Exam 2 Solutions. Note that there are several variations of some problems, indicated by choices in parentheses. + R 1 2

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1 PHY49 Fll 5 Acot, Woodd Exm Solution Note tht thee e evel vition of ome poblem, indicted by choice in penthee. Poblem : Two light bulb hve eitnce of Ω nd 3Ω. They e connected in pllel co V line. Wht i the totl powe dpted by the two bulb (o the Ω o 3Ω bulb)? Thi poblem w imil to Execie 6-C done in cl. o 44 W nd 48 W individully. P IE E R E (R + R ) () ( + )W 9W ; (.) 3 Poblem : In the cicuit hown in the figue both btteie hve inignificnt intenl eitnce nd the idelized mmete ed I 4.A in the diection hown. Find the E.M.F. of the bttey ( negtive nwe indicte tht the E.M.F. polity i oppoite to wht i hown). Thi poblem w imil to Execie 6-D done in cl. 75. V E RI + R ( I I ); E R I + R ( I I ); I 4. A; R. Ω ; R 48. Ω ; R 5. Ω; (.) 3 3 We note tht the fit eqution in (.) implie nd eqution to yield E,, I (( R + R ) I E ), which cn be immeditely put into the R R+ R3 3 R E RI ( R + R) I RI (( R+ R ) I E ) V ; (.3) Poblem 3: A cpcito with n initil potentil diffeence of V() 5V i dichged though eito when witch between them i cloed t t. At t., the potentil diffeence co the cpcito i V.5V. Wht i the potentil diffeence co the cpcito t t? Thi i bed on homewok poblem 7.64

2 PHY49 Fll 5 Acot, Woodd t/ Reclling tht q() t CV () t q + q e τ in which the dichging-condition q() CE CV() nd q( ) t/ give q nd q CV(), we hve Vt ( ) V() e τ. The thid condition V(. ).5 V (5 V) e.5 detemine τ to be τ. / ( ln ).75, which detemine the voltge t ll time, nd o we (. )/(.75 ) clculte diectly (. ) (5 ) V V e.5v. 5 (. )/ τ Poblem 4: In the figue E 4V, R R3.Ω, nd R Ω. Wht i the potentil diffeence VA V? (.4) Thi i bed on Ch.7 homewok poblem (7.35), nd Execie 6-F done in cl Let the cuent though the EMF be i, tht though R be i, nd the cuent though R be i. Junction ule: i i + i i i + i 3 i 3 i i Left loop ule, ubtituting R Ω nd R Ω: 4 i i i 4 i Zig-zg loop ule: 4 i i 3 i ( ) 4 i i i 4 3i + i ( ) + i i 8 + 7i i 4A i 4 i 6A i i + i A Now V A V i R ( 6A) ( Ω) 6V

3 PHY49 Fll 5 Acot, Woodd Poblem 5: A poton tvel though unifom mgnetic nd electic field. The mgnetic field i.5 i mt. At one intnt the velocity of the poton i v jm/. At tht intnt wht i the net foce cting on the poton if the electic field i 4. k V/m? Thi i poblem 8., which w igned in homewok F q( E + v ) q(e k + vĵ ( î )) q(e + v) k (.6 9 )(4. + ()(.5 3 )) kn.44 8 N k (.5) Poblem 6: In the figue chged pticle move into egion of unifom mgnetic field, goe though hlf cicle, nd then exit tht egion. The pticle i eithe poton o n electon (you mut decide which). It pend 3 n in the egion. Wht i the mgnitude of? Thi i poblem 8.6, which w igned in homewok We immeditely notice tht the pticle velocity v v( j) i deflected in the k pointing out of the pge. Looking t the vecto-vlue of the foce, (.6) î diection by the mgnetic field F F ( î ) q v q(v( ĵ)) ( k) qv( î ) qv > q > q +e.6 9 C; m m p.67 7 kg; While in the cicul egion, the chged pticle h contnt peed v v, nd mintin thi velocity. Thi i becue the mgnetic foce F (due to mgnetic field k nd velocity v v dθ Rω dθ ) i pependicul to the diplcement, nd thu doe no wok. Letting the hlf-cicle hve diu R, nd letting the pticle hve chge q, we hve, dk dw F d (q v ) Rd θ qvr( d θ ) qvr() dk dw K i K f K ; (.8) (.7) Hence, no kinetic enegy i dded o ubtcted to the pticle of m m. the velocity v i of contnt mgnitude. In the cicul tjectoy, Newton nd Lw then i, v F m p m p R F v m p R q v v m p R q v v in9 " m p R qvinθ v m v p Rq ; (.9) πr πr In (.9), the velocity i given by v, o, T (/) T

