Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015

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1 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be infinitely fr wy. (ii) Four point chrges re fixed t corners of sure centered t origin, s shown in Fig. 1. The length of ech side of sure is. The chrges re locted s follows: is t (, ), is t (, ), 3 is t (, ), nd 6 is t (, ). A fifth prticle tht hs mss m nd chrge is plced t origin nd relesed from rest. Find its speed when it is very fr from origin.. Five identicl point chrges re rrnged in two different mnners s shown in Fig. : in once cse s fce-centered sure, in or s regulr pentgon. Find potentil energy of ech system of chrges, tking zero of potentil energy to be infinitely fr wy. Express your nswer in terms of constnt times energy of two chrges seprted by distnce. 3. Consider system of two chrges shown in Fig. 3. Find electric potentil t n rbitrry point on x xis nd mke plot of electric potentil s function of x/. 4. A point prticle tht hs chrge of 11.1 nc is t origin. (i) Wht is (re) shpes of euipotentil surfces in region round this chrge? (ii) Assuming potentil to be zero t r =, clculte rdii of five surfces tht hve potentils eul to. V, 4. V, 6. V, 8. V nd 1. V, nd sketch m to scle centered on chrge. (iii) Are se surfces eully spced? Explin your nswer. (iv) Estimte electric field strength between 4.-V nd 6.-V euipotentil surfces by dividing difference between two potentils by difference between two rdii. Compre this estimte to exct vlue t loction midwy between se two surfces. 5. Two coxil conducting cylindricl shells hve eul nd opposite chrges. The inner shell hs chrge nd n outer rdius, nd outer shell hs chrge nd n inner rdius b. The length of ech cylindricl shell is L, nd L is very long compred with b. Find potentil difference, V V b between shells. 6. An electric potentil V (z) is described by function V m 1 z 4 V, z >. m, 1. m < z <. m V (z) = 3 V 3 V m 3 z 3, m < z < 1. m 3 V 3 V m 3 z 3, 1. m < z < m,. m < z < 1. m V m 1 z 4 V, z <. m The grph in Fig. 4 shows vrition of n electric potentil V (z) s function of z. (i) Give electric field vector E for ech of six regions. (ii) Mke plot of z-component of electric field, E z, s function of z. Mke sure you lbel xes to indicte numeric mgnitude of field. (iii) Qulittively describe distribution of chrges tht gives rise to this potentil lndscpe nd hence electric fields you clculted. Tht is, where re chrges, wht sign re y, wht shpe re y (plne, slb...)?

2 7. Two conducting, concentric spheres hve rdii nd b. The outer sphere is given chrge Q. Wht is chrge on inner sphere if it is erd?. 8. Consider two nested, sphericl conducting shells. The first hs inner rdius nd outer rdius b. The second hs inner rdius c nd outer rdius d. The system is shown in Fig. 5. In following four situtions, determine totl chrge on ech of fces of conducting spheres (inner nd outer for ech), s well s electric field nd potentil everywhere in spce (s function of distnce r from center of sphericl shells). In ll cses shells begin unchrged, nd chrge is n instntly introduced somewhere. (i) Both shells re not connected to ny or conductors (floting) tht is, ir net chrge will remin fixed. A positive chrge Q is introduced into center of inner sphericl shell. Tke zero of potentil to be t infinity. (ii) The inner shell is not connected to ground (floting) but outer shell is grounded tht is, it is fixed t V = nd hs whtever chrge is necessry on it to mintin this potentil. A negtive chrge Q is introduced into center of inner sphericl shell. (iii) The inner shell is grounded but outer shell is floting. A positive chrge Q is introduced into center of inner sphericl shell. (iv) Finlly, outer shell is grounded nd inner shell is floting. This time positive chrge Q is introduced into region in between two shells. In this cse uestion Wht re E(r) nd V (r)? cnnot be nswered nlyticlly in some regions of spce. In regions where se uestions cn be nswered nlyticlly, give nswers. In regions where y cnnot be nswered nlyticlly, explin why, but try to drw wht you think electric field should look like nd give s much informtion bout potentil s possible. 9. The hydrogen tom in its ground stte cn be modeled s positive point chrge of mgnitude e ( proton) surrounded by negtive chrge distribution tht hs chrge density ( electron) tht vries with distnce from center of proton r s: ρ(r) = ρ e r/ ( result obtined from untum mechnics), where =.53 nm is most probble distnce of electron from proton. (i) Clculte vlue of ρ needed for hydrogen tom to be neutrl. (ii) Clculte electrosttic potentil (reltive to infinity) of this system s function of distnce r from proton. 1. A prticle tht hs mss m nd positive chrge is constrined to move long x-xis. At x = L nd x = L re two ring chrges of rdius L. Ech ring is centered on x-xis nd lies in plne perpendiculr to it. Ech ring hs totl positive chrge Q uniformly distributed on it. (i) Obtin n expression for potentil V (x) on x xis due to chrge on rings. (ii) Show tht V (x) hs minimum t x =. (iii) Show tht for x << L, potentil pproches form V (x) = V () αx. (iv) Use result of Prt (iii) to derive n expression for ngulr freuency of oscilltion of mss m if it is displced slightly from origin nd relesed. (Assume potentil euls zero t points fr from rings.)

