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M~~6-:ed Name: Unid: _ _ _ _ _ _ _ _ _ _ __ 1 Exam 3 Discussion TA: _ _ _ _ _ _ _ _ __ Show all work!! Report all numbers to three (3) significant figures. A wire carrying current I has two long straight sections connected by an arc of radius R centered at the origin as shown. Using unit vectors (t, j, and k [z-axis is out of the pageD, radius R, angle e, infinitesimal angle element de, and current I, as needed, find the infinitesimal element of magnetic field, dB at the origin as produced by the current in the infinitesimal element of length, Rde. A. )Jo~JSXR df--~l fl-J~ K~ -Alole ~ tfrl RZ B. Arc L{nf<2. yl1R Assuming that the straight section ofwire in the figure above extend to infmite distance from the origin, find the magnitude ofthe total magnetic field B • at the origin. to. Bto•• = - - - - - - Alt~.--n~!tv~ (/';J kMW.'~'t £'M'ft l( - S1'Ivli1rt S hurt vi 0'("" B,'o t dB' ~ (z-axis: out) l{jI\~ th J~ -::: R~ (} V ~,'v1" i h~ V'l (~h+ haV! Ct{ rule w{ h~l)f- F Vector dB = J !Je I... tOdf- ci J.(o I W~ ~ bo lie. p ; tAAoL C. 4 R .).1 (J(' I f'\.Q. fA 11 Wi re V\J.{ ViP"JL is/( r J 01 (j ),-0 W/'f'.f5 r"/~) titS Kr -=- 0 = 0 , Us i.-,t de -f',.o_____ t01 K~rr 8~ 'd ~ =: An infinite slab of thickness L has its mid-plane in the x y plane. It contains a uniform current density J flowing in the +x direction. Find the magnitude of the magnetic field above the slab at a distance z>U2 away from the midplane. 11 B= ---~~~=-------- two sfyq"4ht VtCltV( V1J ols ~; t( \\ loop;: ~ , "L 7-"kf- t:: I JJAoIde() Ll rr R. -lAol rt '-frr~ l~:= AoI If if( z J l h W~ V<,~ A"'fe('e~ LI7JIN q~ ~k~ t\ '1>f/~re f/lj-~ 4. for ouv-- Irflr..,,,ier, {OcJf', T~(/s jr3,Jl -::::){iJ I.eI\(lo~() tJ-l'((' jf~. J..R =::: B·~, + f$, hi_fi t6· WIJuJ'/<,1,+B. ~f'1I1t -: : . e"VII ft>tis w+0 .= 2. II? W r;o "" OV( (<<se If'll - :JA "- J twol ti)uS L~W::: J'W L.Mo 3:::.}= Ph~ SJlrin~ ... it·... 2220 Name: _ _---'\.tF-}J~~4-lt-l_ _ __ Vnid: _________________ 20 J3 Ikn 2 Exam 3 Discussion TA: ______________ Show all work!! Report all numbers to three (3) significant figures. A. y A square loop (length a to a side) lies in the x-y plane and is bisected by the x axis. SI as shown. The loop is pushed in the +x direction at speed v into a region-the 1 quadrant of the x-y plane---containing a uniform magnetic field with strength B directed into the page. as shown. Find the magnitude of the induced EMF in the loop and the direction in which induced current would flow (clockwise or t. f = ~ .Jj ) counterclockwise). 1 r:) <::L V j 5 at z... a -1 (n .Jl E= Bn)lnlc~ 17 curre-n";:;t-d-ir-ec-t-io-n-:----,C....-c.:='" W ------~-------- 1 !3 ==-(-B~x) z L.eM..1;'> B. Iu.w t-ft .:L ClA.rr~1 tv off'ASe decr.e a 5.1A.jfjeld.. ow couvJ-ev- c )"ckf.Jt'Je An inductor (L = 0.003 H) and resistors (R1 = 1 Q and ~ = 2 Q) are connected to a 12 V DC power supply as shown. The switch has been closed for a long time. What is the current I across resistor R2 at time a time t = 0.001 s after the switch is opened? -t/r: L 1= r((J~""I/~ G./(pAICt):::<Jo e L = R-z. -::v."ol5"---"9 f\{U5 I ...Je: '1 - ':J ;:: 12 ft C. A series circuit has an AC power supply [V(t)= 180 cos (100 t) V]. a resistor (R = 3 Q). a capacitor (C = 0.005 F) and an inductor (L = 0.05 H). Find the current I as a function of time and determine the average power Pavg dissipated by the circuit. 