Formula sheet 2 - l J A r ‘ 431m ghm’ehauz Ana 5 W a...

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Unformatted text preview: l- ' J A r ‘ 431m ghm’ehauz : Ana-+5 W a yr“, q aanncj saw! u an a «4%; ,4, a H can ls! magnuqu Ip-va qwfi- mhm W ?*'¥““‘aa"“*"'"5 Va : :34 in HM : h . W MAM. Awwaxawncfiwww’flws‘fi;_ ] IA __ WNW” *” { Suwammaw-cmg SEJIE m1, 3M“ m - ‘ ; - a»! amt-mm (ma-«mn- N 10115 W». 1%.: Ma ms: m Wma?nv4 musk “Mu-am! mm row a: fume-HM) m 1m 9% MM mom". 6, rm,“ \ 4 . 1‘ ~1 , "mm: m. *Mfimumn at m “Hm!” “Hm “wank, Wm”, 4..st ma“. w.) ‘2 a“. PM ‘n‘w\=%D(Ac]AT~J Wamux Xmauu 1 x (m 7&er film: )/u.m ‘ :Tl __ (M A” " mans: mum-am A 4-9 M‘L-rym‘ F . EA“: pr “.44 4.“, mg a k; ,nqe um“ (*fm‘) W M an. n,’ MM “‘1 tor-fig mqu (him awn La\\l:|=n§, -= a, 4km M m a) I“, “Am-fl A5“: A :Dfi‘rifi (“WW r‘hfinI1) w wmn twmu cu‘hehu. w Man-u— “ has numwum 4-1 a... u. MOI/ta” T - c 1 J ‘ , v m - . : 1. = A T x c!) AV hTA-n _ “man” .3?“ m “LENA Ru“, k“ .. hm Wm ; a, m m aha,“ , “av/oiflnfifhe ) q, C / n ) _ 1 “mm a A, h W «my, Wu.“ w 6 " Ezhgn aw JINNM);{Mika‘allwa}Magma “runway 1) x . ‘ x a: u‘gslg' w * v rxg , 1 = L. , Laimu x3\ ‘ . - ‘ _ a (s 3/3 «an/9 Tavim (Ta w/QMAE/ a; (“mm-nng mm. ar- 5.1-) Anu- :wmhmuk finch-HEW L A B HwDUAfmz) gv‘ 4‘» PM H, Man ’7? ' ‘ may: a. 5 nueLumunJ’Cl‘fi- m7,“ :u‘sm‘gmg afr‘wwfivmhm J/ql: M r - 7 a ‘ _ __ _ _ _ _ _ _ \ 1 '_ _‘ _ g '“' m- "‘1 ‘ud #81:th ins-53m muffin“. 5n” gum“ mummy durum, j: x C“ a: rm n\.+.m m Lyman»: LT 5: we in) r r , sol-95‘qu ., j, : T2051“)! ‘ _ 4 FM fin: Vmwmmac inn-«m» mam-m flesh-w" m m: wflflvnmm In: a! ; are : SE w: mm Am, -_\r« .355-mmknne:u§q_ mi"? 4% -‘mw wazfes Md N, w”; .- ' ' MMMU "My wamw/mv more as w «m at (Howa‘ av inn} A . 1‘1/u‘mm e\==hwr. me u was ram-ma gym” 1 Ana warm at u run-H4 j“ , gov“ m Q ) ¢ M‘v'vw “*va wwmnv a? a Md 5; (1)4 w, Mum‘mv‘vw “ “I’m” ‘I‘\""' A“"’“”“‘/ v “a ’r“ “‘ “ “mm” “" °“ ° “‘L“*‘1W‘I«‘°=°M'\"AD 1‘ ’1' s‘ M 3 vaur“ . =._. *«nu ‘ ‘ Th: (4 many m mu {firm Adm“ 5M “KL ‘M {m mam (M ML “bk i. a w’é’m 31, is , 91—? W311 ‘ : um. :4, AV v... v.“ . m 51.55“, (cm). Mlle am mmewj uA® ‘w “hm; @w W ” mm" “"1" Mm m 95:. v? ike mmqu and «ma 'r: alflimdéun ‘ 1 " M ‘”““‘“M "WM “M'W‘flm‘ WWW” d‘Ktvr-«u; Wm _ . ‘ A ' JJS' " “4'” L “A a IBM-v, _ F _ _ _ _ _ 3m.an 1x Wm -. / .m n-\ (M: 03.. A [2" =0] — i i ' ‘ MK "'(B‘wnflngk‘qi'fiu) = 4314(qu ‘- ‘M " H bvm- (m ujh51‘§°fiaj: m4, (WV) - .Hw.