3 ving L. A perugh negred )zk. Q is n if. C. C. C. C 4. m 4. m 69. Eight point chrges, ech of mgnitude, re locted on. C. C. C. C corners of cube of edge s, s shown in Figure P m () Determine x, y, nd z components of resultnt 4. m force exerted by or chrges on chrge locted 3. mj t point A. (b) Wht re mgnitude nd direction of this resultnt force? 69 [SSM] Four point chrges re fixed t corners of sure centered t origin. The length of ech side of sure is. The chrges re locted s follows: is t (, ), is t (, ), 3 is t (, ), nd 6 z is t (, ). A fifth prticle tht hs mss m nd chrge is plced t origin nd relesed from rest. Find its speed when it is very fr from origin. Picture Problem y The digrm shows four point chrges fixed t corners of sure nd fifth chrged prticle Pointtht is relesed from m, rest t Aorigin. We cn use s conservtion of energy to relte s initil potentil energy of prticle to y its kinetic energy when it is t gret s distnce from origin nd x 6 electrosttic potentil t origin to express Ui. 69 nd 7. Figure P3.69 Problems Figure 1: Problem 1. K U Use conservtion of energy to relte 7. Consider chrge distribution shownenergy in Figure initil potentil of P3.69. or, becuse Ki = Uf =, () Show tht mgnitude electric prticle toof its kinetic energyfield whentit K f U i center of ny fce of is cube hs vlue of.18k e /s. t gret distnce from origin: (b) Wht is direction of electric field t center of top fce of cube? U V Express initil potentil energy C is d of ced nduion. prt e. fricd t De- point chrges re rrnged in two differe sure, in or s regulr pentgon. Fi e zero of potentil energy to be infinitely fr e energy of two chrges seprted by dis chrged prticle # is 71. Review problem. A negtively of prticle to its chrge nd plced t center of uniformly chrged ring, where electrosttic potentil t origin: ring hs totl positive chrge Q s shown in Exmple 3.8. The prticle, confined to for move long x xis, is Ui to obtin: Substitute Kf nd displced smll distnce x long xis (where x '' ) nd relesed. Show tht prticle oscilltes in simple hrmonic motion with freuency given by f$ 1 ( " ke Q m 3 x 3 i 1 mv V v V m # 1/ 7. A line of chrge with uniform density 35. nc/m lies long line y $ # 15. cm, between points with coordintes x $ nd x $ 4. cm. Find electric field it cretes t origin. 73. Review problem. An electric dipole in uniform electric field is displced slightly from its euilibrium position, s shown in Figure P3.73, where! is smll. The seprtion of chrges is, nd moment of inerti of dipole is I. Assuming dipole is relesed from this Figure : Problem. erposition, we know tht potentil energy of

4 tric Potentil Due to System of Two Chrges MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics system of two chrges shown in Figure Exm Prctice Problems Prt 1 Solutions Problem 1 Electric Field nd Chrge Distributions from Electric Potentil An electric potentil V ( z ) is described by function )(!V "m -1 )z 4V ; z >. m ; 1. m < z <. m 3 V! # 3 V & $ % "m-3 ' ( z3 ; m < z < 1. m V (z) = * 3 V # 3 V & $ % "m-3 ' ( z3 ;!1. m < z < m Figure ; 3:! The. m electric < z < dipole!1. mof problem 3.,(V "m -1 )z 4V ; z <!. m Figure Electric dipole The grph below shows vrition of n electric potentil V ( z ) s function of z. lectric potentil t n rbitrry point on x xis nd mke plot. ric potentil cn be found by superposition principle. At poi ve V( x) 1 1 ( ) x 4 x 4 x x expression my be rewritten s ) Give electric field vector E! for ech of six regions in (i) to (vi) below? V( x) =!" V 1 1 V x/ 1 x / 1 Solution: We shll fct tht E!!. Since electric potentil only depends on vrible z, we hve tht z -component of electric field is given by Figure 4: Problem 6. dv

5 Figure 5: The Frdy cge of problem 8.

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