112-20062-MT2 - Q.1. A cylindrical wire of radius a and...

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Unformatted text preview: Q.1. A cylindrical wire of radius a and length L is lying 7 along the x—axis as shown in the figure. The resistivity of the wire varies as p(x) = p0 (1+ x2/2L2) where pc is a constant. A potential difference of V is applied between the ends of the wire. a) What is the resistance dR of the wire segment of very short length dx, at the position x, in terms of po, x, a, L and dx? chat '2 SCY)‘PQX/A (lRL'gJa ((4- )%§—L b) Calculate the resistance of the whole wire in terms of pm a and L. L Rzgoffl. 6 L ‘5 .35. 29 5 .1: QB ‘ - Z _ T4136“ £519?“ :_E_ EL 5 "Trqq- % 0) Calculate thecurrent passing through the wire in terms of V, pa, a, and L. :1; = a... a V R 3: t, L. 6 tr at 1 6 \I we“ . ‘1?- ‘g 9 L ' 3 d) Calculate the current density in the wire in terms of V, p0, and L. )5 -:.. E. A a; it); 7t L51 L‘ e) Calculate the electric field in the wire at the point where x = LB, in terms of V and L. Eb!” : (W E u_ : Aggy“. (3‘ go<4+ L1: qL’YU 2 l‘ Q_\ |_ 1 t, 6.?Wm. J V817“ ‘1— L- ex < 'iLS’o A E.a\8V B £._u2u0\{ C .115 ' Q.2. Directions of the‘cu'n‘ents and the numerical values ‘i‘n‘i-‘J 3' ~ 5 5' VF? 5‘: w "5* ‘ " r" u 5 ' u a -, ;. 7 - _ . 1 =_ . of emfs and resmtances of the mulltlloop curcmt are n sit-1:9 ‘ - - ~ ' ,-- u .. -- » I ,v ,1 given on the figure. fezss‘mamrm ew- ajgglusga cs: x 1.5.: 7: .;;.,-,-3;,,-;r;.«, ‘J ‘ A: :21: £3121.qu .- ' ' " l ‘ ' an a) Write down the Ioop equatmn for the loop ‘1: 5m m, .e . . = , , _ “f: ABEFA (Use the numerical values ofemf’s and _ '45? MM.“ .! _,_ “Angle; 3.1% {5 W fl. W. b resistances). .l . .3 i Us: 7:. V . _ _, , .1 :ij. Us???” I“ _; fig. 1, ‘ 4: +4.11 ‘+"«:,“I, =0 ' ' ' ~ ' - - ‘ I b) write down the 100? equation for the loop BCDBB. (Usetthelnhmerical values and i resistances). Jz~§44_r323_1;ez=o Z—ZIJH‘IZEO z—zr3_61;=-o c) Write down thejunction equation for the junction E. (In terms of currents I;,I;,i3). 2; ""‘ r3 " r2. I; 4- 2:, 2 I; d) Calculate the currents 11, [2, and i3. _¢e+aIL-eézri-fsl=° get—“l -::E£-=-.—‘LA z -1? +102“ ~41" =-o z .5 I‘fz—fan‘fa-(HZIESA *4I4I‘3jllfzro I I;ng _______‘_____________._.__.—. 1',1A -12. +u1‘z- D I;-rA II 2‘.._.2A q .al‘ e) Calculate the potential difference VA — VD, between the points A and D. VA_V0 ___. g’r’_[_3r3 ._._ 4‘3 -{x{..2.}=l 421-2 e/4v wig—v0 arr—1+529—r3—1-rl—1JH = “V “V 1) Calculate the total power dissipated as heat in the resistors of the circuit. A: .3sz, w 3‘” flufa‘firéw [92:17:21, 1W P4213224=4w F550“? #7. : 5‘0 W g) Calculate the total power delivered by the batteries. Is the energy conservation satisfied? 3} a £5, ,3“: a H w rrvrflw'ns) 2;; :- uffal EL 5: ~4W («bray-bang) _ i F- few (2.3. A wire of mass m = 10 g, length L = 20 cm and resistance - 0 ~ .0 R x 0.10 9 is connected to two identical springs each of spring . constant k : 4.0 me. Other ends of the springs are connected to a switch S and a battery of emf 8 = 10 V as shown in the k k figure. The system is in the horizontal plane, points C and D are ' fixed and the wire is on a frictionless, horizontal table. 0 o E The wire is in a uniform magnetic field of B = 2.0mT, which is in vertically upward direction. When the springs have their natural 9 0 lengths, switch S is ciosed.