Problems set # 3 Physics 169 February 24, 2015

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1 Prof. Anhordoqui Problems set # 3 Physis 169 Februry 4, A point hrge q is loted t the enter of uniform ring hving liner hrge density λ nd rdius, s shown in Fig. 1. Determine the totl eletri flux through sphere entered t the point hrge nd hving rdius R, where R <.. A point hrge Q is loted just bove the enter of the flt fe of hemisphere of rdius R s shown in Fig.. Wht is the eletri flux (i) through the urved surfe nd (ii) through the flt fe? 3. The line g in Fig. 3 is digonl of ube. A point hrge q is loted on the extension of line g, very lose to vertex of the ube. Determine the eletri flux through eh of the sides of the ube whih meet t the point. 4. A sphere of rdius R surrounds point hrge Q, loted t its enter. (i) Show tht the eletri flux through irulr p of hlf-ngle (see Fig. 4) is Φ E = Q ɛ 0 (1 os θ). Wht is the flux for (ii) θ = 90 nd (iii) θ = An insulting solid sphere of rdius hs uniform volume hrge density nd rries totl positive hrge Q. A spheril gussin surfe of rdius r, whih shres ommon enter with the insulting sphere, is inflted strting from r = 0. (i) Find n expression for the eletri flux pssing through the surfe of the gussin sphere s funtion of r for r <. (ii) Find n expression for the eletri flux for r >. (iii) Plot the flux versus r. 6. A solid insulting sphere of rdius rries net positive hrge 3Q, uniformly distributed throughout its volume. Conentri with this sphere is onduting spheril shell with inner rdius b nd outer rdius, nd hving net hrge Q, s shown in Fig. 5. (i) Construt spheril gussin surfe of rdius r > nd find the net hrge enlosed by this surfe. (ii) Wht is the diretion of the eletri field t r >? (iii) Find the eletri field t r. (iv) Find the eletri field in the region with rdius r where b < r < (v) Construt spheril gussin surfe of rdius r, where b < r <, nd find the net hrge enlosed by this surfe. (vi) Construt spheril gussin surfe of rdius r, where < r < b, nd find the net hrge enlosed by this surfe. (vii) Find the eletri field in the region < r < b. (viii) Construt spheril gussin surfe of rdius r <, nd find n expression for the net hrge enlosed by this surfe, s funtion of r. Note tht the hrge inside this surfe is less thn 3Q. (ix) Find the eletri field in the region r <. ( x) Determine the hrge on the inner surfe of the onduting shell. (xi) Determine the hrge on the outer surfe of the onduting shell. (xii) Mke plot of the mgnitude of the eletri field versus r. 7.Consider long ylindril hrge distribution of rdius R with uniform hrge density ρ. Find the eletri field t distne r from the xis where r < R. 8. A solid, insulting sphere of rdius hs uniform hrge density ρ nd totl hrge Q. Conentri with this sphere is n unhrged, onduting hollow sphere whose inner nd outer rdii re b nd, s shown in Fig. 6. (i) Find the mgnitude of the eletri field in the regions r <, < r < b, b < r <, nd r >. (ii) Determine the indued hrge per unit re on the inner nd outer surfes of the hollow sphere.

2 9. An erly (inorret) model of the hydrogen tom, suggested by J. J. Thomson, proposed tht positive loud of hrge e ws uniformly distributed throughout the volume of sphere of rdius R, with the eletron n equl-mgnitude negtive point hrge e t the enter. (i) Using Guss lw, show tht the eletron would be in equilibrium t the enter nd, if displed from the enter distne r < R, would experiene restoring fore of the form F = kr, where k is onstnt. (ii) Show tht k = e 4πɛ 0. (iii) Find n expression for the frequeny f of simple R 3 hrmoni osilltions tht n eletron of mss m e would undergo if displed smll distne (< R) from the enter nd relesed. (iv) Clulte numeril vlue for R tht would result in frequeny of Hz, the frequeny of the light rdited in the most intense line in the hydrogen spetrum. 10. An infinitely long ylindril insulting shell of inner rdius nd outer rdius b hs uniform volume hrge density ρ. A line of uniform liner hrge density λ is pled long the xis of the shell. Determine the eletri field everywhere. 11. A prtile of mss m nd hrge q moves t high speed long the x xis. It is initilly ner x =, nd it ends up ner x = +. A seond hrge Q is fixed t the point x = 0, y = d. As the moving hrge psses the sttionry hrge, its x omponent of veloity does not hnge ppreibly, but it quires smll veloity in the y diretion. Determine the ngle through whih the moving hrge is defleted. [Hint: The integrl you enounter in determining v y n be evluted by pplying Guss lw to long ylinder of rdius d, entered on the sttionry hrge.] 1. Two infinite, nononduting sheets of hrge re prllel to eh other, s shown in Fig. 7. The sheet on the left hs uniform surfe hrge density σ, nd the one on the right hs uniform hrge density σ. Clulte the eletri field t points (i) to the left of, (ii) in between, nd (iii) to the right of the two sheets. (iv) Repet the lultions when both sheets hve positive uniform surfe hrge densities of vlue σ. 13. A sphere of rdius is mde of nononduting mteril tht hs uniform volume hrge density ρ. (Assume tht the mteril does not ffet the eletri field.) A spheril vity of rdius is now removed from the sphere, s shown in Fig. 8. Show tht the eletri field within the vity is uniform nd is given by E x = 0 nd E y = ρ 3ɛ 0. [Hint: The field within the vity is the superposition of the field due to the originl unut sphere, plus the field due to sphere the size of the vity with uniform negtive hrge density ρ]. 14. A solid insulting sphere of rdius R hs nonuniform hrge density tht vries with r ording to the expression ρ = Ar, where A is onstnt nd r < R is mesured from the enter of the sphere. (i) Show tht the mgnitude of the eletri field outside (r > R) the sphere is E = AR5 5ɛ 0. (ii) Show tht the mgnitude of the eletri field inside (r < R) the sphere is E = Ar3 r 5ɛ 0. [Hint: The totl hrge Q on the sphere is equl to the integrl of ρdv, where r extends from 0 to R; lso, the hrge q within rdius r < R is less thn Q. To evlute the integrls, note tht the volume element dv for spheril shell of rdius r nd thikness dr is equl to 4r dr.] 15. A slb of insulting mteril (infinite in two of its three dimensions) hs uniform positive hrge density ρ. An edge view of the slb is shown in Fig.9. (i) Show tht the mgnitude of the eletri field distne x from its enter nd inside the slb is E = ρx/ɛ 0. (ii) Suppose n eletron of hrge e nd mss m e n move freely within the slb. It is relesed from rest t distne x from the enter. Show tht the eletron exhibits simple hrmoni motion with frequeny s = 1 ρe π m eɛ 0. (iii) A slb of insulting mteril hs nonuniform positive hrge density ρ = Cx,

