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PROBLEMS_18-1 Electrical Potential Energy1. A 3 μC charge is brought in from infinity and fixed at the origin of a coordinate system. (a) How much work is done? (b) A second charge, of 5 μC, is brought in form infinity and placed 10 cm away from the first charge. How much does the electric field of the first charge do when the second charge is brought in? (c) How much work does the external agent do to bring the second charge in if that charge moves with unchanging kinetic energy?2. A charge of 4 μC is placed at the point x = 2, y = 3, z = 0 (all distances given in centimeters). Calculate the work done in bring a charge of -8 μC from x =2, y = 15, z = -30 to the point x = 2, y = 12, z = 6, assuming that the charge is moved at a steady speed.3. Derive an expression for the work required to set up the four-charge configuration of Fig.18-15, assuming the charges are initially infinitely far apart. Let V=0 at infinite.4. Two point charges are located on the x -axis, e q -=1 at 0=x and e q +=2 ata x =. (a) Find the work that must be done by an external force to bring a third point charge e q +=3 from infinity to a x 2=. (b) Find the total potential energy of the system of three charges.5. .A particle of positive charge Q is fixed at point P . A second particle of mass m and negative charge –q moves at constant speed in a circle of radius r 1, centered at P . Derive an expression for the work W that must be done by an external agent on the second particle to increase the radius of the circle of motion to r 2.18-2 Electric Potential6. Charges +q , -q , +q , and -q are placed on successive corners of a square in the xy -plane.Plot all locations in the xy -plane where the potential is zero.7. The origin of a coordinate system is at the intersection point of the perpendicular bisectorsof the sides of an equilateral triangle of sides 10 cm. Calculate the potential at the origin due to three identical charges of 0.8 μC placed at the corners of the triangle.8. A charge Q is distributed uniformly over the surface of a spherical shell of radius R . Howmuch work is required to move these charges to a shell with half the radius? The charges are again distributed uniformly.9. Calculate the potential inside and outside a sphere of radius R and charge Q , in which thecharge is distributed uniformly throughout the sphere.10. As a space shuttle moves through the dilute ionized gas of Earth ’s ionosphere, its potentiala ++q--q Fig. 18-15 Problem 3.is typically changed by -1.0 V during one revolution. By assuming that the shuttle is a sphere of radius 10 m, estimate the amount of charge it collects.11. An infinite nonconducting sheet with positive surface charge density σ on one side. (a)Show that the electric potential of an infinite sheet of charge can be written as (/),00V V 2z σε=-where V 0 is the electric potential at the surface of the sheet and z is the perpendicular distance from the sheet. (b) How much work is done by the electric field of the sheet as a small positive test charge q 0 is moved from an initial position on the sheet to a final position located a distance z from the sheet?12. A thick spherical shell of charge Q and uniform volume charge density ρ is bounced byradii r 1 and r 2, where r 2 > r 1. With V = 0 at infinity, find the electric potential V as a function of the distance r from the center of the distribution, considering the regions (a) r > r 2, (b) r 1< r < r 2 and (c) r < r 1. (d) Do these situations agree at r = r 2 and r = r 1?13. An electric field of approximately 100 V/m is often observed near the surface of Earth. Ifthis were the field over the entire surface, what would be the electric potential of a point on the surface? (Set V = 0 at infinity.)14. A plastic rod has been formed into a circle of radius R . It has a positive charge +Quniformly distributed along one-quarter of its circumference and a negative charge of -6Q uniformly distributed along the rest of the circumference (Fig. 18-16). With V = 0 at infinity, what is the electric potential (a) at the center C of the circle and (b) at point P , which is on the central axis of the circle at distance z from the center?15. A plastic disk is charged on one side with a uniform surface charge density σ, and thenthree quadrants of the disk are removed. The remaining quadrant is shown in Fig. 18-17. With V = 0 at infinity, what is the potential due to the remaining quadrant at point P , which is on the central axis of the original disk at a distance z from the original center?16. The plastic rod shown in Fig. 18-18 has length L and a nonuniform liner charge density λCR Pz+Q-6QFig. 18-16 Problem 14.