‘ «2‘ mm wu Linn/k, “war “a .cans 1n Kan-(4;; "MI W, (W, 'M “A - [MIAK inhxlua] L2) 'nn %us 7 E.an mm an -. ‘e-rlmwmwm 4.4% m4 mam. 1 mm “b m. gammy“ {MM TH Emmy“ .2 a AP.“ Wyn—w lad: a) “,5 NM»; argmromsxfla \s :wm 'm‘a 'mCT,‘ =CRJTJ St: A, e m. ‘m «h! immuna J an gmva Jag” E mwmu arr musmg w f u m I _ ‘ ‘ H \ A? 4:“ THIIQCA am; “A gm:ng ‘erem. ' “3* WAR?" k) =1 3W7/C313'TEQ " a y” l” " " “A ' 5*<‘N'""fl"‘/+ 5~~'\“V~ 'AVNQ (a ‘3 01min 0. m4 A ' J. 7(D/1YX4c/Ax‘)‘ A> 'fiub‘vfl army”) 7.241% arm-Wu .; agar." Re 3.. mg... a; no,“ Mm I‘M: b‘ 1;,” V,“ M H a“ qt: ’ 'rw m‘cmficwiL P'urvima—w‘w mm“ 5" «mm: mm“ 5‘ 3/6W m h ma +5 Emu" ‘5) 7MB = *AcR‘T a QR“: = moxwmmam-mww 5“?“ "WWW “a”. w w «mg mm.) ann.( as g“ m A F a A, r m a . I q DA 1 yak mm 51m .045»sz : u ‘ \ , I G_ — 9:! L S, M! wM-«e 1‘ WM Sumgiaw. “A n ‘5 m mama: {Mu km; ,3 a“ (’ ngweum Mn (‘3: A K Y swam“) /(j K + 9W 5 43pm Z) J ‘ A 1,7 :' ’ r: ugh- mm «M53 w-a‘ P‘ov‘a A m a 2mm," W A‘ a? wad-ms. (flmrhegvenI—lnimb-IMAEMMLmmx:fiflxlun{mfm)whmh ' ’ mid-nlaflwdizkcmmswnmm-VJ m m “hummus! w : mm.» :chan (l) which m. “ppm uvn . 1- m pu‘md A we, NM mmj m m, 5%; WE.“ Mia,“ h M u A5 mammsniNanmmmADP-mvwimpxwau:mxw", a: Rat-1m “m aw magma puum— ‘ rE‘bW 53mg: :r—r; m5 \W Iadnncz Lg“). «m “w WWW AW": W 51mm“ 9? 1 “ms mum :MhLSN‘i?‘ w H Ltm‘; = §1Sin>seh -\) » hmmkummmdzuw.‘m,=msu=nlamismmmxmlmunarpu Jfinb'sarhmAanbeommmlmlmfimimmffimegh I _‘ w *0 “WWW www.mmmmmmm 5:2;10 =9 :9 ,9 [Ho (uyfimmymjnycj mmamwmmfimw Wm mmmmmmflmmw. n wmnmu-Iynmyan-u mwafdlflvinuhol may. 0am “~"5W awn" ,5 Mm :wr 9“ A5 M argmn‘. [45w Tm"er "T. «A w: y ww- , ‘ , , 51? 4* '5: ‘ . . ‘ . ‘ . . t‘jmrMMa-ww «ma-mews Wm“ M W .\. W 4H,, 3‘9 W 4;, mm“: mm mm AhcreCMewenmnofmsamphmmhmmmmdfimmm “w °‘ W" ‘W‘ W mung ‘bflfimn’r K 0.." Lu! u'hmmr‘ an“ .s -: 14 miumfi H MW: L M, U Wm“ WM g mm; A re L _ .-mmm-mmdwmmk‘m‘tmummmimu: ("manmmwmtmcmmn - xzjmflmdimmymmmmfmmbingmfimmfind _,uj,_ln10 ._ 1 ' (u- f?