4 PHY49 Fll 5 Acot, Woodd π R 7 πr πr mp ((/) T) mpπ (.67 kg) π kg N v T ;(. 9 9 m T T R( + e) e T (.6 C)(3 ) C C In the lt tep of (.), we illutte the unit of mgnetic field. Poblem 7: In cetin cycloton poton move in cicle of diu.5 m. The mgnitude of the mgnetic field i. T. Wht i the kinetic enegy of the poton in million electon-volt (MeV)? Thi i poblem 8-38, which w igned in homewok. 9 e 7 (.6 C)(. T) ωc m R kg)( 7. ) 3.67 kg.6 J MeV K mv m( R) m( ) (.67 5 m 7.67MeV ;(.) Poblem 8: The figue how wood cylinde of m m.5 kg nd length L. m, with N tun of wie wpped ound it longitudinlly, o tht the plne of the wie contin the long centl xi of the cylinde. The cylinde i eleed on plne inclined t n ngle θ to the hoizontl, with the plne of the coil pllel to the incline plne. If thee i veticl unifom mgnetic field of mgnitude.5 T, wht i the let cuent i though the coil tht keep the cylinde fom olling down the plne? Thi i poblem 8-5, which w igned in homewok. à (.) Let the wood-cylinde be of m m, nd hve moment of ineti I. Newton nd Lw fo tnltionl equilibium between the foce of ttic 3 fiction f nd mgnetic foce F nd ottionl equilibium between the toque of ttic fiction Rf nd mgnetic toque τ i, (.3) m m F f mg in θ; Iα τ R F inθ fr R F in 9 fr R F Rf ; i i i i i i i i i i A tel i unit of foce pe unit velocity pe unit chge; eentilly the unit of electic field divided by velocity. The diu of the wooden cylinde i R, but the wooden-mteil my be inhomogeneou, o ume I mr. 3 CAUTION: The foce of ttic fiction i ection foce, nd it mgnitude i unknown. The mximum vlue the foce of fiction could tke on, if we knew the ttic-fiction-coefficient contibution mg co µ, i mx f µ S N, whee N i ection foce which h known S θ fom gvity, but n unknown contibution fom the net mgnetic foce.

5 PHY49 Fll 5 Acot, Woodd The poblem i to find F, F, F3, F : the mgnetic foce upon the fou ection of the que-loop hown in the 4 Figue. The mgnitude of the foce upon wie of length l mking ngle θ with mgnetic field cying cuent I dq/ dt i given by F Il inθ. Thee e N uch wie poducing identicl nd upeimpoing foce. Thu, let the wood-cylinde be of diu R. Section nd 3 e of length R, while Section nd 4 e of length L. Then, R F Ri N R i R N ir N l in θ ( + )( ) in(9 θ) + co θ ; R F Ri l in θ N R( + i)( R) in(7 θ) N ir co θn R F ; R F Ri l in θ N ( + R)( + ilin θ) N RiLin θn; R F Ri l in θ N ( R)( ilin θ) N RiLin θn; Combining (.3) nd the explicit foce (.4), nd noting the implifiction RF RF, we hve, 3 3 (.4) f mg inθ ## "## $ 4 Rf Rmg inθ R i F i R F + R F + R 3 F 3 + R 4 F 4 RiLinθ N + RiLinθ N + RiLinθ N; (.5) i Solving (.5) fo i, we hve, mg (.5 kg)(9.8 ) C Rmg RiL N i (.6) m olve fo i inθ inθ.453 ; N LN (. m)(.5 ( mc / ))(.) Aftewod: We note tht the net toque τ due to the fou bnche of the loop h the popety, τ 4 i RLe A/ N # "# $ R i F i RiLN inθ RL i N inθ i N inθ A i inθ % µ inθ % % µ ; (.7) We intoduced the mgnetic dipole moment vecto, µ i A in in n, whee n k coθ + jinθ i the plne-noml defining the vecto-e A N Nn NLRn. Recll, lo, tht we encounteed vecto e in ou tudy of the flux tht ntully occued in Gu lw. Poblem 9: The figue how, in co ection, two long tight wie held gint pltic cylinde of diu R cm. Wie cie cuent i 6mA out of the pge nd i fixed in plce t the left ide of the cylinde. Wie cie cuent i 4mA out of the pge nd cn be moved ound the cylinde. At wht (poitive) ngle θ hould wie be poitioned uch tht, t the oigin, the net mgnetic field due to the two cuent h mgnitude 8nT? Thi i poblem 9-34, which w woked in cl on Oct.4

6 PHY49 Fll 5 Acot, Woodd The two mgnetic field decompoe j (inθ i co θ j), o the eultnt of thi, uing nd µ i µ i 3 Ampèe lw to y nd (in which we clely hve i i ), i, π R π R µ i 9 3 π R 3.(8 T ) π R 4 θ θ 3 4 µ 3 i 3 4 (4 A) (.8) µ i 3 µ i 3 3 ( in θ) + ( co θ) πr in θ + ( co θ) πr in θ+ ( ) + co θ coθ (.9) + 3co co ( ( ) ) co ( ( ) ) ; Poblem : In the figue long tight wie cie cuent i 3.A nd ectngul loop cie cuenti.a. Tke the dimenion to be.cm, b 8.cm, nd L 3.cm. In unit vecto nottion, wht i the foce on the loop due toi? Thi i poblem 9-4, which w igned in homewok. (.) The hoizontl wie: Conide two typicl wie, nd b, ditnce d pt, nd cying epective cuent i nd i b. A diffeentil element of foce df ct upon wie-b nd i due to wie-, b (.)

