is the potential difference between the plates of the capacitor, before and after the dielectric has been inserted? (b) What is the electrical field at the point midway between the plates before and after the dielectric is inserted?Some cell walls in the human body have a layer of negative charge on the inside surface. Suppose that the surface charge densities are 0.50103C/m2 the cell wall is 5.0109m thick, and the cell wall material has a dielectric constant of = 5.4. (a) Find the magnitude of the electric field in the wall between two charge layers, (b) Find the potential difference between the inside and the outside of the cell. Which is at higher potential? (c) A typical cell in die human body has volume 1016m3 . Estimate the total electrical field energy stored in the wall of a cell of this size when assuming that the cell is spherical. (Hint: Calculate the volume of the cell wall.)A parallel-plate capacitor with only air between its plates is charged by connecting the capacitor to a battery. The capacitor is then disconnected from the battery, without any of the charge leaving the plates, (a) A voltmeter reads 45.0 V when placed across the capacitor. When a dielectric is inserted between die plates, completely filling the space, the voltmeter reads 11.5 V. What is the dielectric constant of the material? (b) What will the voltmeter read if the dielectric is now pulled away out so it fills only one-third of the space between the plates?Two flat plates containing equal and opposite charges are separated by material 4.0 mm thick with a dielectric constant of 5.0. If the electrical field in die dielectric is 1.5 MV/m, what are (a) the charge density on the capacitor plates, and (b) the induced charge density on the surfaces of the dielectric?For a Teflon ™-filled, parallel-plate capacitor, the area of the plate is 50.0 cm and the spacing between the plates is 0.50 mm. If the capacitor is connected to a 200-V battery, find (a) the free charge on the capacitor plates, (b) the electrical field in the dielectric, and (c) the induced charge on the dielectric surfaces.Find the capacitance of a parallel-plate capacitor having plates with a surface area of 5.00 m2 and separated by 0.100 mm of Teflon™.(a) What is the capacitance of a parallel-plate capacitor with plates of area 1.50 m that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?Two parallel plates have equal and opposite charges. When the space between the plates is evacuated, the electrical field is E=3.20105 V/m . When the space is filled with dielectric, the electrical field is E=2.50105 V/m. (a) What is the surface charge density on each surface of the dielectric? (b) What is the dielectric constant?The dielectric to be used in a parallel-plate capacitor has a dielectric constant of 3.60 and a dielectric strength of 1.60107 V/m. The capacitor has to have a capacitance of 1.25 nF and must be able to withstand a maximum potential difference 5.5 kV. What is the minimum area the plates of the capacitor may have?When a 360-nF air capacitor is connected to a power supply, the energy stored in the capacitor is 18.5J . While the capacitor is connected to the power supply, a slab of dielectric is insetted that completely fills die space between the plates. This increases the stored energy by 23.2J . (a) What is the potential difference between the capacitor plates? (b) What is die dielectric constant of the slab?A parallel-plate capacitor has square plates that are 8.00 cm on each side and 3.80 mm apart. The space between the plates is completely filled with two square slabs of dielectric, each 8.00 cm on a side and 1.90 mm thick. One slab is Pyrex glass and the other slab is polystyrene. If the potential difference between the plates is 86.0 V, find how much electrical energy can be stored in this capacitor.A capacitor is made from two flat parallel plates placed 0.40 mm apart. When a charge of 0.020C is placed oil the plates the potential difference between them is 250 V. (a) What is the capacitance of the plates? (b) What is the area of each plate? (c) What is the charge on the plates when the potential difference between them is 500 V? (d) What maximum potential difference can be applied between the plates so that the magnitude of electrical fields between the plates does not exceed 3.0 MV/m?An air-filled (empty) parallel-plate capacitor is made from two square plates that are 25 cm on each side and 1.0 mm apart. The capacitor is connected to a 50-V battery and fully charged. It is then disconnected from the battery and its plates are pulled apart to a separation of 2.00 mm. (a) What is the capacitance of this new capacitor? (b) What is the charge on each plate? (c) What is the electrical field between the plates?Suppose that the capacitance of a variable capacitor can be manually changed from 100 to 800 pF by