‘; gawlzgazazgfu'mmamamcmaidum:Huwumumnmuflknfl’mfi IthishuhwnitnfimillimnldlimMindlemd{cm'1)audlis&clighwavelmghin “""' W W I f'scspmeinthmmmll - ""*’”‘T"T’ sug‘rsg'pmvitharflnh'nny. :zjumbxw'wmu mwmhtgpwhmm RWIy-nlivsflumflml‘nrmmyudI—pwmmylivclfvw>IMIUP‘<L mm mm aim h V. w Ear Mm =5‘4wrl-Dx mu“ "M Q“ T“ «kmvmfi; mam, S: Lug-,5) Sim Q“ ' 0 w M I... By my MW pmm = Fem-my n1 w puma. um _p_,~wm;iupu_wtwrmw4)yu a '5) Km. w: MKMKrnuA my WWW Th: h PH» Su " ‘ . ‘ W‘anrasmn 1 5 “gt-“L‘hl’u‘w’v -r.._ ‘ ‘ A ,3 . ‘ r ' - : mo: u émwl m mm. “1.41 mom a 1. 5 ma :1 ~ “5 a“ >§$A ‘Ithmglmmm m ,7 "an". EM‘_F‘._U_J~YWN amndwdm Ami \w'vr L mmfiu’) mane «:5»: W511»! ' 4» swarm \qu "udsw VN.|5AR mm s‘ ‘. ‘A'Jhr :brnrgw E1 “Smrflwf, M) as m um;sz a; 1 mam, fi‘qbg-E-m»: (my? ‘ "‘2‘ «WW-«1 «3? § '"Ww‘": W «New, Pam him,” ah llmmnummum-minmmfi mmmlllmmdmlmm-l ithnmmumywwmmcuhuznfr-rmIL-c-l’nn-nmslhzums) Mmfl,wfimm%4 “$J:R5fi$fl“w”” -~ 'mmmm mmmmw“mwwmm"m"* s) iflhqme q) , 9 $9 you”: given”: mm‘beewu (awn “on.me rarW-"T ‘dmmgmm w y ' m Hm:Fab-PimdenmmeNmmbfilfiMKM":U—HP H “ ’Ju T a magma” flammsw 55815:“;er 5- “#1704” 0, elsewmw, F31 ' —’@—’ 5 wfl=llrur=9')’]" IndWIJA.I=F!nnnbe|aw R Wm”: :1 i.(')+fifllf.(r) :0 w] r H! mm. .5 a. um: a: If! L-Ipllne gum. mm: cinch-k will of lpoint chug: simngu r20. Sou): mum 1.! given by am: M. u, ggéggggo. :Iaplaciancmbeminmas NW Wegoblckm [he mural DDE-ndchack ha 5:: iflnmdifiuhm Mfmcanh-mal sun-um, qr): “l. “335,731. mwmummmd we oauin 31:3(i.'=‘ “#79 rd, . _:r{:j_'."+22-"['H-B"u"f, Pill them back In “IE-11E of EIDE: d f+3L_£f dr‘ r d? D =¢"(f' -+3;;'+ 25}.'+H’f-+g&',)iif r y D hunky-value pmblam is Ihm given hy 2A3 , 2A}? a ,. ‘v-Bln- #1—31 arr-gr we um immly me mu um um soman an snn‘xfy tbs ODEifwahB’de, ‘( “,6 Using in: above luau“ and n-hinx. «as w.“ nanny use an even vial soluu'm I: "u:As;nnEI:A: 1—»; _ F0 ,. mmI-mnr‘mrremmmu l;A - -t __A ,-_‘-- w 2"“ ’5‘ ’ 22‘ ‘ g-§(y.-aw)-If,(a.-+..r-,-$m»+.m+§[a4-) , “a "‘ inmte[011=f(rj. Wemdfimm Emlyn whattypeoffimotimm g+§gi% Namath:pmpasakl’sluokatasimplctODEby:emngw 11.2.-JI .-A-.-A..VA ,5; =1'(-(.'78£ )rr’(s('+m 3+7”: 7: )*’z(fls oBe")-y,(z"-z )uD .m L. (b) We will magnum! a general solution oflhc same diffusion equalion with a aflgfllm E DV'C- rC +1 = 0. Given the spherical symmetry for C(r)=C(r), wccancxpmsa thcdifiusion equation mdimaynnnafim) " m DddC __ fi;“?