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\ h>tjiRYb pTHVWx=ijp\|xUhYqSJk QSx x=iVUjiRcTQExcxUgp\|x:xygvh~ \AWTHVWQ x TJfiq`\JRU\igih>jiTH~hYji\n^\Jj 9eh\Jn>niVU\|xyh>qS NvhYRUn>VUQ:9ehYN+9 Q`JjiTHVUQSji QSjiNiQ P_Q`Np\|q[p\JV xyQ`n>q`h~TJxyQSTHj \JjvN \|iiqS_QSji xUgihxUT_THq`RTJ iQSN ~hYnogp\JjvQ`nYRxyT xygih RWx=iNv TJ[vq \JRW \Nv_jp\|QSnYRYb1 giQSRzxygih>THV Hk jiTutOj6\|R] \JJjihLxyTJg _NiVUT_NEEji\JQSnYR ? lazZ L kvQSj xUhYHVU\|xyh>R xygihzh64p\|xUQ`THjiRfTJ nYq`\JRURWQ`nY\Jqmh>q`h>n>xyVWTH\JHjih xyQSRU#QSj xUTxygvh]R xo\JjvNp\JVUN vQ`N T_Nih>qeQ`j TJVUNih>V xyTwNihYRWnYVUQ:9eh]iq`\JRU\c~TJxyQSTHjb pTHV n>hYVWxy\JQ`j q`TGt NihYjvRUQSx iq`\JRU\JRYkQ`THj6\JjiN6hYqShYn>xUVUTHj PHhYqST_nYQ x NiQSRWxyVWQ9vxUQ`TJjiR \G jiTJx 9eh l\B_tfhYqSq`Q`\Jjk \JjiN l aZn^\|jjiT2qSTHjiHh>V 9eh\Jviq`QShYNb M ~THVUh nYTH~iqShL xUVUhY\|xy~hYjCx'_jvTutOj\JRiq \|RU\1_QSjih>xUQ`n xUgihYTJVWgp\JR19eh>hYj6Nih>PHh>q`THeh>N THVzxygih>RUh Q`jiR xo\JjvnYhYR>k 9vx C tOQ`qSq TEniR THjl aZ \|jiN xygih VUhY\Jq`~RzQ`jtOgiQSn gQSxwiVUT_N,inYh>R~\Jn>niVU\|xyhVUh>R=iq xyRYb gih THqSq`TGtOQSjiNiQ`RWniRWRUQSTHjNiVy\^tOR VUTH ? L k ? DHDGL k\JjiN ?AFCL b

l aZ \|q`qSTutOR iRxUTcNvhYRUn>VUQ:9ehOxygihORWxy\|xyhOT|\:iq`\JRU\]Q`j xUhYVU~RT|\ h tN9iq:2P|\JVUQ`\9iqShYR>Xxygvh Nih>jiRUQ x

nk iVUh>RUR=vVUh

pkh>q`h>n>xyVWQ`n\JjvN \JHjvh>xyQSnMVihYq`NvR

E\JjiN

BkniVWVUh>j x Nih>jiRUQ x

Jk

\JjiN pTGt PHh>q`T_n>QSx vbzdfTQij xUQ`jih^\Jnog TJ0xygihcPHhYn xyTHVWR \JRxygiVWhYh2P|\JVWQ \9vq`h>RYk tfhwRWhYh2xygp\|xt h

D jih>hYN D S Q`jiNih>ehYjiNvhYjCx h64p\|xUQ`THjvR2xyT NvhVpjih\/vq`qS n>THjiRWxUVy\JQSjihYN T_NvhYq b<MR2tOQSxUg \JjCiVWTQ9iq`h>R.Q`j<h>q`hYn xyVWTH \|Hjih xyQ`n>RYkvQSx \JhYRORUh>jiRUh:xyTM9ehYHQSj<tOQ xygl \B_tfh>q`q ROh6i\|xyQSTHjiRYX

∇ · E =ρ

ε0

? b DuL

∇× E = −∂B

∂t

? b L

∇ · B = 0? b @CL

∇× B = µ0(J + ε0∂E

∂t)

? b S L gih]nogp\JVWHhzNihYjiRWQSx

ρQSR I THjq \JVWHhRWn^\JqShYRfQ`j iq`\JRU\JR[NihzxUT Zzh9 Hh RUgvQ`hYqSNiQSji;_giTGtfh>PJhYVYk

xygvh NiQSPHh>VUHh>jinYh T|xUgih hYq`h>n>xUVUQSnKVphYqSN \|x \JjC THjvh eTHQ`jCx]QSR]jvTJx jvhYnYh>RURy\|VUQ`q I b1 gvQ`R]QSRxygihhYRWRUh>jinYh T| xygih NiQ`qShY~\xygp\ x:tfh A\|nYh tOgih>j \|xWxyhY~vxUQ`jixyT<nYTHjiR xyV5in>xw\Jj\JviVUT<vQS\|xyhxygvhYTHV TJ vq \JRW \]Nv_jp\J~Q`n>RYb pTHVWx=ijp\|xUhYq HkCQ x x=iVUjvR[TQExxygi\|xt hnY\JjNvQ`RUnY\JVWN c\vRUR R[q`\Gt? h64p\|xyQSTHj b DuL \JjiN RWxUQ`q`q NvhYVUQ PHh \ n>TH~iq`h xyh2RUh x TJh64p\|xyQSTHjiR>b iVWxUgihYVWTHVWhJkmQ 0xo\Jh2xygihNiQ PHhYVUJhYjin>hcTJ \JVy\JNi\G R q \^t ? h64p\|xUQ`THj b L kvt hcHh x

∇ · ∇ × E = −∇ · ∂B

∂t

? b JHL

0 =∂

∂t(∇ · B)

? b LtOgiQSn gQSvq`Q hYRcxUgp\|x

∇ · BQSRwnYTHjvRWxo\|j x Q`j xUQ`~hJkRUT h64p\|xUQ`THj b @ QSR \Jn xp\JqSqS6\JjQSjiQSxUQ`\Jq

nYTJjiNiQSxUQ`TJj<VU\|xygvhYV xUgp\Jj\Jj<Q`jiNvhYehYjvNihYjCxzh64p\|xUQ`THjb hn^\|jcNihYVWQSPHh RUh PHhYVy\|q ~THVUh0h64p\ xyQ`TJjiR 9C n>THjiRWQ`Nih>VUQ`jv xygvhiq`\JRU\ R iQ`N iVUTHeh>VWxyQShYR>b

h>HqShYn>xUQ`ji xygih2h# h>n>xUR]T| QSTHjiQ ^\|xUQSTHj\JjiN VWhYn>TH*9iQ`ji\|xyQSTHjkvxygihcxUTJxo\Jqji*9eh>VTJ0p\JV xyQ`n>q`h>RNQ`j \~PHTHq:i~h

VTJ iq \JRW \~nY\Jj THjiq nogp\JjvHh Q 0xUgihYVWh QSRz\ jih x ! TJ p\JV xyQ`n>q`h>Rz\Jn>VUTHRWR

xygvhcR5iVWA\|nYhS9mTijiNiQSji xygp\|x PHTHq:i~hJb gvhYVUh THVUh|k 9 xUgih:NiQ PHhYVUJhYjin>h:xygih>THVUh>

∂N

∂t=

∫

V

∂n

∂tdV = −

∮

S

nv · dS = −∫

V

∇ · (nv)dV? b KJL

giQSR*vRWxgiTJq`N THV\|j PHTHq:i~h . kpRUT xygih:Q`jCxyh>HVy\JjvNiROTJ xUgih:PJTHqi~h Q`jCxyh>HVy\|q`R.*iRWx)9ehh64p\|q kp\JjiN<tfh:gp\GPHh xygvhch64p\|xyQSTHj<TJn>THj xUQ`j4iQSx