= cx , where c is a positive constant. With V = 0 at infinity, find the electric potential (a) at point P 1 on the x axis, a distance d from one end and (b) point P 2 on the y axis, a distance y from one end of the rod.17. An insulated spherical conductor of radius r 1 carries a charge Q . A second conductingsphere of radius r 2 and initially uncharged is then connected to the first by a long conducting wire. (a) After connection, what can you say about the electric potential of each sphere? (b) How much charge is transferred to the second sphere? Assume the connected spheres are far apart compared to their radii.18. A very long conducting cylinder (length L ) of radius R 0 (R 0 << L ) carries a uniformsurface charge density σ. The cylinder is at an electric potential V 0. What is the potential, at points far from the end, at a distance r from the center of the cylinder? Determine for (a) r > R 0 and (b) r < R 0.19. Suppose the flat circular disk of Fig. 18-5 (Example 18-4) has a nonuniform surfacecharge density σ = ar 2, where r is measured from the center of the disk. Find the potential V (x ) at points along the x axis, relative to V = 0 at x = ∞.18-4 Calculating the Field from the Potential20．The electric potential of a charge distribution within some region of space is V (x , y , z ) =Q /4πε0x . Find the electric field in his region.21. Find the electric field of a charge distribution if the electric potential of the distribution isV = Ax 3z - By 2z 2 + C , where A , B , and C are constants.22. Use the result of Problem 16 to find the electric field component E x at point P 1 and E y atpoint P 2.18-5 Potentials and Fields around Conductors23. The same charges are placed on two identical drops of mercury. The drops are isolatedand take perfectly spherical shapes, and the electric potential at the surface of each drop is 900 V . The drops coalesce into a larger drop with a net charge double that of either smaller charge. What is the potential at the surface of this larger charge?24. An electric field of 3⨯106 V/m is sufficiently large to cause sparking in air. Find thehighest potential to which a conductor of radius 10 cm can be raised before breakdown occurs in the air surrounding it. Assumed that zero potential is taken at infinity.25. Two spherical conductors of radii 20 mm and 100 mm are connected by a thin wire andcarry charges q 1 and q 2, respectively. If the wire is cut and the centers of the spheres are250 mm apart, there is a repulsive force of 3.5 N between them. Use this information to calculate (a) q 1 and q 2 and (b) the electric fields at the surfaces when they are connected by the wire.Problems1. (a) 0; (b) -1.35 J; (c) +1.35 J.2. -1.77 J.3. ..20021q a ε-4. (a) ;20e W 8a πε=+(b) .20e U 8a πε=-5. ().012qQ 11W 8r r πε=- 6.7. +3.7⨯105 V .8. .20Q 8R πε9. r < R : ();230Q 3r V 8R R πε=- r > R : .0QV 4r πε=10. ∆q = -1.1⨯10-9 C.11. (b) .00q z W 2σε= 12. (a) ;0Q4rπεxy(b) ();322120r 31r r 322rρε--;()334213Q r r πρ=- (c) ();22210r r 2ρε- (d) yes. 13. 6.4⨯108 V .14. (a) ;05Q 4R πε- (b) ().221205Q 4z R πε-+15. [()].22120z R z 8σε+- 16. (a) [ln()];0cL d 1L/d 4πε-+(b) ).0cy 4πε17. (a) The same; (b) ().2212Q r Q r r =+18. (a) ln ;0000R R V rσε+(b) V = V 0. 19. [()()].22122230a x R R 2x 2x 6ε+-+ 20. .20QE=i 4x πεr r 21. ().2232E 3Ax zi 2Byz j Ax 2By z k =-+--r r r r22. P 1: E x = ,()0Q 4d d L πε+leftward. P 2: E y= (0c14πεupward.23. 1430 V .24. 3⨯105 V .25. (a) 2.2 μC, 11 μC; (b) ˆˆ.;..761122E 49510r E 99010r =⨯=⨯r r r r。