(Assnme that the springs and ' connection cables have no resistance). a 0 a a D @ 0 L’- a) Will the springs compress or extend? Explain briefly how you determined your answEi‘B; .Tlne Spring/s an“ axiemdia bacaaet l‘ “T: i a I ilk dives/from 6:] i]; 15 awn! from #18 Eating. 0U!“ extend q . 0 S) b) How much is the final compression or extension when the equilibrium is reached? ‘0 __ i __ l0 .. i a a. fierLBsme—UHOJHZMO :)3m90=]+,‘(m N In €?uihbfi tum. scale.) {23-} (force (M the wire ,‘3 mam. Hence, lit)“: F3 -2 _3 =5 7%: F“ = “to N ==5Xl0m EL: 2K4N/m Xa=5xI03m 0 it c) What is the total stored energy in the springs when the equilibrium is reached? -.?2 ii =Z/3<(-%)<x’”)w= tN/Mxfim m) a: [0-11 éouiex ll II ‘4 I 2 l0 you. as 0 if d) Now the switch 3 is opened suddenly. What is the initial acceleration of the wire? when Hie 5W\L+Ci/k is crested) F3 30% 4:: zero. Mai orce examine wire. is ,zkxo. Home. 3 _ _ 231; SM)" ztxaqizao =3 ao-jfi—w-rmal scam/Sal G) e 6 I 3’ Q.4. proton enters into a uniform magnetic feild of B a: 1.0 k T I i I I e 9 ° _ . f with a velocity ofv =1.9 x :07; m/s as shown in the figure. I .6 I IEII I I I II a) Find the magnetic force acting on the proton at the moment it ‘ _I I II enters into the magnetic field. Give it in unit vector notation. (a 9 e I A ,I ‘7 I A - I . l . F a C1 v x 13 II . III a a e a A? m I x @ 4mm cflI-‘Mo /5)(10T) if I If a o o .— {L A . N» a ,m ' b) Draw the orbit of the proton in the magnetic field on the above figure. Calculate the radius of its orbit. : a *1? 1. I? F m 4. WW Unix“? x10 “/.) I r ‘15 (tamo'mc) (4.0T) @ “i N lxlo'lm 90‘2M = . x10 m ’” H: c) Calculate the angular frequency of the proton. q I. . O “1/5 3’ a): 5L: Ml" : 93w) ract/s r 0.1m @ d) Find the period of the proton. , .l“ 7~5x lo"f fora/S _ "l flew" 6; 066% l0 s CUT; ’- '- _.— @ i w (“1‘31 [0} 6" "T --8 6 6x10 3 e) What is the work dOne by the magnetic field on the proton during the four complete turns? * 1"" “" -» “were: (dam-Jam [Fiat-t: hfigtt ., {2ch MCIIIIIIIE «at for-’1"? - flhb --5 Power: Fa V=O because Fix: Q.5. The3 current in a cylindrical wire of radius R : is not .1 :u. 325 an: utter 3:4: w uniformly distributed, but the current density varies pi -. :3 -‘- ,:E W- 2:12. .m m ‘ linearly from the center as j = A (r/R), where A is a .3."constant and r is the distance from the axis of the wire. 2e : ® a) Determine the magnetic field ( in terms of ye, A, R and r) 3 inside the wire, where r < R. . 2 ~- ': Em”: .I r v 3m? .=> glean v.1- r=53ts=get~map §BAQ=B§AQ=ZWVE 1:2!!5. Efflzém ‘8‘ O A WA wwg=fl°fec V3 :1“; Y: A 1.. B=fl03R r B:fi.\r" ® 1)) Determine the magnetic field ( in terms of no, A, R and r) outside the wire, where r > R. at: . g " “I mfi°£ 3%. L I: g—gv zwwév B =. VLL‘R “L— ° 3 r x_ 2.813: F;— :i“ g ‘3 AR“- Bg/N 35“ G) Show that the equations of part a and b yieId the same result for the points on the surface of the wire, where r = R. FH'OVVK M'k 7 = “L 1' ‘w. = = ch: 3' a) B y 3K'r 5. ? st=~> e ju 3 A FMM b) _¢£‘3—R; §nV-‘ Y‘='P\‘ B: 0T3 ...
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112-20062-MT2 - Q.1. A cylindrical wire of radius a and...

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