∂N

∂t+ ∇ · (nv) = 0

? b L

DGF gihwNih>jiRUQ x

nnY\Jj%9eh2VUh>q \|xUhYNxyTxUgih2iVUh>RUR=vVUh

p9C<xygihcxygih>VU~T_Nv_jp\J~Q`n:h64p\ xyQ`TJj

TJ RWxo\ xyh:TJ xygih:iq`\JRU\vkEtOgvQ`nog<xo\Jh>ROxygih THVW

p = Cnγ? b FCL

tOgih>VUhγQ`Rzxygih Vy\|xyQSTT|[xygih RUehYn>Q:Vin~gihY\|xc\|x:nYTJjiRWxy\Jj x:iVUh>RUR=vVUh xyTxygvh RUeh>nYQ Vpn gih^\|xc\|x

nYTJjiRWxy\Jj x PHTHq:i~h \|jiN dQ`R \n>THjiRWxy\Jj xYb pTHV Q`RUT|xygih>VU\JqCn>THvVUhYRWRUQSTHjwTJV hLEp\JjiRWQ`THj kγ = 1

ktOgiQSq`hO THVf\|NiQ \9i\|xyQSnn>TH~iVUh>RURUQSTHj THV h$vi\JjiRUQSTHj tOQSxUgxygiVWhYhzNihYJVUh>hYR.TJ VUh>hYNiTJk

γ = 5/3b

TwRUh>x.TJHTuPHh>VUjiQSjiwh64p\ xyQ`TJjiR THVf\wR _RWxyh> t TQiq`N 9ehznYTH~iqSh>xyhOtOQSxygvTQvx h t xyTHj RRUh>nYTHjvN q`\Gt:b h 9ehYJQ`j 9 n>THjiRUQSNihYVWQ`jixUgih2 THVUn>hYRzTHj h>q`h>n>xyVWTHjiR]\JjiN Q`THjiRQSjxygihwvq \JRW \RUh>p\JVU\|xyh>qSJb h>Hq`h>n>xUQ`ji HVU\GP_Q x Jkp\|j<hYq`h>n>xUVUTHj<QSj\Jj<QSTHjiQ Yh>N<vq \JRW \wn^\|j< hYh>q \ TJVUn>h VWTHiVWhYRUR5iVUh:HVy\|NiQ`h>j xURYkvhYqShYn>xUVUQSn Vph>q`NiR>kp\JjiN<\JHjvh>xyQSn Vph>q`NiR ? "THVUh>j x ] THVUn>h L b gvhYjQ`j xyh>VU~RTJ xygih:h>q`hYn xyVWTHj\JRUR

me

kiNihYjiRWQSx ne

kiPHh>q`T_nYQ x ve

k \|jiNiVUh>RUR5iVUhpe

menedve

dt= −ene(E + ve × B) −∇pe

? b D^ICL

gihNih>VUQ P|\|xUQSPHhOTHj~xUgihq`h x[gp\JjvN RWQ`NihOTJ xUgiQ`Rh6i\|xyQSTHjNih>RUn>VUQ:9mh>R[xygih\Jn>nYhYqShYVU\|xyQSTHj~T|hYqShYn xyVUTJjiR]Q`jxygih>Q`VzTGtOj Vy\J~h;Q xtfTQiqSN%9mhwTHVWhwn>THj PHh>jiQ`h>j x]xyTgp\^PHh \|j h6i\|xyQSTHj xygp\|x\Jivq`Q`h>Rf TJV iQSN<h>q`h>h>j xyRfxygp\ x\|VUh V!vh>NQ`jRWp\Jn>hJb C jTJjih NiQSh>jiRUQSTHjk_xygvh n gi\JjiHh Q`j\Jj\JV59iQSxUVy\JV iQ`N iVUTJmh>VWx

KQSj \ Vy\|h~TGPEQSjitOQSxygxygih vQ`N nY\Jj 9eh tOVWQSxUxUhYj\|R2xygih

R=v T| x tfT xyh>VU~R

dK(x, t)

dt=

∂K

∂t+

∂K

∂x

dx

dt=

∂K

∂t+ ux

∂K

∂x

? b DJDuL

giQSROHhYjvhYVy\|q`Q YhYR Q`jxUgiVUh>hcNvQ`~hYjiRWQ`THjvR.xyT

dK

dt=

∂K

∂t+ (v · ∇)K

? b D L

giQSRQ`R jiTGtOj \JR<\n>THj PJhYn>xUQSPHh Nih>VUQSP \|xyQ PHhJb T+ijiNvhYVUR xo\JjvN Q xyR~h^\JjiQSji Q`jCxiQ xyQ PHh>qSJknYTJjiRUQSNihYV\ mT|xTJ t.\|xUhYVOTHj\ RWxyTGPHhJb' gvh:xyh>eh>Vy\|x=iVUhc\|x\&V!vh>Nq`T_n^\ xyQ`TJjQSj<xUgih t.\|xUhYV

I n^\|jnogp\JjiHh 9ehYn^\iRUh gihY\|x.Q`R xUVy\JjvRW hYVWVUh>NxUTwxygiht.\|xUhYV VUTH xygihzRWxyTGPHh]hYqShY~hYjCx^k_t.\JVWQSjixygvh tf\|xyh>V h PHhYV tgihYVWh~QSj xygih eTJx ? xUgih p\JVWxUQ \Jq NvhYVUQ P|\|xUQSPHh2xyh>VU L k THV 9ehYnY\iRWh giTJxzt.\|xUhYV~TuPHh>R2Q`jCxyT<xygihqST_n^\ xyQ`TJj6T| QSj xUhYVUh>RWxw\JjiN NvQ`RUvq \Jn>hYR:xygihn>T_THqShYV:tf\|xyh>Vcxygp\ xct.\JRw\JqSVUh^\|NvxygvhYVUh ? xygvh n>THj PHh>n>xUQSPHh xyh>VU L b C jvRUhYV xyQ`jvxygih nYTHjCPHhYn xyQ PHh6Nih>VUQ P|\|xyQ PHh Q`j xUT xygih h>q`hYn xyVWTHjh64p\ xyQ`TJjTJ T|xyQ`TJjk_tfhcgi\GPHh

mene

[

∂ve

∂t+ (ve · ∇)ve

]

= −ene(E + ve × B) −∇pe

? b D^@CLMzj\Jji\Jq`TJHTQiR.h64p\ xyQ`TJjn^\|j 9eh tOVWQSxUxUhYj THV.xygih:QSTHjiR Q`jxygih:iq`\JRU\

mini

[

∂vi

∂t+ (vi · ∇)vi

]

= eni(E + vi × B) −∇pi

? b D S LMzRWR=i~Q`jv

ne = ni

kit hcnY\Jj\JNiNh6i\|xyQSTHj b DG@ \JjvN b D S xyT Hh>x

ρ

[

∂v

∂t+ (v · ∇)v

]

= J × B −∇p? b DJHL

9ehYnY\iRWhen(vi − ve) = J

b M&~THVUh]n>TH~iq`h xyh]NvhYVUQ P|\|xUQ`THj tfTiq`N<\JqSq`TGt THVfxygih eTJRURUQ:9iQ`qSQ x TJ RWgih^\JVWQ`ji ? THV h$v\Jvq`h|kiT|xyQSTHjQ`jxygih 4 NiQ`VWhYn xyQ`TJj<qSh^\JNvQ`jiwxyT ~TH~hYj x=i)xyVU\JjiR hYV Q`jxygvh:? NiQ`VWhYn>xUQ`THj L 9 VUh>iq \|nYQSji

∇p9Cxygih:NiQ PHhYVUJhYjin>hcTJ xygih:iVWhYRUR5iVUh:xyhYjvRUTHV

Pb

gih&Vpji\JqxygvVUhYh jih>nYhYRWRy\JVW h64p\|xUQ`THjvR:n>TH~h VWTH 4]gi R q`\Gt:k tOgvQ`nog6NvhYRUn>VUQ:9ehYR xygihhYqShYn xyVUQSn^\|qmiVUTJehYVWxUQ`h>R.TJ xUgih]iq \|RU\vb C j<Q xyRf~THRWx.JhYjih>Vy\JqSQ >hYN THVUk04]gv R.q`\Gt Q`RfVU\|xygih>VNp\vj xyQSjiik,9vxOtfh:n^\JjTJ xUhYjjvhYHqShYn xRUh PHhYVy\|qxyhYVWRO\JjiN<tOVWQSxyh

