784 CHAPTER 23 Electric Potential

and direction) when acted onby the force in part (a)? (c)How far below the axis has theelectron moved when it reachesthe end of the plates? (d) Atwhat angle with the axis is itmoving as it leaves the plates? (e) How far below the axis will itstrike the fluorescent screen S?23.66 .. CP Deflecting Plates of an Oscilloscope. The verticaldeflecting plates of a typical classroom oscilloscope are a pair ofparallel square metal plates carrying equal but opposite charges.Typical dimensions are about on a side, with a separationof about The potential difference between the plates is25.0 V. The plates are close enough that we can ignore fringing atthe ends. Under these conditions: (a) how much charge is on eachplate, and (b) how strong is the electric field between the plates?(c) If an electron is ejected at rest from the negative plate, how fastis it moving when it reaches the positive plate?23.67 .. Electrostatic precipi-tators use electric forces toremove pollutant particles fromsmoke, in particular in thesmokestacks of coal-burningpower plants. One form of pre-cipitator consists of a vertical,hollow, metal cylinder with athin wire, insulated from thecylinder, running along its axis(Fig. P23.67). A large potentialdifference is established betweenthe wire and the outer cylinder,with the wire at lower poten-tial. This sets up a strong radialelectric field directed inward.The field produces a region of ionized air near the wire. Smokeenters the precipitator at the bottom, ash and dust in it pick up elec-trons, and the charged pollutants are accelerated toward the outercylinder wall by the electric field. Suppose the radius of the centralwire is the radius of the cylinder is and apotential difference of is established between the wire andthe cylinder. Also assume that the wire and cylinder are both verylong in comparison to the cylinder radius, so the results of Problem23.63 apply. (a) What is the magnitude of the electric field midwaybetween the wire and the cylinder wall? (b) What magnitude ofcharge must a ash particle have if the electric fieldcomputed in part (a) is to exert a force ten times the weight of theparticle?23.68 .. CALC A disk with radius has uniform surface chargedensity (a) By regarding the disk as a series of thin concentricrings, calculate the electric potential at a point on the disk’s axisa distance from the center of the disk. Assume that the potentialis zero at infinity. (Hint: Use the result of Example 23.11 in Sec-tion 23.3.) (b) Calculate Show that the result agrees withthe expression for calculated in Example 21.11 (Section 21.5).23.69 .. CALC (a) From the expression for obtained in Problem22.42, find the expressions for the electric potential as a functionof both inside and outside the cylinder. Let at the surfaceof the cylinder. In each case, express your result in terms of thecharge per unit length of the charge distribution. (b) Graph and

as functions of from to 23.70 . CALC A thin insulating rod is bent into a semicircular arcof radius and a total electric charge is distributed uniformlyQa,

r = 3R.r = 0rEVl

V = 0r,V

EEx

-0V>0x.

xV

s.R

30.0-mg

50.0 kV14.0 cm,90.0 mm,

5.0 mm.3.0 cm

Air flow14.0 cm

50.0 kV

+

6.0 cm 12.0 cm

v0

2.0 cm S

along the rod. Calculate the potential at the center of curvature ofthe arc if the potential is assumed to be zero at infinity.23.71 ... CALC Self-Energy of a Sphere of Charge. A solidsphere of radius contains a total charge distributed uniformlythroughout its volume. Find the energy needed to assemble thischarge by bringing infinitesimal charges from far away. Thisenergy is called the “self-energy” of the charge distribution. (Hint:After you have assembled a charge in a sphere of radius howmuch energy would it take to add a spherical shell of thickness having charge Then integrate to get the total energy.)23.72 .. CALC (a) From the expression for obtained in Example22.9 (Section 22.4), find the expression for the electric potential as a function of both inside and outside the uniformly chargedsphere. Assume that at infinity. (b) Graph and as func-tions of from to 23.73 .. Charge is distributed uniformly over thevolume of an insulating sphere that has radius . Whatis the potential difference between the center of the sphere and thesurface of the sphere?23.74 . An insulating spherical shell with inner radius and outer radius carries a charge of uni-formly distributed over its outer surface (see Exercise 23.41).Point is at the center of the shell, point is on the inner surface,and point is on the outer surface. (a) What will a voltmeter read ifit is connected between the following points: (i) and (ii) and (iii) and infinity; (iv) and (b) Which is at higherpotential: (i) or (ii) or (iii) or (c) Which, if any, of theanswers would change sign if the charge were 23.75 .. Exercise 23.41 shows that, outside a spherical shell withuniform surface charge, the potential is the same as if all thecharge were concentrated into a point charge at the center of thesphere. (a) Use this result to show that for two uniformly chargedinsulating shells, the force they exert on each other and theirmutual electrical energy are the same as if all the charge were con-centrated at their centers. (Hint: See Section 13.6.) (b) Does thissame result hold for solid insulating spheres, with charge distrib-uted uniformly throughout their volume? (c) Does this same resulthold for the force between two charged conducting shells?Between two charged solid conductors? Explain.23.76 .. CP Two plastic spheres, each carrying charge uniformlydistributed throughout its interior, are initially placed in contactand then released. One sphere is in diameter, has mass

, and contains of charge. The other sphere isin diameter, has mass and contains of

charge. Find the maximum acceleration and the maximum speedachieved by each sphere (relative to the fixed point of their initiallocation in space), assuming that no other forces are acting onthem. (Hint: The uniformly distributed charges behave as thoughthey were concentrated at the centers of the two spheres.)23.77 . CALC Use the electric field calculated in Problem 22.45 tocalculate the potential difference between the solid conductingsphere and the thin insulating shell.23.78 . CALC Consider a solid conducting sphere inside a hollowconducting sphere, with radii and charges specified in Problem22.44. Take as Use the electric field calculated inProblem 22.44 to calculate the potential at the following valuesof (a) (at the outer surface of the hollow sphere); (b)

(at the inner surface of the hollow sphere); (c) (at thesurface of the solid sphere); (d) (at the center of the solidsphere).23.79 . CALC Electric charge is distributed uniformly along a thinrod of length with total charge Take the potential to be zero atQ.a,

r = 0r = ar = b

r = cr:V

r S q .V = 0

-30.0 mC150.0 g,40.0 cm-10.0 mC50.0 g

60.0 cm

-150 mC?c?ac;bb;a

c?acc;bb;a

cba

+150.0 mC60.0 cm25.0 cm

R = 5.00 cmQ = +4.00 mC

r = 3R.r = 0rEVV = 0

rV

Edq?

drr,q

QR

Figure P23.67

Figure P23.65