3 is pplied ugh ressuming the plne the y xis, x xis. N/C exists erstorm is by 3.00 m d t of the r. letri field ound. The 5 N m /C. mrine. ing the number horizon- 0 4 N/C s x through f e slnted thin to be of the sphere s ture nd l? ith the P4.11. surfes h eh e of re urfe lies e xy plne? hving liner hrge density nd rdius, s shown in Figure P4.6. Determine the totl eletri flux through sphere entered t the point hrge nd hving rdius R, where R. λ q 7. A pyrmid with horizontl squre bse, 6.00 m on eh side, nd height of 4.00 m is pled in vertil eletri field of 5.0 N/C. Clulte the totl eletri flux through the pyrmid s four slnted surfes. Q 8. A one with bse δ rdius 0 R nd height h is loted on horizontl tble. A horizontl uniform field E penetrtes the one, s shown in Figure P4.8. R Determine the eletri flux tht enters the left-hnd side of the one. E 16. In the ir over prtiulr R region t n ltitude of 500 m bove the ground the eletri field is 10 N/C direted downwrd. At 600 m bove Figure the P4.8 ground the eletri field is 100 N/C downwrd. Wht is the verge volume hrge density in the lyer of ir between these two elevtions? Is Setion 4. Guss s Lw it positive or negtive? 9. The following hrges re loted inside submrine: 5.00 C, 9.00 C, 7.0 C, nd 84.0 C. () Clulte 17. A point hrge Q 5.00 C is loted t the enter of ube of edge L m. In ddition, six other identil point hrges hving q 1.00 C re positioned symmetrilly round Q s shown in Figure P4.17. Determine the eletri flux through one fe of the ube. R Figure P4.6 Figure 1: Problem 1. Figure P4.15 h Figure : Problem. where x is mesured from the enter of the slb s shown in Fig. 9, nd C is onstnt. The slb is infinite in the y nd z diretions. Derive expressions for the eletri field in the exterior regions nd the interior region of the slb ( d/ < x < d/).