E + v × B = ηJ? b D L

tOgih>VUhxygih~xyh>VU~R2THjxUgihq`h> xcVUhYvVUhYRWhYjCx xygihhYqShYn xyVUTJRWxo\ xyQ`n THVUn>h\|jiNxygih "THVUh>j x TJVUnYh|k\JjiN

ηQSRfxUgih]iq \|RU\wVWhYRUQSRWxUQSP_QSx Hb C j kEt h \Jh]TJjih]THVWh]\JivVUTEQS \|xUQSTHj \JjiN

\JRWR=i~h xygp\|x xygvh:iq \JRW \ QSR mh>VW h>n>xUqSn>THjiNin xyQSjiikpRWT

E + v × B = 0? b DuKHL

64p\|xyQSTHjiR b k b Sik b vk b F k b DJ kv\JjiN b DuK xiVWjTQvxfxyT&9eh]hL_xUVUhY~hYq iRWh>/iqm THV(ijiNih>VRRWxy\JjiNiQSji q \JVWHh]RUn^\|q`h:vq \JRW \*9eh>gp\GP_Q`TJVYb

D

MzjQ`jCxyh>VUhYR xyQSjiwnYTJjiRUh6vhYjinYh:TJxygihzQSNih^\Jq lazZ&\JRWR=i~vxyQSTHjiR Q`R[xygi\|x.\JHjvh>xyQSn#VphYqSNq`QSjihYRO\|VUh VWT! Yh>j! QSj xyT xUgih:iq \JRW \Eki~h^\|jiQ`jv xygp\ xOxygvh><~TuPJh:THjiq \Jq`THjv tOQ xyg<xUgih 9vqiq`\JRU\ pTGt p\ xUxyh>VUjiR>b gih~ VUT YhYj Q`j 6xygih>THVUh>k THVcMqS P>j Rcxygih>THVUh>k RWxy\|xyh>Rcxygp\|xxygvh ! xygiVWTQiHg \R5iVWA\|nYh~xygi\|x2~TuPHh>RctOQ xyg6xUgihiq`\JRU\Q`R:nYTHjiR xo\JjCxwQSj6xyQS~hJb@ T RWhYhtOg Jk[tfh THq`qSTGt ? @ L \|jiN ? DHDGL \JjiN R59iR xyQSx=vxyh<Meh>VUh R q`\Gt ? h64p\|xUQ`THj b S L QSj xUT xygihVUh>RUQSRWxyQ PHh 4]gi R]q`\Gt ? h6i\|xyQSTHj b D L b1 gih NiQ`RWiq \|nYhY~h>j x]nvVUVUh>j x ∂E

∂t

n^\|j 9eh jihYJq`hYn xyh>NQ`j6l azZ 9ehYn^\iRUh xygih xyQ`~h RWn^\JqSh TJV NiQ# iRWQ`THj QSR *inog6q`TJjiHhYVzxUgp\Jj xUgih xyQShwRUnY\Jq`h THVq`QSHgCx.xUT~iVWTHeTHC\ xyh Q`jxUgih:iq \|RU\ ?AFCL kpRWT tfhcJh>x

E = −v × B +η

µ0

∇× B? b D L

\JRWR=i~Q`jixygp\ x]xUgih VUh>RUQSRWxyQ P_QSx QSRznYTJjiRWxy\Jj x]xygvVUTQiJgiTQvxzxygvh iq \|RU\vb \ QSji xygih niVUqT|9eTJxUgRUQSNihYR

∇× E = −∇× v × B + ∇× η

µ0

∇× B? b D^FCL

TGt tfhcnY\Jj iRUh \|Vy\JNp\^ R q \Gt ? h64p\|xUQ`THj b L xyT hYqSQ`~QSjp\ xyhEkiq`hY\GP_Q`jvMiR tOQSxUg

∂B

∂t= ∇× v × B +

η

µ0

∇2B? b ICL

giQSRQSR jiTGtOj \JR0xygvhO\JHjih xyQ`nfQ`jvNin>xUQ`THj h64p\|xyQSTHjb0rOhYnY\Jq`qSQSjizxygvh\JRWR=i~vxyQSTHjiR T|mQ`NihY\Jql aZwkptfh:n^\JjRWh>xOxygih:VUh>RUQSRWxyQ P_QSx h64p\JqxUT >hYVWTikpRUT

∂B

∂t= ∇× v × B

? b DuL

TGt n>THjiRWQ`Nih>V \OnvVWPHhC9eTQijiNiQSji\OR5iVWA\Jn>h

StOgvQ`nog2QSRTGP_Q`ji.xygvVUTQiJg:xygvh[vq \JRW \

? RWhYh Q`iVUh KHL b C j\xyQShdt\|j6h>q`hY~h>j x

dlTJ

CRWt hYh>iR2TQvxc\Jj6\JVWh^\

dS = v × dldtb

gih>VUh> TJVUhJk 9 xygvhcvVUT_Nin xV=iqShJkvxygvh:Vy\|xyh TJ n gi\JjiHh:TJ\JJjih>xUQ`n !xygvVUTQiJgCQ`R

dΦ

dt=

d

dt

∫

S

B · dS =

∫

S

∂B

∂t· dS +

∫

C

B · v × dl? b H L

H

Plasma motion C1 C2

t1 t2

QSib K b l \JJjih>xUQ`n ! nYTJjiRUh>VWP|\|xUQ`THjQSj Q`NihY\Jq l aZwb C \2nvVWPHhC1

Q`R NiQSRWxUTHVWxUhYNQSj xUTC29 <iq`\JRU\ TJxUQ`TJjkvxygih>jxygvh xygvVUTQiJg

C1

\|xt1\JjiNxygvh xygvVUTQiJg

C2

\ xt2tOQSq`q

9eh:h64p\Jq 9ehYnY\iRWh2 \|Hjih xyQ`n Vph>q`N<q`Q`jvhYRO\|VUh VUT! >hYj! Q`jb !

giQSRwh$EiVUh>RURUQSTHjQSRw\|jp\JqSTHHTQiR xyTxygihnYTJj PHhYn xyQ PHhNih>VUQ P|\|xUQSPHh~NihYRWnYVUQ:9ehYN h^\JVWq`QShYVYbMzRCTGPHhYR>kxygvh xUgiVUTQvHg xygih qST_TH n^\|j6nogp\JjvHh h>QSxygvhYV19eh>n^\vRUh xUgih \JJjih>xUQ`nKVphYqSN Q`R

nogp\JjiJQ`jiQ`jxyQ`~h ? xygvhKVpVWRWxxyhYVWTJjxygihwVWQ`Jg x L THV 9eh>n^\vRUhwxygihK9eTQijiNi\JVWQ`RTGPEQSji~Q`jRUi\JnYh ? xUgihRWhYn>THjiNxUhYVUTJj~xUgihOVUQ`Jg x L b 0 RUQSji2\ PHh>n>xUTHV[Q`NvhYj xUQSx Hk

B ·v×dl = −v×B ·dlk

RUTdΦ

dt=

∫

S

∂B

∂t· dS −

∫

C

v × B · dl? b @CL

QSjp\JqSq Jki\JviqS_Q`jv xUTQJhYR pxygih>THVWhY xyT xygih:RWhYnYTJjiNxUhYVW

dΦ

dt=

∫

S

[

∂B

∂t−∇× (v × B)

]