turning a dial connected to one set of plates by a shaft, from 0 to 180 . With the dial set at 180 (corresponding to C = 800 pF), the capacitor is connected to a 500-V source. After charging, the capacitor is disconnected from the source, and the dial is turned to 0°. (a) What is the charge on the capacitor? (b) What is the voltage across the capacitor when the dial is set to 0°?Earth can be considered as a spherical capacitor with two plates, where the negative plate is the surface of Earth and the positive plate is the bottom of the ionosphere, which is located at an altitude of approximately 70 km. The potential difference between Earth’s surface and the ionosphere is about 350,000 V. (a) Calculate the capacitance of this system, (b) Find the total charge on this capacitor, (c) Find the energy stored in this system.A 4.00F capacitor and a 6.00F capacitor are connected in parallel across a 600-V supply line, (a) Find the charge on each capacitor and voltage across each, (b) The charged capacitors are disconnected from the line and from each other. They are then reconnected to each other with terminals of unlike sign together. Find the final charge on each capacitor and the voltage across each.Three capacitors having capacitances of 8.40, 8.40, and 4.20F , respectively, are connected in series across a 36.0-V potential difference. (a) What is the charge on the 4.20F capacitor? (b) The capacitors are disconnected from the potential difference without allowing them to discharge. They are then reconnected in parallel with each other with the positively charged plates connected together. What is the voltage across each capacitor in the parallel combination?A parallel-plate capacitor with capacitance 5.0F is charged with a 12.0-V battery, after which the battery is disconnected. Determine the minimum work required to increase the separation between the plates by a factor of 3.(a) How much energy is stored in the electrical fields in the capacitors (in total) shown below? (b) Is this energy equal to the work done by the 400-V source in charging the capacitors?Three capacitors having capacitances 8.4, 8.4, and 4.2 F are connected in series across a 36.0-V potential difference, (a) What is the total energy stored in all three capacitors? (b) The capacitors are disconnected from the potential difference without allowing them to discharge. They are then reconnected in parallel with each other with the positively charged plates connected together. What is the total energy now stored in the capacitors?(a) An 8.00/F capacitor is connected in parallel to another capacitor, producing a total capacitance of 5.00/F . What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?(a) On a particular day, it takes 9.60103 J of electrical energy to start a truck’s engine. Calculate the capacitance of a capacitor that could store that amount of energy at 12.0 V. (b) What is unreasonable about this result? (c) Which assumptions are responsible?(a) A certain parallel-plate capacitor has plates of area 4.00 m2 , separated by 0.0100 mm of nylon, and stores 0.170 C of charge. What is the applied voltage? (b) What is unreasonable about this result? (c) Which assumptions are responsible or inconsistent?A prankster applies 450 V to an 80.0F capacitor and then tosses it to an unsuspecting victim. The victim’s finger is burned by the discharge of the capacitor through 0.200 g of flesh. Estimate, what is the temperature increase of the flesh? Is it reasonable to assume that no thermodynamic phase change happened?A spherical capacitor is formed from two concentric spherical conducting spheres separated by vacuum. Tire inner sphere has radius 12.5 cm and the outer sphere has radius 14.8 cm. A potential difference of 120 V is applied to the capacitor, (a) What is the capacitance of the capacitor? tb) What is the magnitude of the electrical field at r = 12.6 cm, just outside the inner sphere? (c) What is the magnitude of the electrical field at r = 14.7 cm, just inside the outer sphere? (d) For a parallel-plate capacitor the electrical field is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?The network of capacitors shown below are all uncharged when a 300-V potential is applied between points A and B with the switch S open, (a) What is the Potential difference VE-VD? (b) What is the potential at point E after the switch is closed? (c) How much charge flows through the switch after it is closed?Electronic flash units for cameras contain a capacitor for storing the energy used to produce the flash. In one such unit the flash lasts for 1/675 fraction of a second with an average light power output of 270 kW. (a) If the conversion of electrical energy to light is 95% efficient (because the rest of the energy goes to thermal energy), how much energy must be stored in the capacitor for one flash? (b) The