Phys A Solutons to Practce Probles hapter Inucton an Maxwell s uatons (a) At t s, the ef has a agntue of t ag t Wb s t Wb s Wb s t Wb s V t 5 (a) Table - gves the resstvty of copper Thus, L A 8 9 5 (b) The current nuce n the loop has a agntue of t t ag r, where r s the raus of the loop Therefore, to prouce a current of A reures that the fel change at a rate of t A r 5 4T s The prary fference between ths an the stuaton escrbe n Touchstone xaple - s n the uantty A The area through whch there s agnetc flux s not the area of the short col, n ths case, but s the area of the soleno (snce there s no fel outse an eal soleno) 4 So n ths case, A 8 The agntue of the nuce ef s ag N N na t t 4 4 T A 8 5A 5 s V Fro Oh's law, the nuce current has a agntue of
V 5 A 9 Snce a suare's area s relate to the length of ts ses by Thus, Faraay's law gves the agntue of the ef as A, we have A t t ag A 4T 5 /s 9 V t t t (a) In the regon of the saller loop the agnetc fel prouce by the larger loop ay be taken to be unfor an eual to ts value at the center of the saller loop, on the axs Fro -9 for the fel ue to a agnetc pole on the pole axs, x Gven the approxaton that ths fel s approxately constant over the loop area, the agnetc flux through the sall loop s (b) The ef s gven by Faraay s law: ag A r x r r r v t t x x t x ag x 4 4 (c) The flux through the saller loop s prouce by a agnetc fel that ponts up through the loop As the saller loop oves upwar, the flux through t ecreases, an the nuce current wll flow n a recton such that the resultng agnetc fel woul oppose that ecrease So the current wll flow n the sae sense as t oes n the larger loop, proucng a fel that ponts upwar all the angle between the agnetc fel vector an the loop's area vector Snce the col s rotate at a freuency f, the angle ft, where we've chosen ra at the te t s The ef nuce n the col has an apltue of ag N NA cos ft fna sn ft t t Snce the area of the rectangular col s value of ths ef s A 5 5, the axu
fna rev s 5 T 5 55 V ax 9 (a) The agnetc flux through the loop (ro rals en strp) s ncreasng because the area s ncreasng Faraay's law gves the agntue of the nuce ef as t ag A t Lv 5 T 5 55 s 48 V (b) y Oh s law, the nuce current s 48 V 8 A y Lenz s law, the current s clockwse n Fg -4 (c) Mechancal energy s transfore to theral energy at a rate of P A 8 9 W (a) The agnetc flux through the loop (ro rals en strp) s ncreasng because the area s ncreasng Faraay's law gves the agntue of the nuce ef as t ag A t Lv T 5 s V (b) y Oh s law, the nuce current s V 4 5 A y Lenz s law, the current s clockwse n Fg -4 (c) Mechancal energy s transfore to theral energy at a rate of P 5A 4 9 W () The agnetc fel apples a force to the current n the ro The force s recte rghtwar an has agntue F L (5 A)( )( T) 8 N To keep the ro ovng at constant velocty, an external agent ust apply a leftwar force of that sae agntue (e) The power exerte by the external agent s P Fv 8 N 5 s 9W, whch s the sae as our result fro part (c)
hapter Magnetc Fels ue to currents Usng the result of proble an -, we wsh to show that a a, or, 4 where a s the length of a se of the suare an s the raus of the crcle oth wres have the sae length L, so a L 4 an L Therefore, what we wsh to show s that L L 8 or, whch s true, snce 8 5 Let the x axs pont to the rght an the y axs pont up on the page The wre on the top wll prouce a agnetc fel wth a postve y-coponent an a postve x-coponent The wre on the botto wll prouce a agnetc fel wth a postve y-coponent an a negatve x- coponent Snce the wres carry the sae current an are eustant fro the pont P, the x- coponents of the fels a to zero Wth as the angle between x axs an the lne between P an one of the wres, the y-coponents a to gve the agntue of the fel: cos 4 T A 4 A 4 5 T 4 (a) If the currents are parallel, the two fels are n opposte rectons n the regon between the wres Snce the currents are the sae, the total fel s zero along the lne that runs halfway between the wres There s no possble current for whch the fel oes not vansh (b) If the currents are antparallel, the fels are n the sae recton n the regon between the wres The total fel at the pont (a stance r 4 fro each wre) has agntue so the reure current s, r r r 4 T 4 T A A 5 Usng the rght-han rule, we see that the current carre by wre ust be out of the page The fel ue to wre s r where 5 A an r 5 c The fel ue to wre