· dS = 0? b S L

VUTJ h64p\|xUQ`THj b D b T xygih !xUgiVUTQvHgCQ`R n>THjiRWxy\Jj xQSjxyQSh|b

@

MzR THjih ~Q`HgCx]h$Emh>n>xYk VUhY\Jq0iq`\JRU\JR]\JVUh jvTJx:eh>VW h>n>xyq nYTJjiNin xyQ`jvikRUT<Q`Nvh^\Jq l aZA\JQSq`RxyTziVWhYNiQSn>xRUh PHhYVU\JqvQ`jCxyh>VUhYR xyQSjiziq`\JRU\vgihYjiTJh>jp\vb C j i\JVWxUQ`nvq \JV>kutOgihYj VWh^\Jq_vq \JRW \nYTJeTHjvhYjCxyR:n^\|VUVW_Q`jv<THieTHRWQSxUhYqS NiQSVUh>n>xyh>N6 \|Hjih>xUQ`n VihYq`NvR:V=ij6QSj xUT<TJjih~\JjiT|xygih>VYk giQSHgVph>q`N HVy\JNvQ`hYjCxyR n^\|jn^\iRWh Vph>q`NqSQ`jvhYR[xUT NvQ # iRUhxUgiVUTiHg xUgihziq`\JRU\vk n gi\JjiHhxygih>Q`V[xUTHeTHqOTHJJkE\JjvN VUh>nYTJjijihYn x^kEnYTHjCPHhYVWxUQ`ji2R xyTHVWhYN \JHjih xyQSnOhYjvhYVU|~Q`jCxyT Q`jih xyQSnh>jihYVWJ~THV[gvh^\|xYb C jTJxUgihYVtfTHVUNvRYk_xygvh VUT! >hYj! Q`j! ! nYTHjiNvQSxyQSTHj~QSRP_Q`THq`\|xyh>Nb xiNv_QSjiw\JHjih xyQSnOVUhYn>THjijvhYn>xUQ`THjQ`R.xUgih:iVUQS \|VW JTC\JqT| i b

"h>xMiR VUh xiVWj xyT xygvh<Q`jiN,in>xUQ`THjh64p\ xyQSTHj ? b ICL kxUgiQ`RwxyQ`~h\|q`qSTutOQSji xUgih<vq \JRW \VUh>RUQSRWxyQ P_QSx xUTU9eh:jiTHj" Yh>VUTib

∂B

∂t= ∇× v × B +

η

µ0

∇2B? b JHL

C tfh]nYTHjvRUQ`NvhYV.THjiq ~xUgih VpVUR x xUhYVW THjxUgih]VUQSHg xfgp\|jiNRUQ`NvhJkEtfh]Hh>x.h64p\|xUQ`THj b D k_tOgiQSn gNih>RUnYVWQ9eh>R n gi\JjiHh>R QSj:xygvh[ \|Hjih xyQ`nVphYqSNK9vVUTQiJg x \9eTQEx 9 nYTHjCPHhYn xyQSPHh[TJxUQ`TJjiRTJvvq \JRW \tOQSxUg< VWT! Yh>j! QSj !mb dfTHjiRWQ`Nih>VUQ`jv~TJjiqSxygih:RUh>nYTHjvNxUhYVUkvxUgih:h64p\|xUQ`THj 9ehYn>THh>R

∂B

∂t=

η

µ0

∇2B? b L

giQSROgp\JR xygih:RWxy\JjiNp\|VUN< THVU TJ\ NiQ# iRWQ`THjh64p\|xUQ`THj kpRUT tfh:RUh>hcxUgp\|xOQ`j<VUh>RUQSRWxyQ PHh2l aZwk\JHjih xyQSn VihYq`NvROn^\Jjnogp\JjiJh:xUgiVUTQvHgNiQ# vRUQ`TJj\JR t hYqSq \JROn>THj PHh>n>xUQ`THj b

hn^\|jnogp\JVy\|n>xyh>VUQ YhOxUgihQS~mTJVWxo\|jinYh.TJ nYTHjCPHhYn xyQSTHj~PHhYVWR=iR NiQ# vRUQ`TJj Q`j~\:p\JVWxUQ`niq \|Viq`\JRU\ 9C"xy\ Q`ji xygvh Vy\ xyQ`T TJcxUgih x tfT nYTH~eTHjvhYj xUR<TJcxUgih QSjiNin xyQSTHj h64p\|xUQ`THj b C LQ`R xygvh<n gi\JVy\Jn xyh>VUQ`R xyQSnq`h>jiJxyg RWn^\|q`h<QSjxygvh<iq \|RU\vk xUgih<RUp\ xyQ`\Jq.Nih>VUQ P|\|xUQSPHh

∇n^\Jj>9eh

\JivVUTEQS \ xyhYN 9 1

L

b gih>jconvection

diffusion≈ (1/L)vB

ηB/µ0L2=

µ0Lv

η≡ RM

? b KHL

gih 64p\|j xyQ x RM

QSR3_jvTutOj \JR xUgih: \|Hjih xyQ`n rOh>_jiTJq`NiROj4i*9ehYVYb

S

2l

2L

x

y

QSib vb tfhYh x5 \JV=Jh>V2VUhYn>THjijvhYn>xUQ`THj~T_Nih>q HhYTJh xyV HbU gih~giTHqSq`Tut*\JVUVWTutOR:VUhYvVUhYRWhYjCxxygvh]NvQ`VUh>n>xUQ`THjTJiq \|RU\ iTut:k_tOgiQ`qShxygvh]RW \|q`qShYV 9iq \|n$~\|VUVUTGtORfRUgiTGt xygihzNvQ`VUh>n>xUQ`THjTJxygihhY*9eh>NiNih>N \JHjvh>xyQSn VphYqSNiRYb[rOh>nYTHjijvhYn>xUQ`THjT_n>niVWROtOQSxygvQ`jxygih:RUgi\JNihYNVWhYHQSTHjb

TGt xygp\ xtfh gp\GPJh6VWhYq`\BEhYN TiV<\JRUR5i~vxyQSTHjiRRUTxygp\ x< \|Hjih>xUQ`n VihYqSN qSQ`jih>R \^NiQ# iRWhxUgiVUTQvHg xUgihiq`\JRU\6\JjiN VUhYn>THjijih>n>xYk tfhtfTiq`N q`Q:JhxyTgp\GPHh \6T_NvhYq THV xygihjp\|x=iVUh<TJxygiQSR vVUT_nYh>RUR\JjiN xygih<Vy\ xyh\ x tOgiQSn g Q x tOQ`qSqfT_nYniVYb> gih VpVURWx\JjvN RWQ`~iq`h>RWx~T_NihYq TJ[ \|Hjih xyQ`n2VUh>nYTHjijvhYn xyQ`TJj t.\JR iVWTHeTHRUh>N 9 tfhYh x2\JjvN6\JV5JhYV ?A@H@HL Q`j xygih q \|xUhDGF?J|I RYb gih THq`qSTGtOQSji NiQ`RWniRWRUQ`TJjNvVy\^tORO VUTJ ? S L k ? @ L kp\JjiN ? DHDGL b

gih t hYh>xR \JV5JhYV~T_Nih>q \JRUR5ih>R xygp\ x~VUh>nYTHjvjihYn xyQ`TJj T_nYniVUR~h>j xUQ`VUh>qS Q`jx t T6NiQ ~hYjiRWQ`THjiR>b Q`h>q`N q`QSjihYR \JVWh\JRWR=i~hYN xUT 9eh\JiiVWTEQ`\|xUhYq 2RWxUVy\JQSHg xYk?9,vxtOQ xyg A5iR x[h>jiTQiHgnvVWP|\ xiVUhfxUgp\|xt h.n^\Jj NihVijihO\VUhYn xo\JjvQiq \|V VUhYn>THjijih>n>xUQ`THj VUhYJQ`THjwTJ qShYji|xyg "\|jiNwxygiQSn$Qjih>RUR q ? RUhYhQ`QvVUh L bOq`\JRU\ nYTH~eTHjih>j xyRn>THj xy\JQ`jiQSji~THvmTJRUQSji \JJjih>xUQ`n1VphYqSNiR pTGtQ`j VUTJ)xUgih

±yNiQSVUhYn xyQSTHjiR tOQSxUgPHhYqST_nYQ x

vin

b C jiRWQ`Nihzxygih]VUh>nYTHjvjihYn xyQ`TJj<VUh>HQ`TJjkE \JJjih>xUQ`nVph>q`NJVy\JNiQShYjCxyR 9ehYn>THhzgvQ`Hg h>jiTQiJgxygi\|x(VihYq`N q`QSjihYR[xUgp\|x tfh>VUh]iVWh>P_Q`TQvRUqS VUT Yh>jQSj n^\Jj