capacitor has a potential difference between its plates of 125 V when the stored energy equals the value stored in part (a). What is the capacitance?A spherical capacitor is formed from two concentric spherical conducting shells separated by a vacuum. The inner sphere has radius 12.5 cm and the outer sphere has radius 14.8 cm. A potential difference of 120 V is appLied to the capacitor, (a) What is the energy density at r = 12.6 cm, just outside the inner sphere? (b) What is the energy density at r = 14.7 cm, just inside the outer sphere? (c) For the parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for the spherical capacitor?81CPA parallel-plate capacitor is filled with two dielectrics, as shown below. When the plate area is A and separation between plates is d, show that the capacitance is given by C=0Adk1+k22 C=0Adk1+k22 C=0Adk1+k22A parallel-plate capacitor is filled with two dielectrics, as shown below. Show that the capacitance is give by C=20Adk1k2k1+k2A capacitor has parallel plates of area 12 cm2 separated by 2.0 mm. The space between the plates is filled with polystyrene, (a) Find the maximum permissible voltage across the capacitor to avoid dielectric breakdown, (b) When the voltage equals the value found in part (a), find the surface charge density on the surface of the dielectric.Check Your Understanding Handheld calculators often use small solar cells to supply the energy required to complete the calculations needed to complete your next physics exam. The current needed to run your calculator can be as small as 0.30 mA. How long would it take for 1.00 C of charge to flow from the solar cells? Can solar cells be used, instead of batteries, to stall traditional internal combustion engines presently used in most cars and trucks?Check Your Understanding Circuit breakers in a home are rated in amperes, normally in a range from 10 amps to 30 amps, and are used to protect the residents from harm and their appliances from damage due to large currents. A single 15-amp circuit breaker may be used to protect several outlets in the living room, whereas a single 20-amp circuit breaker may be used to protect the refrigerator in the kitchen. What can you deduce from this about current used by the various appliances?Check Your Understanding In Example 9.4, the drift velocity was calculated for a 2.053-mm diameter (12-gauge) copper wire carrying a 20-amp current. Would the drift velocity change for a 1.628-mm diameter (14-gauge) wire carrying the same 20-amp current?Check Your Understanding The current density is proportional to the current and inversely proportional to the area. If the current density in a conducting wire increases, what would happen to the drift velocity of the charges in the wire?Check Your Understanding Copper wires use routinely used for extension cords and house wiring for several reasons. Copper has the highest electrical conductivity rating, and therefore the lowest resistivity rating, of all nonprecious metals. Also important is the tensile strength, where the tensile strength is a measure of the force required to pull an object to the point where it breaks. The tensile strength of a material is the maximumamount of tensile stress it can take before breaking. Copper has a high tensile strength, 2108Nm2 A thirdimportant characteristic is ductility. Ductility is a measure of a material’s ability to be drawn into wires and a measure of the flexibility of the material, and copper has a high ductility. Summarizing, for a conductor to be a suitable candidate for making wire, there are at least three important characteristics: low resistivity, high tensile strength, and high ductility. What other materials are used for wiring and what are the advantages and disadvantages?Check Your Understanding A strain gauge is an electrical device to measure strain, as shown below. It consists of a flexible, insulating backing that supports a conduction foil pattern. The resistance of the foil changes as the backing is stretched. How does the strain gauge resistance change? Is the strain gauge affected by temperature changes?Check Your Understanding The resistance between the two conductors of a coaxial cable depends on the resistivity of the material separating the two conductors, tire length of the cable and the inner and outer radius of the two conductor. If you are designing a coaxial cable, how does the resistance between the two conductors depend on these variables?Check Your Understanding The voltage supplied to your house varies as V( t )= V max sin( 2ft ) If a resistor is connected across this voltage, will Ohm’s law V = IR still be valid?Check Your Understanding Electric motors have a reasonably high efficiency. A 100-hp motor can have an efficiency of 90% and a 1-hp motor can have an efficiency of 80%. Why is it important to use high- performance motors?