*”‘* F+7E—FIC=“IZ WiThP2=meqz ‘4”)- Th: mu] solution of Eq. (2) far (.2li and 1' >0 is given by 1 L2" +13?” C(r) = % r um; um: mummy 534mm m. we puhnhiliry In unildmac-qupulmng In: am“ we; «asunmzmmmmwmm 1 «(:pomgim by _ n = (ms affirming bum)vrpmmuny o! populatingupper izval as: m bumm_ = w...» 4r FM I I m mi—meJ-vm‘" furT=310 K This yields ‘hflmrw me am mum» Im— Mfl“ I DO mined m2 Is 1 1 = (up) =ap{m.1:r~r —9‘3xm"(x) ~ 2.9110‘ (yr) mm. ma :mpcndmth: m ilmizafuroocummuflpummm: ammo-um -... Inn-l wan — nnnrlj' impouibdr. [mu mu Eltn+riz €i’2-IJ e4? '3. Jlocn+¢d at His min rifle puma). gardenut; Sulfma} r» rs iii/am 53 E, (fl *- E- {1:359} +m9é) ‘lWE-r‘ die-u in ~ (new). I; Hermie s gum-J 4.94.: f, The Mdgimum frilnoili: ('mna‘lTei/ES afifimf‘ Ul r... Sui wlxibln NDUA/dr‘o ml 3 Ho. ianvyfi e? ‘6‘ in a. is gulkn is.n _ i [Wei e , w e m 5-. . PM ' {—2— L flood Hum/a.) . Tameka, ‘Hu. some if mimmm gnu-m —> TM m, =0 55A eeiegieiw s4; (3‘ r, nzkguzd whtn pl rs RN15! hE‘ PMM‘L (l'fiya'm-bilgeo‘ Assume“ ‘5. i, leg/+4 .x e «and; w). m 1a,‘ Mi 4" 5“” ‘ wile ' (.)(M«.+‘ia.‘l+ rm‘illa-‘W/ll (0.704“, is i. iieJ mm...) on 44.; dirgogon fa, Amide” M 444. .3- agrs Puflm £J: emu. .3 MM 2 D -_- waif clf (. )l‘f-i/m a up Lia. 441A w {MP‘ Hinge” Hm.) - ow; ln-nyUL Via A w an; = eerie)“ (c-m per HLD. xenon) = "snu’fimb, w: end/Mt m l Gianni) {NAN m-ili-l- (ll/l 1“aner “add M mh wafnm s 77.. mygjfnfl u lew (I) Rum“! M1 LL ‘ 1L J .9 f? (llmbdnmimfn lemmas {up an. mm. .in, m if - ed in a We w “glwl‘ymirblh‘ wilrsimr .4 umpdr pzr kl, L1 +9 es4.ma2&. Hm lide Emma“ “kipwi “in” “$13.43 “madam 5 AMA,“ .L b: 44.; law . ‘ ' " “WW ,4; Hip “And” M i K: ii; “31. (“him awhdwhrwmniinuiiididi [q Folsom r15nl“; e-i mm 72 7Kr+ lulu. I {Ase ‘ an or.) “44-. H. 7 W44»... 64? dim-L we £1. (AM/v M ‘i Examp onsidar a anepam’cle system wrdi onlytwo mates of 21:0 and 51:2. The" die panition funme 2 is qiven by : Z : 251.3; : Heep-m OX «14:11)? 4U! 81}; l 1 21 i “we in: 77+7 7 r e or { zni-erDi J4HDF(4DI)) l + l ( x” m..- “i=9 xiii/Kim @er: 4Dr‘ a: Mr H: [G{Y—Y‘7HI‘(Y‘ mid!” for r; p(x,l)dx: probability of findm die particle at MHZ a = W(m,n)dx/2l Consider die light dismbulion L(r. s) in a Sfl'l'llrll'lfil’lFEE turbid medium occupyim die haifspaee ofz > ii Widi an lnten‘ace w'rdi fl ridl I 10-, min (W/ml) (r): I' ilr mm (me3) £3 1‘ Mn: =1(..iris:=ixio"‘lsi his ha ,3 am Qui- rneem ulcwldiB—A imimr: “My: i-i-Lunh .r iii—i in owl AF (IT/rmluJAl. (KS/run M (Kn-ii lM-ll art hand hr emu/r33 uiLeLAi-M mind”...3 Mei-es is “Md .F mind» .e M .n who lmnlz (mi) lcsw‘ri-n) in L.I)11)li'u“ "James, 4 I (keel/mil = HHlei/mi‘l :i m )(ln'l‘ (y/miemil A wuurm'WIV/m/uwfll ~ L-qfl‘FD" (~j/Hafepu1‘l a i (id/mil = Lb‘xm“ (“j/More‘s“ h T‘ ‘l U : {Le} , flatworm L; fl, 1519 .3u116035 r [u dr P in :0 whrd Mis. e mm.