J NiQ# iRWh \JjiN VUh>nYTHjvjihYn x<;[\| xyh>V2VUh>nYTHjvjihYn xyQ`TJj Vph>q`N qSQ`jih>R2\JjiN iq`\JRU\ iTut#TvxwQSjxUgih

±xNiQSVUhYn xyQSTHjiR tOQ xygPHh>q`T_n>QSx

vout

b( gih2~T_Nih>q \|RUR=vh>Rxygp\|xVUh>nYTHjvjihYn xyQ`TJj QSR\ RWxyhY\JNv? R xo\|xUhiVWT_nYhYRWRYkpRUT

vin

\JjiNvout

NiT jvTJxOnogp\JjvHh tOQ xygxyQSh|b C j<TJxygih>V.tfTHVWNiRYkp\JqSxUgiTQiJgiq`\JRU\wQ`RnYTJjiRWxy\Jj xUqS ~TuP_Q`ji\JjiN6QSjiNiQ PEQSNp\|qVihYq`N qSQ`jih>R:\JVWh nogp\|jiHQ`jvxygih>Q`VzxyTHeTHqSTHJJkQStfh xyT_TQ\RUjp\JvRUgiTJx TJ.xygih VUh>nYTHjvjihYn xyQSTHjVWhYJQ`THj \ x \|j xyQ`~hJk Q x2t TQiqSN\Jq t.\^ER2q`T_TQ xUgih Ry\J~hJb QSxUgxygiQSROQSj<~Q`jiNkEt hcnY\JjRUhYh:xUgp\|xO \|RUROn>THj xUQ`j4iQSx Q`~iqSQSh>R

4ρLvin = 4ρlvout

? b L

Lvin = lvout

? b FCL

QS tfh2\|RUR=vh xygi\|xOxUgih:iq`\JRU\ Q`R Q`jvnYTH~iVUh>RURWQ9iqShJb4 vxyRWQ`Nih xUgih VUh>nYTHjvjihYn xyQ`TJjVUh>HQ`TJjk xygvh \JHjvh>xyQSn VphYqSN*q`QSjihYR \JVUh \|iiVUT<EQ`\|xyh>qS

RWxUVy\JQSHg xYk RUT∇×B = 0

\JjiN Meh>VUh Rq`\Gt ? h64p\|xUQ`THj b S L xyh>q`q`RUiR xygp\ x xygih>VUh QSR jiTnvVUVUh>j xYb' gih L n>TH~eTHjihYjCxT| xygvh:VUhYRWQ`RWxUQSPHh 4 gv ROq`\Gt ? h64p\|xyQSTHj b D L xygvhYj 9mh>nYTJh>R

Ez + vinBin = 0? b @JICL

C jiRWQ`NihxUgih VUh>HQ`TJjk xUgih jih x \|Hjih xyQ`nMVihYq`N Q`R YhYVWTik0RUTxygih $ n>THeTJjihYjCx T|&4]gi Rwq`\GtHQ PHhYR

Ez = ηJz

? b @EDuL hwnY\Jj nYTH*9vQ`jih:xUgihYRWhwx tfT VWhYq`\|xyQSTHjiR#9 jvTJxyQSji~xUgp\|xxygih2RWxUh^\JNE4 RWxo\ xyh \JRWR=i~vxyQSTHjQ`&iqSQ`hYR dB

dt= 0

kRWT 9 \JVy\|Np\G R2q \^t ? h64p\|xUQ`THj b L k ∇×E = 0\JjiNxUgihhYq`h>n>xUVUQ`nMVph>q`NvR

Q`jvRUQ`Nvh:\JjiNTQExyRUQSNih xUgih:VUh>HQ`TJj\JVUh]xygih:RU\Jh|b T tfh:gp\GPHh

vinBin = ηJz

? b @ L

TGtt hf\G:\Jiiq :Mz~ehYVWh Rq \^t xyTxygih VWhYnYTJjijihYn xyQSTHj VWhYJQ`THjw\|R\tOgiTJq`h|b 64p\ xyQ`jvxygvhzqSQ`jihOQ`jCxyhYJVy\JqpTJ

B\JVWTQijiNxygvh 9eTHVWNihYV TJxUgihVUhYJQ`THj~xUT2xygihhYjvnYq`TJRUhYNniVUVWhYjCx.NihYjiRWQSx Hk

tfh1VpjiN

Bin(4L) = µ0Jz(2L)(2l)? b @H@CL

Bin = µ0Jzl? b @ S L

9vRWxyQ xvxUQ`ji xygiQSROh$EiVUh>RURUQSTHjQ`jCxyT TiVOVWhYR5iqSx VUTH 4]gi ROq \^t

vin =η

µ0l

? b @QJHLvinµ0l

η= Rm = 1

? b @ L T<xygvh~\JHjih xyQ`n rh _jiTHq`NvRcj4i*9ehYV:Q`R D NiVUQSji tfhYh x5 \JV=Jh>V2VUhYn>THjijvhYn>xUQ`THj k QSvqS_QSjixygi\|x2xUgih QSjpTGtPHhYqST_nYQ x 6\JjvN xUgiQ`n7 jih>RUR T|.xygihVUh>nYTHjvjihYn xyQ`TJj VUh>HQ`TJj \JNBA5iR xwxUgihY~RUh>qSPHh>RijCxyQ`qm\JHjih xyQ`n !Q`R29eh>Q`ji \JjvjiQ`giQSq`\|xUhYN\|x.xUgih Ry\|h]Vy\ xyhzxygp\|x iq`\JRU\wQSR.hYRUnY\JiQSji TQvxxygvhcRWQ`Nih>ROTJ xUgih:VUh>HQ`THj b

Z iVWQ`jiVWhYnYTJjijihYn xyQSTHjk\JHjih xyQSnwhYjih>VUJ RWxUTHVUh>N Q`j xUgih Q`jvnYTH~QSji@VphYqSN QSR]nYTJj PHhYV xyh>NQ`jCxyT Q`jih xyQ`nh>jih>VUJTJxUgih<TQvxUHTHQSji iq`\JRU\vkRWT tfhnY\Jj nYTJjiRUh>VWPHhh>jihYVWJ xyT VpjiN xygihTQvx pTut PHh>q`T_nYQ x Q`jxUgih:Q`NihY\JqnY\JRUh:tOQ xyg<jiT~iq`\JRU\ gihY\|xyQSjiib

B2in

2µ0

=ρv2

out

2

? b @HKHL

vout =Bin√µ0ρ

? b @ L giQSR PHhYqST_nYQ x VUhYvVUhYRWhYjCxyRO\wRWTHVWxfTJRUehYh>N<qSQ`~Q x[Q`jxygvhziq \JRW \!;_QSx Q`R jiTGtOj<\JR xygvhzMzqS P>jRUeh>hYN ?

vA

L bvA

gp\JR \|NiNiQSxUQ`THji\Jq Q`~eTHVWxy\Jjin>h~Q`jl aZ xygvhYTHV \JR2xUgih igp\|RUhPHhYq`T_n>QSx 6T|q \|VUHh$ RUnY\JqShJkvqSTut2 VWh64ih>jin>THRWnYQ`qSq \ xyQ THjvR2_jvTutOj \JROMqS P>j<t.\GPHh>RYb

TGt t hcnY\JjnYTH*9iQSjih h6i\|xyQSTHjiR xyT VUh>q \ xyhvin

\JjvNvout

v2

in = (vin)(vin) = (voutl

L)(

η

µ0l) = v2

out(η

voutµ0L)

? b @JFCL

gih VUh>nYTHjijvhYn xyQ`TJjwVy\|xUh ? l L Q`R TJ xyh>jwh$vvVUhYRWRUhYN Q`j:xyh>VU~R TJvxygih[Mzq PYj2RUehYh>N Q`jcTHVUNih>VxyT\|q`qSTut THVOR xyVy\JQSHgCxU THV t.\JVUN<n>TH~p\JVWQ`RUTJjiR#9eh x t hYh>j iq`\JRU\JRYb[rOhYnY\JqSq`QSji xUgp\|x

vout = vA

k

K tfh:gp\GPHh

M ≡ vin

vA

? b S ICL

M =

√

vAµ0L

η=

1√S

? b S DuL

tOgih>VUh QSR \ RUehYn>Q \Jq n^\|RUh:T| xUgih:rOh>_jiTHqSN R.ji*9eh>V ? jiTGtOj\JRfxygih " ijvN64iQ`R xj4i*9ehYV Lxygi\|xOVUh>iVUh>RUhYjCxyRgiTut tfh>q`q\JHjvh>xyQSn Vph>q`N<q`QSjihYR \JVWh VWT! Yh>j<xyT xUgih:iq`\JRU\vb Eeh>VUQ`~hYjCxo\JqVUh>R=iq xyR0RUgiTGt xUgp\|x0xUgih