4 hs rfes. er the mpre upper rge of.0 m, Wht were o net field. Find b) the hrge rdius th the nd the ) r f this ius of ndute onhrge in the hped lulte urfe retest l yline wire hs tion,. hs The hrge line g Q in, nd Figure the P4. outer is shell digonl hs net of hrge ube. 3Q. A The point hrges qre is loted in eletrostti on the extension equilibrium. of line g, Using very Guss s lose to lw, vertex find of the the hrges ube. Determine nd the the eletri eletri fields flux everywhere. through eh of the sides of the ube whih meet t the 5. A point positive. point hrge is t distne R/ from the enter of n unhrged thin onduting spheril shell of rdius R. Sketh the eletri field dlines set up by this rrngement both inside nd qoutside the shell. b Setion 4.5 Forml Derivtion of Guss s Lw 53. A sphere of rdius R surrounds h point hrge Q, loted g t its enter. () Show tht the eletri flux through irulr p of hlf-ngle (Fig. P4.53) is e f (1 os ) Figure P4. Q 0 Figure 3: Problem 3. Wht is the flux for (b) 90 nd () 180? Setion 4.3 Applition of Guss s Lw to Vrious Chrge Distributions 3. Determine the mgnitude of the eletri field t the surfe of led-08 nuleus, whih ontins 8 protons nd 16 neutrons. Assume the led nuleus hs volume 08 times tht of one proton, nd onsider proton to be sphere of rdius θ R m. 4. A solid sphere of rdius 40.0 m hs totl positive hrge of 6.0 C uniformly distributed Q throughout its volume. Clulte the mgnitude of the eletri field () 0 m, (b) 10.0 m, () 40.0 m, nd (d) 60.0 m from the enter of the sphere. 5. A 10.0-g piee of Styrofom rries net hrge of C nd flots bove the enter of lrge horizontl sheet of plsti tht Figure hs P4.53 uniform hrge density on its Figure 4: Problem 4. surfe. Wht is the hrge per unit re on the plsti sheet? 6. A ylindril shell of rdius 7.00 m nd length 40 m hs Additionl its hrge Problems uniformly distributed on its urved surfe. The 54. A mgnitude nonuniform of the eletri eletri field field is t given point by 19.0 the m expression rdilly the m Consi totl surf from 3. In nu ont eh Wht pushi 33. Fill tw from equl they h other. on e t the field qunt estim 34. An in volum Q. A s omm strtin flux p fun eletr 35. A uni tot ylind the fi the y eletr eletr 36. An in 5.7 interi one

5 tht the hrge inside this surfe is less thn 3Q. (i) Find the eletri field in the region r. ( j) Determine the (k) Determine the hrge on the outer surfe mining vof y n t hrge on the inner surfe of the onduting shell. long ylinder (k) Determine the hrge on the outer surfe of the onduting shell. (l) Mke plot of the mgnitude hrge. of t onduting shell. (l) Mke plot of the mgnitude of the eletri field versus eletri field r. versus r. 3Q Q 56. Consider two identil onduting spheres whose surfes re seprted by smll distne. One sphere is given lrge net positive hrge while the other is given smll net positive hrge. It is found tht the fore between them is ttrtive even though both spheres hve net hrges of the sme sign. Explin bhow this is possible Q Q A solid, insulting sphere of rdius hs uniform hrge density nd totl hrge Q. Conentri with this sphere is n unhrged, onduting hollow sphere whose Figure 5: Problem P defleted. Sugge 60. Review problem gen tom, sugge itive loud of throughout the eletron n equ the enter. () would be in eq from the ente restoring fore (b) Show tht K frequeny f o eletron of m distne ( R) f numeril vl Hz, most intense lin 61. An infinitely l rdius nd ou density. A lin pled long th field everywhere 6. Two infinite, no to eh other, left hs unifo inner nd outer rdii re b nd, s shown in Figure P4.57. () Find the mgnitude of the eletri field in the regions r, r b, b r, nd r. (b) Determine the indued hrge per unit re on the inner nd Consider two outer identil surfes of the onduting hollow sphere. spheres whose surf re seprted by smll distne. One sphere is given Insultor lrge net positive hrge while the other is given sm Condutor net positive hrge. It is found tht the fore betwe them is ttrtive even though both spheres hve n b hrges of the sme sign. Explin how this is possible. σ A solid, insulting Figure P4.57 sphere Problems 57 of nd 58. rdius hs unifo Figure 6: Problem 8. hrge density nd totl hrge Q. Conentri with t sphere is n unhrged, onduting hollow sphere who inner nd outer rdii re b nd, s shown in Figu b Figure P4.55

6 ible. uniform with this ere whose in Figure eld in the (b) Deterinner nd 760 pled CHAPTER long 4 the Guss s xis of Lwthe shell. Determine the eletri field everywhere Two infinite, nononduting sheets of hrge re prllel to eh other, s shown in Figure P4.6. The sheet on the left hs uniform surfe hrge density, nd the one on the right hs uniform hrge density. Clulte the eletri field t points () to the left of, (b) in between, nd () to the right of the two sheets. Wht If? Repet the lultions for Problem 6 when both sheets hve positive uniform surfe hrge densities of vlue. 64. A sphere of rdius is mde of nononduting mteril tht hs uniform volume hrge density. (Assume tht the mteril does not ffet the eletri field.) A spheril vity of rdius is now removed from the sphere, s shown in Figure P4.64. Show tht the eletri field within the vity is uniform σ nd is given by E x 0 nd E y /3 0. (Suggestion: The field σ within the vity is the superposition of the field due Figure to the P4.6 originl unut sphere, plus Figure 7: Problem 1. the field due to sphere the size of the vity with uniform negtive hrge density.) 67. A th e (b (r r r vo th 68. A R S y x Figure P4.64 Figure 8: Problem A uniformly hrged spheril shell with surfe hrge 69. A

7 A y O d x Figure P4.71 Problems 71 nd 7. Figure 9: Problem A slb of insulting mteril hs nonuniform positive hrge density Cx, where x is mesured from the enter of the slb s shown in Figure P4.71, nd C is onstnt. The slb is infinite in the y nd z diretions. Derive expressions for the eletri field in () the exterior regions nd (b) the interior region of the slb ( d/ x d/). 73. () Using the mthemtil similrity between Coulomb s