7 PHY49 Fll 5 Acot, Woodd y the Loentz foce lw, the diffeentil element of foce df pe unit length dx i due to mgnetic field 4 b µ i dfb b π d ( j), o we clculte the foce pe unit length fb, dx f b d F b dz dq v b b dq d z µ i dq b dt ( ( ĵ) πd ) b ( k dz) µ i dt ( ( ĵ) πd ) µ i i dz dz dz b k ( ĵ) µ i i b î; (.) πd πd Evidently, the foce between wie nd b i in the + î diection, nd thu i ttctive. Uing thi eult (.) upon the two hoizontl wie in the Figue (numbeed nd 3 (note the diffeent coodinte)), F d F d F L L dx dx µ i i π ĵ dx µ i i L π ĵ; F 3 d L L F 3 dx dx µ i i ( ĵ) π( + b) dx µ i i L ĵ; (.3) π( + b) The veticl wie: The totl foce upon wie nd 4 due to wie (of infinite length) e, F d F d F dy dy +b dq v dy; F dy 4 +b d F dy dy The integnd in (.4) (i.e., the foce pe unit y -length) e, d F dy dq v i ( ĵ dy) µ i ( k) π y µ i i dy dy π y î; d F 4 dy dq v 4 4 dy +b dq 4 v 4 dy; (.4) dy +b i (+ ĵ dy) µ i ( k) π y µ i i dy π y î;(.5) Combining (.4) nd (.5), we hve, + b + b + b + b µ µ dy µ + b µ µ dy µ + b F i dy i i ln ; F4 i dy i i ln ; πy π y π πy π (.6) y π Looking t (.6), we ee F F4, o the upepoition of thee two foce mke no contibution. Hence, F F + F + F 3 + F 4 F + F + F 3 F F + F 3 µ i i L π ĵ µ i i L π( + b) ĵ µ i i L π (4π 7 T m A )(3.A)(.A)(3.cm) π ĵ + b ĵ.cm.cm + 8.cm.3N ĵ ; (.7) Poblem : The cuent denity J 3 inide long, olid, cylindicl wie of diu 3. m i in the diection of the centl xi, nd it mgnitude vie linely with dil ditnce fom the xi ccoding to J J() J ( / ) k, whee J 3 A. Wht i the mgnitude of the mgnetic field t / /? You my m need the Jcobin tem ddθ fo integtion in pol coodinte. Thi i poblem 9-47, which w igned in homewok. d i d i d i µ i encl wie µ encl wie µ l encl wie π µ encl wie πd 4 Poof: fom Ampèe lw: i poblem, encl i nd (by the ight hnd ule) j.. In thi

8 PHY49 Fll 5 Acot, Woodd The cuent encloed by n Ampèin-loop of diu b i, πj πj πj b i i ( b) di da J ( / ) d d d ( b ) ; encl encl encl b π b 3 di 3 3 φ da 3 3 wie x.. (.8) The mgnetic field t ditnce b fom the cente of the wie, then, i given fom the encloed cuent (.8) vi the lw of Ampèe, " d µ i encl (b) " d π J µ b 3 π J φ πb φ µ b olve fo J µ b 3 ; (.9) Fom (.9) follow, µ J ( ) 7 Tm 3 A 7 b ( ) µ J (4π A )(3. m)(3 ).64 T ; (.3) m 3 Poblem : A olenoid tht i L 95cmlong h diu R.cmnd winding of N tun; it cie cuent of i 3.6A. Wht i the mgnitude of the mgnetic field inide the olenoid? Thi i poblem 9-5, which w igned in homewok. The mgnetic field t the cente of olenoid mde of wie cying cuent i with n N / L tun pe mete i of mgnitude µ ni. Theefoe, µ ( N/ L) i 4π (3.6 A) 5.744mT. 7 Tm A.95cm T Poblem 3: A wie loop of length L 4cmnd W 5cm lie in mgnetic field (.8 ) yt k. Wht e the mgnitude nd diection of the induced E.M.F.? Thi i poblem 3-, which w woked in cl on Oct.9 m W L d dt dy b yt dx b W L bw L mv E ( )( ). ; clockwie; Poblem 4: The figue how od of length L cmtht i foced to move t contnt peed v 5 m/ long the hoizontl il. The od, il nd connecting tip t the ight fom conducting loop. The od h eitnce R.4Ω; the et of the loop h negligible eitnce. A cuent i A though the long tight wie t ditnce mmfom the loop et up nonunifom mgnetic field though the loop. At wht te (in µw) i theml enegy geneted in the od? Thi i poblem 3-33, which w woked in cl on Oct.