1√S

VUhYn>THjijih>n>xUQ`THj VU\|xyh.vVUhYNvQ`n xyhYNM9C tfhYh x5 \JV=Jh>VQ`R xyT_T]RUq`TGt:kC\JjiN~THVUh:RUTHigvQ`RWxUQ`nY\|xyh>N ~T_NihYqSRTJ0VUh>nYTHjijvhYn xyQ`TJj gp\^PHhwRWQ`jin>hK9ehYh>j Nih>PHh>q`THeh>Nb& iV xygih>VU~THVUh|kVUh>R=iq xyR \|x i ?GJJL \|jiN hYqSRUh>t gihYVWhgp\^PHh<RUgvTutOj xygp\ x xUgihQ`jCxyhYVU\Jn>xUQ`THjvR*9eh>x tfh>hYj VWhYnYTJj!jih>n>xyQSTHj VphYqSNiR\JVWhcjvTJxnYTHj,VijihYNxUT \ iq`\Jjp\JV QSj xUhYVWA\|nYh|bfaTutfh PHhYV>k t hYh x5 \|V=JhYVVWhY\JQ`jvR\iRUh /iq QSj xUVUT_Nin xyQSTHj xyTxUgih xygvhYTHV TJ[VUh>nYTHjijvhYn>xUQ`THj k \JjiN xUgih x tfT NiQ`~hYjiRWQ`THji\Jq ~T_NihYqiVWTuP_Q`Nih>R\JjQSNih^\|q Vy\J~h>t THV= THV3vjiNihYVWRWxy\JjiNiQSji xygih 9p\JRWQ`n iVUT_n>hYRWRUhYRz\|x tfTHV5 b

$&% /<,2 ( !2FQ

MzRNiQSRUnvRURUh>N h^\|VUq`QShYV>k|hL_xUhYVUji\Jq kJigiTJxUTHj! 9p\JRUh>N NvQ \JJjiTHRWxUQ`n>R iq`\G2\|j Q`~eTHVWxy\JjCx VUTJq`h[Q`jxygvh\Jjp\Jq ERWQ`R T| q`\9eTHVU\|xyTHV wiq`\JRU\]iVWTHehYV xyQ`h>RYb T ijvNihYVWRWxo\|jiNxUgihO~h^\JjiQSji TJmhY\JR=vVUhL~hYjCxyR:\JNih*9 R=inogNiQ`\JHjiTJRWxyQSnYR>kt h~*iR x:TGPHh&9eh>HTJjiN6xygih q \JVWHh$ RUnY\Jq`hwNih>RUnYVWQ`vxUQ`THjvRiVWTuP_Q`Nih>N 9C l azZxygih>THVW \JjiN T_niR THj xygih vVUT_nYh>RURUh>R2\|x]xygvh \|xyTJQSnwqSh>PHh>qxygi\|x n^\vRUhiq`\JRU\JRfxyT Vy\JNiQ`\|xyh|b gvh: THq`qSTGtOQSji NiQ`RWniRWRUQSTHjNiVy\^tORO VUTJ ? @HFHL k ? FHL kp\JjiN ? FHL b

pTHViq \|RU\JRxygp\ x \JVUhwTHvxUQ`nY\Jq`q xygiQSj ? ~h^\|jiQ`jvxygp\ xzxygih2x _iQSn^\Jq igiTJxUTHj vVUT_Nin>hYNQ`j"xygih iq \|RU\ hYRWn^\|mh>RtOQ xygiTvx 9eh>Q`ji \9iRUTHV59ehYN THVRWn^\|xWxyhYVWhYN L kzxygvh ~THRWxQ`~eTHVWxy\Jj xVy\|NiQ \|xUQ PHhiVUT_n>hYRURWhYR\JVWhqSQ`jihh>QSRURUQSTHjk[VU\JNiQ \ xyQ PHhVUh>nYTH*9iQSjp\|xUQ`TJjk \|jiN9iVWhY~RURWxUVy\Jgvqiji

Vy\|NiQ \|xUQSTHjb gih i iq`\JRU\ Q`R TJvxyQSn^\JqSq 6xygiQSj \|x TJRWx igvTJxyTHj hYjih>VUHQShYR>kf\JqSxUgiTQiHgQSx\G9eh>nYTJh:THExyQ`nY\Jq`q xUgiQ`n7\|x xygih t.\^PHhYqShYjiJxUgT|RUTJh VUh>RUTHjp\|j x\|xyTJQSnzxUVy\JjiRWQSxUQ`THjiR>b

sfVUh>~RURWxUVy\Jgiq:ijiVy\|NiQ \|xUQ`TJj ? 9iVU\ Q`ji Vy\JNvQ \|xUQ`TJj! L THVUQSHQSjp\|xUhYR VUTH xUgih \JnYn>hYqShYVy\ xyQSTHjTJ\ nogp\JVWHhYNp\JV xyQSnYq`hN,ih xyT xygihhYqShYn xyVUQSn@Vph>q`NT|\|jiTJxygvhYV p\|VWxyQSnYqShJb% gvhxyTJxy\Jq eTGtfhYVVy\|NiQ \|xUhYN 9C<\ R ER xyh> TJ\JnYn>hYqShYVU\|xyQSji nogp\JVWHhYROn^\Jj 9ehc\JiiVWTEQS \|xUhYN 9

P =2d2

3c3

? b S LtOgih>VUh

dQSR.xygih:h>q`hYn xyVWQ`n NiQSeTHq`h]~TH~hYj x TJ xygvhcR _RWxyh>kpNihVijihYN 9

d =∑

i

qiri

? b S @CLdfTHqSq`QSRUQSTJjiR 9eh x tfh>hYj6x tfT<Q`Nih>j xUQ`n^\|q0i\JVWxUQ`n>q`hYR]nYVWh^\|xUh~jvTjih x2\JnYn>hYqShYVU\|xyQSTHj T| xUgih RW_RWxUhY RNiQSeTHq`h TJh>j x^kRWT xUgih xUTJxo\JqzVU\JNiQ \ xyhYN eTut hYVP|\|jiQ`RWgihYR<Q`j"xygvh NiQSeTHq`h q`QS~Q x\JjiN n^\JjiR5p\Jq`q T9eh jihYHqShYn xyh>Nb gvhYVUh THVUh|k xUgih \JQSjnYTHjCxyVUQ:9vxUhYV2xUT%9iVWhY~RURWxUVy\Jgvqiji Vy\JNiQ`\|xyQSTJjQ`j iq`\JRU\JR Q`R hYqShYn xyVUTHj QSTHj n>THq`qSQSTHjiR>b C j vVy\Jn xyQ`n>hJk[xUgihVy\|NiQ \ xyQSTHjQ`R NiTH~Q`ji\|xyh>N+9 xygih\Jn>nYh>q`hYVU\|xyQSTHjcTJih>q`hYn xyVWTHjiR;JxUgih q`\JVUJh \|RUR NiQSRUnYVWhYeh>jin>K9eh>x tfh>hYj hYqShYn>xUVUTHjvR\JjiN2Q`TJjiR ~h^\|jiRxygi\|x xUgih nogp\JjvHh QSj xygih Q`THj R~PHhYqST_nYQSx NvVUQ`jv\|j h>q`h>n>xUVUTHj! Q`THj nYTJq`qSQSRUQSTHj Q`RjihYJq`Q`JQ:9iqSh|bsfVWhY~RURWxUVy\Jq:iji VU\JNiQ`\|xyQSTHjQ`RO\|q`RUT& jiTutOj \|R VWhYhL VUh>h!]hY~Q`RWRUQ`TJj 9eh>n^\vRUh2Q x Q`R nY\iRUh>N 9 hYqShYn xyVUTJj<xyVU\JjiRUQ xyQSTHjiR29eh>x tfh>hYjx tfTUij49eTQijiNkpTHV VWhYh!ykiRWxy\|xyh>RTJ xygih xy\JVUHh xOQ`THjb