s r, where r 5 c The agntues of the fels ust be eual f they are to cancel at pont P Ths reures r r 5 c 5A 4A 5 c The agnetc fel prouce by the current n the long straght wre exerts a force on each of the four segents of the rectangle The segent on the rght se of the rectangle experences a force to the rght, whle the segent on the left se of the rectangle experences an eual force to the left These forces su to zero The top an botto of the rectangle experence fferent agntues of force because they are at fferent stances fro the straght wre The net force has a agntue of F net L L b a a b a a b 4 T A A A 8 9 N, an F net ponts towar the wre (a) Two of the currents are out of the page an one s nto the page, so the net current enclose by the path s A, out of the page Snce the path s traverse n the clockwse sense, a current nto the page s postve an a current out of the page s negatve, as ncate by the rght-han rule assocate wth Apere s law Thus, s enc 4 T A A 5 T (b) The net current enclose by the path s zero (two currents are out of the page an two are nto the page), so s enc 4 It s clear that the unknown current ust be recte nto the page n orer that ther fels be antparallel at P The value of the current s set by the reureent that the two fels have eual agntues Wth c, A, an the unknown current, that eans A 4 The agnetc fel nse a soleno has a agntue gven by N n, where N =, 5, an = A Ths yels = T
45 Snce the toro has a suare cross-secton, there are unue nner an outer ra We use - to evaluate the fel (a) The nner raus s r = 5 c, so the fel there s N 5 4 T /A 8A 5 4 T r 5 (b) The outer raus s r = c The fel there s N 5 4 T /A 8 A 4 r 4 T 5 (a) The agnetc fel at the center of a col of raus has a agntue of N Therefore, the rato s b b a a a b 4 (b) The pole oent of a col of raus has a agntue of N, so the rato of the agnetc pole oents s b a b a hapter 9 Magnetc Fels (a) The spee of the alpha partcle s foun fro 9-: v er 9 T 45 4 u kg u s (b) The pero s T e 4 u kg u 9 T 9 s (c) The knetc energy of the alpha partcle s
4 u kg u s 5 9 K v J ev 4 ev () The reure acceleratng potental s V K e 4 5eV e 4V 5 The partcles wll travel n crcular paths wth a spee obtane fro 9- as v r The knetc energy of each partcle s K r v In orer for the other partcles to crculate n the sae path as the proton, (a) K alpha proton alpha Kproton 4 proton alpha MeV MeV; (b) K euteron proton euteron Kproton proton euteron MeV 5 MeV (a) After acceleratng through the potental fference V, the partcles have a velocty gven by v K V The agnetc fel reure to cause the partcles to ove n the state path s foun fro 9- as V 9 5 kg V v r r 9 495 T (b) Let N be the nuber of ons that are separate by the achne per unt te The current s = N an the ass that s separate per unt te s M = N, where s the ass of a sngle on M has the value M kg s 8 8 kg s Snce N = M/ we have