9 PHY49 Fll 5 Acot, Woodd The mgnetic field µ i due to the long tight wie t ditnce y [, L+ ] i π y k, nd the diffeentil vecto e though which it fluxe i da( y) dy( vt + x d ) k, nd o the E.M.F. geneted i E da, yielding the powe P I / R E E. Putting ll thi togethe, L+ L+ µ i µ iv µ iv ( ) ln π π π d dy L+ P vt + x dy R dt y R y R. 4π 7 Tm m ( π A A ) ( )( )(5 ) ln.4375 µ W ;.4Ω. dt (.3) (.3) Poblem 5: The cuent i i() t though L 4.6H inducto vie with time t hown in the gph, whee the veticl xi cle i et by i 8.A nd the hoizontl xi cle i et by eitnce of R Ω. Wht i the mgnitude of the induced E.M.F. duing the time intevl t The inducto h < t? Thi i poblem 3-46, which w igned in homewok. Recll the definition of inductnce: L i the E.M.F.-mgnitude E pe unit te-of-chnge of cuent di dt, o pe unit i, nd pecilize it to the ce of di Δ i, Within the time-intevl dt 5 Δt E Φ di Δi L ; ; i i dt Δ t d dt 3 3 < t, we hve, ( ) ( ) ( ) ( ) ( ) Φ (.33) Δi A Δi A 3 L 4.6 H 3. V ; 3 3 Δt 5.. E Δt 5.. (.34)

10 PHY49 Fll 5 Acot, Woodd Poblem 6: The witch in the figue i cloed on t time t. Wht fction of the totl voltge dop E occu co the inducto t time t L/ R? Thi i poblem 3-5, which w woked in cl on Oct.3 (.35) t / Recll tht n LR-cicuit h cuent it () i + ie τ, whee τ L/ R. The condition i () nd t/ i( ) E / R epectively yield i i nd i E / R, yielding it () ( E / R)( e τ ), mening tht t time of t L/ R τ the inducto voltge in unit of E i, vl di L E d t / τ L τ/ τ L L L ( e ) ( e ) e e e.353 ; (.36) E E dt E R dt R τ R τ R L / R Poblem 7: An LC cicuit h cpcitnce of µf nd n inductnce of mh. At time t the chge on the cpcito i 7µC nd the cuent i 8 ma. Wht i the mximum poible chge in µc (o wht i the mximum poible cuent)? q Q q di UC V( Q) dq dq ; UL P dt I dt L C C E dt I dt ( ) LI di Li ; (.37) q i Thu, the totl enegy in the cicuit, by enegy-conevtion, i U U + U q + Li, yielding mximum chge on the cpcito of C mx mx C L C mxu U q q CU, which, explicitly, i, C q CU C( q + Li ) q + LCi (7 µ C) + ( mh)( µ F)(8 ma) 44.8 µ C; (.38) mx C i mx U / L ma Poblem 8: A inuoidlly vying ouce of E.M.F. with n mplitude of V nd cyclic fequency of 5GHz i pplied co µ H inducto. Wht i the cuent mplitude though the inducto? Thi i bed on HITT clicke quetion given in cl Oct.3 I V V V V mx mx mx 3 6 mx 3.8 A; (.39) XL ωdl π fdl π (5 GHz)( µ H) π(5)()

11 PHY49 Fll 5 Acot, Woodd Poblem 9: A 8Ω eito, L.775H inducto, nd 6.5µ F cpcito e connected in eie co inuoidlly vying ouce of E.M.F. tht h voltge mplitude 3.V nd cyclic fequency of 37.5Hz. Wht i the mgnitude of the phe diffeence between the cuent in the eito nd the E.M.F.? Thi i imil to Exmple 3-E done in cl. d d XL XC ωdl ( ωdc) (35 )(.775 H) ((35 )(6.5 µ F)) φ tn tn tn ; R R 8Ω RMS RMS Poblem : A tnfome connected to V V AC line i to upply V,V fo neon ign. p To educe hock hzd, fue i to be ineted in the pimy cicuit; the fue i to blow when the R.M.S. RMS cuent in the econdy cicuit exceed i 3.mA. Wht cuent ting hould the fue in the pimy cicuit hve? The powe deliveed into the pimy i the me tht on the econdy (o le fo elitic tnfome) RMS 4 RMS N RMS V RMS. V p RMS Np Vp V i i i (3. ma) 3 ma; (.4)