gih2RUeh>n>xyVU\Jq RUQSHjp\|x=iVUh:TJ 9iVWhY~RURWxUVy\Jq:iji~Vy\JNiQ`\|xyQSTJj<Q`R\ RU~T_TJxygnYTHjCxyQSjikpRUQ`jvnYhhYqShYn xyVUTJj \JjiN QSTHj PHh>q`T_n>QSxUQ`hYRQ`j \ iq`\JRU\ \JVUhwnYTHjCxyQSjiTQvRUqSNiQ`R xyVUQ:9vxUhYNb1 gvh xyTJxy\Jq VWhYh$ VUh>h6h>QSRURWQ`THj \|x\ HQ PHhYj h>jihYVWJ n^\|jH9eh6T9vxo\|Q`jih>N 9 Q`jCxyh>HVy\|xUQ`ji TuPHh>V<xygvh6h>q`hYn xyVWTHjPHhYqST_nYQ x NiQ`R xyVUQ:9vxUQ`TJjki\|q`q eTHRURWQ9iqSh Q`~p\Jn x p\JVU\Jh xyh>VURYkp\JjiN<xUgih:p\JV xyQ`n>q`h]NvQ`RWxUVUQ9,vxyQSTHjQ`jxygvhcvq \JRW \vbMzRUR5i~Q`ji~\l\B_tfhYqSq`Q`\JjhYqShYn>xUVUTHj<PJhYq`T_n>QSx NiQSRWxyVWQ9ExyQSTHjk_xygih:h>QSRURUQ P_QSx QSR

ε =25πe6

3mec3

√

2π

emekTZ2nenie

−hν/kT gff

? b SS LtOgih>VUh

ZQSRzxUgih \|xUTH~Q`n2j4i*9ehYV]TJxUgih n>THq`qSQSNiQSjiQ`THjiR>k

hQ`R q`\Jjin7 R n>THjiRWxy\Jj xYk \JjiN

gffQ`R \ 64p\|j x=i!~hYnogp\JjvQ`n^\|qnYTHVWVUhYn xyQSTHj jiTGtOj\JR]xygih bK gih NiQ`~h>jiRUQSTHjiR]T|

F xygvhchY~Q`RWRUQ PEQ x \JVWh:mTGtfh>V B ijvQSxOPHTHq:i~h B ijvQSx hYjih>VUJ<VU\JjiHh|kpTHVOQ`jxygih ijiQ xyR xygi\|xOtOQ`qSq 9ehiRWhYNq \|xUhYV.THjkihYVU B R B n>

3B h . b

gih i4 iq`\JRU\]Q`R \JNih TJ gCENvVUTHHh>jk_jihY\JVUq DGIHI T|mtOgiQSn g~Q`R Q`TJjiQ >hYNbaTGtfh>PJhYVYkh>PHh>j QSj \|j h64iQ`qSQ9iVWQ:i R xo\|xUhTJfjih^\JVWqS nYTJiqSh>xUh~QSTHjiQ ^\|xUQSTJjk h>q`h>n>xyVWTHjiR2\JVUh~n>THjiR xo\JjCxyqSVUh>nYTH*9vQ`jiQSji tOQSxUg iVWTJxyTJjiR19eh> TJVUh&9ehYQSji jiT_n7JhYNqST_THRUh \|C\JQ`j%9C n>THqSq`QSRUQSTJjiRYbK jih>VUJ Q`RnYTJjiRUh>VWPHh>N QSjxygiQSROvVUT_nYh>RUROxygiVWTQiHgxygih:h>QSRURUQSTHjTJ0\ igiTJxUTHjTJ hYjvhYVU|

E =13.6 eV

n2+ K

? b S JHL

tOgih>VUhnQ`R2xUgihgCENvVUTHJhYj h>jih>VUJ q`h PHhYq Q`jCxyTtOgiQ`nog xUgihh>q`hYn xyVWTHjVWhYn>TH*9iQ`jvhYR \JjiN

KQ`R

xygvh Q`jiQ xyQ`\|q Q`jvh>xyQSnhYjvhYVU|TJ.xygih h>q`h>n>xyVWTHjb Q`jvnYhnn^\|j Vy\JjvHh~ VUTH D xUT

∞k0\JjiNxygih

Vy\|jiHh TJOeTHRWRUQ:9iq`hhYqShYn>xUVUTHj Q`jvh>xyQSn h>jihYVWHQShYR QSRwn>THj xUQ`j4iTQiR>k Vy\|NiQ \ xyQ PHhVUh>nYTJ&9vQ`jp\|xUQ`TJjQ`R\6nYTHjCxyQSji iVWT_nYh>RURYb q`hYn xyVWTHjiR~tOQSxUg PHhYqST_nYQSxUQ`h>R\ xxUgihqSTuth>jiN"TJzxUgihNiQSRWxyVWQ9vxUQ`TJj\JVWh<xygihTHR xqSQJh>qS xUTnYTH*9vQ`jih|kfRWTxygvh\AWTHVUQ x TJ]igvTJxyTHjvRhY~QSxWxyh>N\|R \6VWhYR5iqSxT|VUh>nYTH*9vQ`jp\|xUQ`TJjiRQ`jCxyT\Jj h>jihYVWJ qSh>PHh>q

ntOQ`q`q gp\^PHh h>jihYVWHQ`h>R]n>q`THRWh2xyT

13.6 eV/n2bzMRzt h

RUgi\Jq`qRUh>h2Q`jdfgp\JExyhYV)SvkpVy\JNiQ`\|xyQ PHh VWhYn>TH*9iQ`ji\|xyQSTHjQ`R Q`jA\Jn>x\ A\|VOHVWh^\|xUhYVOnYTJj xyVWQ9ExyTHV.xyTxygvhch>QSRURWQ`THjRUeh>n>xUV=i Q`j v xygp\Jj<Q`R39iVUh>RWRWxUVy\Jgiq:iji Vy\JNvQ \|xUQ`TJjb

pTHV0xygihOnYTHjvNiQSxUQ`THjiR vVUhYRWhYjCxfQ`j i kCxUgihNiTJQSjp\JjCxRWehYn xyVy\|q hY\|xvVUhYR[n>TH~hOjiTJx VWTHnYTJj xyQSj,i iVWT_nYh>RURUh>R\|x\JqSq49Ex VUTH&qSQ`jihVy\|NiQ \|xUQ`TJjciVWT_NinYh>N*9 :xyVU\JjiRUQ xyQSTHjiR TJph>q`h>n>xyVWTHjiR9eTQijvN xyTQ`THjvRYb gihYj\Jj h>q`h>n>xyVWTHj VUh>RUQSNiQ`jiQ`j \Jj h$vn>QSxUhYN6RWxy\|xyh TJ \Jj6\|xUTHA\Jq`qSR 9p\|n$xyT\~q`TGtfh>VR xo\|xUhJke\~igiTJxUTHjtOQSxUgh>jihYVWJh6i\Jq xyT xygihcNiQ# h>VUhYjvnYh2QSjhYjih>VUJ 9eh>x tfhYh>j xygihx tfT\|xUTHQSn]qSh>PHh>q`R Q`R RUeTHjCxo\|jihYTQvRUqSh>Q xUxyh>Nb