M 9 8 8 kg s 9 5 kg A (c) ach on eposts energy V n the cup, so the energy eposte n te t s gven by N V t V t For t = h, g 5 kg 98 s NL 5T 45A If the two forces are to a to a zero net force, then v The electrc fel between the two plates has a agntue of V, where s the plate separaton The velocty can be foun fro the knetc energy K kev The reure agnetc fel has a agntue of V v K e V 9 kg ev 9 J ev 4 T The agnetc force on the wre ust be upwar an have a agntue eual to the gravtatonal force g on the wre Applyng the rght-han rule reveals that the current ust be fro left to rght n the fgure Snce the fel an the current are perpencular to each other the agntue of the agnetc force s gven by F ag L, where L s the length of the wre Thus, L g g L kg 98 s 44T 4 A 45 (a) The agntue of the agnetc pole oent s gven by N A, where N s the nuber of turns, s the current n each turn, an A s the area of a loop In ths case the loops are crcular, so A = r, where r s the raus of the col Thus N r A 9 A (b) The axu torue occurs when the pole oent s perpencular to the fel (or the plane of the loop s parallel to the fel) It s gven by A 5 T 85 N hapter 8
apactors an are n parallel, so ther euvalent capactance s 5?F That parallel cobnaton s n seres wth The euvalent capactance of the set of three capactors s 5 F 4 F F (a) The potental fference across s V = V Thus, the charge on capactor s V F V (b) Let = F We frst conser the three-capactor cobnaton consstng of an ts two closest neghbors, each of capactance The euvalent capactance of ths cobnaton s tro 5 5 F Ths tro s n seres wth the capactor on the lower rght; that set of four capactors has an euvalent capactance of e tro F The charge on ths euvalent capactor s e V F V Ths s also eual to the aount of charge on our upper tro of capactors Therefore, the potental fference across that cobnaton s V tro tro 4 V 5 F Snce ths potental fference s ve eually between an the one connecte n seres wth t, the voltage fference across s V V tro V Thus V F V 8 (a) Frst, the euvalent capactance of the two 4 F capactors connecte n seres s a 4 F F Ths cobnaton s then connecte n parallel wth two other F capactors (one on each se), resultng n an euvalent capactance of b a F F F Ths cobnaton s n seres wth another cobnaton (the two F capactors n parallel, wth an euvalent capactance of F F ) Thus, the euvalent capactance of the whole crcut s e b F F (b) Let V = V be the potental fference supple by the battery Then e V F V
(c) The upper set of four capactors has an effectve capactance of b F The lower par of capactors has an effectve capactance of F Snce these two cobnatons are n seres wth each other, they have the sae charge Snce they also have the sae capactance, the potental fference across each ust be the sae an ust be eual to half the potental fference of the battery Hence, V V Snce the lower cobnaton s a par of capactors n parallel, ths s also the potental across capactor : V V The charge on the plates of capactor s V F V () Fro the arguent presente n (c), the potental fference across s gven by V V onseuently, the charge carre by s b V V F V (e) Snce ths voltage fference V b s ve eually between an the other 4 F capactor connecte n seres wth t, the voltage fference across s gven by V V 5 V b Thus, V 4 F 5 V The euvalent capactance of parallel capactors, each wth a capactance of 5 F, s e 5 F F The total energy store n the capactor bank s e F 5,V 5 J U V Thus, the cost s 5 J kw h J kw h 4 5 The capactance wth the electrc n place s gven by ar The energy store s gven by U V V, so ar ar U (4 J) V (4 F)(5 V) 4 Accorng to Table 8-, you shoul use Pyrex t 4 (a) We use e an solve for t: t ln, where = s the capactve te constant Thus, when the capactor has lost / of ts orgnal charge, t ln ln 4 (b) When / of the orgnal charge has been lost,
t ln ln 5 The te t takes for the voltage fference across the capactor to reach V L s gven by e t, where s the ef of the battery We solve for : VL t ln V L 5 s 5 F ln 95 V 95 V V 5 where we use t = 5 s so that ths event woul occur twce per secon 5 The potental fference across the capactor vares as a functon of te t as V V t e, where V 5 V s the ntal potental fference across the capactor The resstance s relate to the te to scharge to V 8 s by t ln V V The nu scharge te s gven as s, so the nu value the resstance shoul take s s n 48 F ln 8 5 The axu scharge te s gven as s, so the axu value the resstor shoul take s n s F ln 8 5 49 k