1(t) 1.(2.1.("3 u.S- (,,,.d Pavg = '2..7DOtJ -1') "" _ I "f - ¢:::. t f{ Ll.trr-u1.7- 10..<;;-> fM-F 11 '2 -r~.x l~jn"L.f-(XL'-Xtl.-) II'I('V' -= I ~:: '-f.LI{~ = L o l(t)= -r~ ..'"'/' (o5(wt - ;:) -I., f,.... ... Jt ;::: R II-;:' ~ L = 5.0 52 .x,,-:::fc=-7.. vSi.. J.t.... -Xc -l --t-tt 10{ L ~ ~\AAA~~- R, t> ,- =- 41Z-4~5t. L./ -z... <-t 3 A by ~ r? I ~V, C c -= ZI.,... ~""'<I.k c"'I.ct.)I. A. {or, Name: ~~-I:';h Unid: ----------------------------- 3 Exam 3 Discussion TA: ____________________ Show all work!! Report all numbers to three (3) significant figures. A. Unusually powerful lightning can develop from the accumulation of positive charges in a cloud during a thunderstorm. In a simple model, the clouds and the surface of the Earth act like a pair of circular conductors (radius R = 2000 m), forming a parallel plate capacitor with separation d=500 m. At time t=0 the charge on each plate starts to accumulate and grows according Q(t) = 5t C, where t>0 is in seconds. What is the strength of the electric field E as a function oft> 0, assuming that E is uniform (as when d«R)? J:,,-t E(t) = ~ R;2000 IV\.;O~ (' :=. __..:...;:...:,=.u._______ Cyl;~Jt;(,.~1 (;,Qu'J~;C\1'1 s4{'-~q[e cOfl-tCiIV1:nd , ~J L\fP~r fl~N' . Ci1t.-Met~cJ: 1,.... 2.;" aQ(i:) j 'dv E' dA =' E ~dA -= E(t) B. U<:x' Mlii-IVC I; Iqw~ C. = n£.. RZ U Q(t) (t-e!o o~ id€O/ ti-pl Qtt {iDO"" 10'1 k -= 0.0000 ~t aGli-lX) b1 T c.:::. 2. qq7~1.IH~ ~ fl - Ltf( 1 ~7 T'Wl '0 17 - A ~ 1L£"~ A storm-tracker helicopter hovers 250 m above the ground just outside the storm at a distance r = 2100 m from the central axis of the storm. What is the strength of the induced magnetic field that it reads as the charges accumulate (t>0)? [Hint: You may give your answer in units ofnT (nano-Teslas). If you do not solve part A, you may attempt this part symbolically-define your variables clearly.] T c.:I'("u!qr AWlp'ti\' Qn p Q-ri, B = 0,4-7(' (A... . f~d, h = 2 t;o Eo. =O,Ol'f~ ~ w::::: '1.44(0 I",f E. ::; To ~ d:: WI :r..,:= 5 :t -C- -=:: f'\1 c U.Se ICjw) ( (J,qu'>s 2. 1DO .c. VI O~E.' --.... J"I\ B'd~ :. fo ICc> ~ B~J~ = tdCe GDVlCev1 on ft.if 1 ~t (~E'JA) d (~\ ) B{r): /!!:!:::21C r ""Jf 1tf~RL j dA fa reo 'yl"~ ~er B( ) _ ~ d ( Tot '(" - l1C'r eN: T(f:.. R~ 7( 2.) R ~~ "" if,7b ' l0- 1D T ;::jO,/.f7b~i1 qoo d IlAc. \K W1.~OSL{,(': ~ .J,4'1 ) B(.) ) i ( 1'\ At after about a minute or so, enough charge builds up to trigger a lightning strike. The transfer of charge (and associated production ofE and B fields) generates electromagnetic radiation with an electric field E(x,t) = 0.0143 cos(0.008 x - 2.4 xl 06 t) V1m, where t is in seconds and x measures length in meters along the path of the wave at some large distance from the lightning bolt .What is the wavelength A. of the radiation? 1..= 785 WI - c. 0 s (I~ x- (» t ) E ( 1<; ) t) :::. Eo "Aw W G -=- AV = 'tIT == k A=~~G = -:::.. ¥ =Ei&s-'~l Ee> R~ ( e((l"x-wt)\ J