“ "Lei u .x ('Ll'rifftf\l:dlflf(-nl Hu l" i.“ “Md.” in it, .limiirr ,m .r morale“ MW don-(rum , lull in ,...re slew. J lime Min .l» Haw! u «hi1: eh Lid ‘(FW' fir Hie minim ' air 31 :0 and nisne-iiisiie irrirlenf hmm nf i Jr, 7\ firm riririi in (u+ << Mr H » 3 l: T AV:p:"_]/":th_m AV=IQ7VM=EECW increase 9 cm. q cm ‘9’ Nat , Ii = QiAVirr AVi) “‘“E b | Maxims 5'" _Z. ‘ “we ‘1 (u "ramp-in 4n] n drive n. _ m ernisiimin The inteng can be evaluated as me followina by admin; Fg/IZ'aXlS )a'ces a = g V-DV¢ 1' 7 <15 1‘ =73 1‘ 7S 1' 9119 - r l r( M: r() P( ) .() an “1 am A _ led ively as s, eiiul 110' omen»in 4| Sp(s,s')li1(l' s‘)dfl‘dn Hr) my): [mi- syneav Dlsmi- sun NOW we can Drooeefl m simpiii‘ilme 1‘" equation spilt out oi me r_ 5mg rediauve uansfei equation on L9 woos)eeir;;(rs)+ir,_[ peso“.- 53.1mm.- s)+£ (m) i. \ Fnr :i <igcnirinfinnfi hmemis irieriiiirn nmi m 7)" and iiiiirriinaieri riy :i miiiiriaied lm‘ldFfll’ beam nf very wall iarneier (0%”) :iinnn ya me 1315, we can wrim me radians of me primary miripment are inside me medium as Since me modified BVPamployim only paint SDLII'CES, so we can find its “Green-fu moon-like" 50le Within line phantom volume (2H1) as q> “FL”: mammalwhw fwlflffl‘l Atthe [:ltigntonivmrfacfl Eif 2:0, ere solution becomes 12ml,“ =¢.<p.o>=fi l WW Substituting] Ii'i'is soluljon and its zderivafive irlln the expression of diffuse reflectance R(p) and takinci 2:“, we find if. FREE“) [mi—’1gfl) 7a“_/‘grfi) H‘ P1 P: H: 91+ W’!‘¢ P137 arrival: .05) l 3‘1’, (1') (pi = fife -I>. mi.“ + w a) A 51> :R—Eh -i>i(r>|__., + db 5;“ Pa} = An '{1 e R I" mfg”) {Ciel +Drsz¢Az +80) emergewfiw why " due in die bound-1w conde E=U so cm 7AM we)? v ' VZ(;)=AnE(r) i ,r I , 50 as re ii, G e) Mr and (die 2nd irrm diverse: iriiidi slower than me 1g menu and is drowed) menu) Ierv a my: erv ...
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Formula sheet 2 - l J A r ‘ 431m ghm’ehauz Ana 5 W a...

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