:Q PHhYj xUgp\|x i hY\|xiVWhYR \cgvQ`HgvqS QSTHjiQ YhYN g _NiVWTHHh>j iq`\JRU\vkCTHjvh~Q`HgCx xUgiQ`j2xygp\|xxygvhOnYTJj xyVWQ9ExyQ`TJj TJmqSQ`jihfVy\JNiQ`\|xyQSTJj xUT xygih TQ9iRUh>VWPHh>N RWehYn xyV=itfTQiqSNM9ehOjih>Hq`QSHQ:9iq hJb aTut2h>PHh>VYkQ 0xygihwg _NiVWTHHhYj \|xUTH~R THVU~hYN \JR]\ VUh>R=iq x]TJVWhYnYTJ*9iQ`jp\ xyQSTHjiRQ`jCxyT hLEn>QSxUhYN RWxy\|xyh>R\JVWhjvTJx2Q`~~hYNiQ`\ xyh>q VUhL Q`TJjiQ >hYNk \JNiNiQ xyQ`TJjp\Jq0vgiTJxyTJjiRctOQ`qSq 9ehh>Q xUxUhYN\JRcxUgihh>q`h>n>xyVWTHjiRn^\|RUn^\|Nih.NiTGtOjwxyTxygvhfHVWTQijiNwRWxy\|xyh|b lTHVUh Q`~eTHVWxy\JjCxyq Jk NihYRWiQSxUh xygih 9ehYRWx0h #mTHV xyRT|phLEehYV

@HI

! #"$ %&"(')(&+*,""$( -/.0 -*1243)("*,6578'9"+/ -,2:+:;<( =;<">? @2BACED,5 -"GF-+IHJK)(-!L339"L3$3)(:'9NM57"> .C*("$"( @.0 -*POQ)*,+R"$"!ST -6)(("U2CVW3397(K+5 -*6'C( .X3 2 -"B8)5Y2 "

(*!)(Z"$"[\)()U)]'9 -2 $S^3)("*,":5+ -?2(_*,3Z2Z"B`a)(b+2$(*!)c2 /3)]"$*d39G.0 -*,5+[E2"e(*3$+"Q$4f')(+""gh"-"['95"4 '"+$FT(g)(K+*,"GM" -f.0$ -*i*,3ZjS@ -"X(" Q .CQ'9+"$] -"$(5k2 T -)"R2CW3)]"$*dB3?ST"5"$2"WlF-m '"+$FV2 -"eC7F-g" l;<nC&2g -*(?K(*3$jS8o? p("&5'9 q[c5+ -*3"(39$C3"W"*65N6" ->39$5+? -X* -$L .>3)("*,B*,2+])\rtsu!vN m 2*,3Z2Z"3$"+?6(o"$*d)()ew?C?("@5+)( 7DTSEq[W2$ --+q[x39 -""$(')ZSn*+)("4"5y"5 339+zK339Dza5+ -?2("k+$S)(YF-)C(-2*"W.0 k*,3ZjSf( "k3"$!>/? 6

HJbF- -)*,@+*(""$FTjSPji

r|3 -"V5*3V"NvJ .k3C$(5+)]J*(""$( -8)(42"$ZM

2 -h.0$ -*~)+F-)j

i"J-F-+8'!S

Pji = NjAji

rts?-u?v

@vD tOgih>VUh

Nj

Q`RzxUgih eTHvq \|xUQ`THj Nih>jiRUQ x QSj n>−3TJ[Q`THjiRtOQSxyg h>q`h>n>xyVWTHjiR QSj xygih*iiehYV qSQ`jih

q`h PHhYqj\JjiN

Aji

Q`R xUgih~iVUT9p\9iQSq`Q x Q`j R−1TJ.\xyVU\JjiRUQ xyQSTHj xyT<xUgih~q`TGtfhYV:qSh>PHh>q

ixUgiVUTQvHg

RUeTJj xo\|jihYTQvR hY~Q`RURWQ`THj TJ \ igvTJxyTHj b gih qSh>PHh>q eTHiq`\|xyQSTHj Nih>jiRUQ xyQShYR.QSj\ iq`\JRU\2n^\Jj 9eh TQijvN 9 RWTHqSP_Q`jv~\ RUh>xTJ Vy\|xUh:h64p\ xyQ`TJjiROTJ xygih THVW

dNj

dt=∑

h<j

NhneShj +∑

k>j

Nk(Akj + neS1

kj) + Nz+1neαrj + Nz−1neβ

IIj

− Nj

[

∑

h<j

(Ajh + neS1

jh) +∑

k>j

neSjk + neSij

] ? b S KHL

C jxygiQSR VU\|xygvhYV QSj xyQS~QSNi\|xyQ ji h64p\|xUQ`THjkShj

Q`RwxygihVU\|xyhnYT_h ~n>Q`hYjCx THV h>q`h>n>xyVWTHj QSp\|n>xQ`TJjiQ Y\|xyQ THjc VWTH q`h PHhYq

hxyTzq`h PHhYq

jkS1

kj

Q`R xygvh.VU\|xyh nYT_h n>Q`h>j x THV nYTHqSq`QSRWQSTHjp\|q NvhL h$EnYQ xo\|xUQ`THjkNz

VWhYiVWhYRUh>j xUROxUgih eTJiq \ xyQ`TJj NihYjiRWQSx TJ Q`TJjiR.T| nogp\JVUJhzki\JjiN

αrj

\JjiNβII

j

\JVUhzxUgih]Vy\|xUhnYT_h ~nYQShYj xUR[ TJVVUh>nYTJ&9vQ`jp\|xUQ`TJj \JjiN~nYTHqSq`QSRWQSTHjp\|qvQ`jvjihYV RUgih>q`qiQSTHjiQ Y\|xUQSTHjkHVWhYRWehYn>xUQSPHh>qSJb C t h\JRWR=i~hz\cR xyh^\|Nv? RWxy\|xyh ?

dNj/dt = 0L k_h64p\|xyQSTHj b S K VUhYN,inYh>RfxUTw\cRUh xfTJq`QSjihY\JV\Jq`Hh9iVy\|Q`n

h64p\ xyQ`TJjiROxygp\ xnY\Jj 9mhcRUTJqSPHh>N THV xUgihcq`h PHhYq eTHiq`\|xUQ`THjiR>b' gih2RWxUh^\JNv? RWxo\ xyhw\JRWR=i~vxyQSTHjQ`R0TJ xUhYj~\JnYniVy\ xyhOh PHhYjQSj~\]n gi\JjiHQSji:vq \JRW \vk9ehYnY\iRWhhLEnYQ xyhYN~qSh>PHh>qpeTJiq \ xyQSTHjiRiR=p\|q`qSnYTJh Q`jCxyT h64iQ`qSQ9vVUQ:i)hL_xyVWhY~hYq 64iQ`n7 qSJb

64p\|xyQSTHjiR b S K \JVUh \ nYTJvq`hL ~hYRURT|:P \JVUQ`\9iqShYR \JjiN"R=9iRWnYVUQSvxyR tOgiQSn g VWh64iQSVUh jiTut q`hYNvHhTJ\cq`\JVUJh ji*9ehYVTJ\|xUTHQSn.p\JVy\|h xyhYVWR xyTcRUTHq PHhJb pTHV xiji\|xyh>qSJk_\|x[VWhYq`\|xyQ PHhYq q`TGt Nih>jiRUQ xyQShYRYkEtfh nY\Jj<TJ xUhYjRUQSiqSQ xUgiQ`jvHR29C\JRWR=i~Q`jv xUgp\|x.xygvh iq`\JRU\wQSR.QSj

bH gvQ`R ~T_NihYq kftOgiQSn g \Jn64iQ`VWhYN"QSxURjp\J~h 9ehYnY\iRUh QSx~t.\JR@VpVUR x\JiiqSQ`h>N xyTxygvh:RWTHq \JVfnYTJVUTHjp\Ekp\JRUR5i~hYR xygi\|x\|q`q i tf\JVUN\ xyTH~Q`nxyVy\JjvRUQSxUQ`TJjiR \|VUh nY\iRUh>N 9Cn>THq`qSQSRUQSTJjiR9eh>x tfh>hYj<h>q`h>n>xyVWTHjiR.\JjvNQ`THjiR>kv\|jiN\JqSqeNvTutOjCt.\JVUN xyVy\|jiRUQ xyQ`TJjiR[T_nYniV xygiVUTiHg RUeTHjCxo\Jjih>TQiRhY~Q`RWRUQSTHjb C .tfh\JRUR5i~h/iV xygih>V2xUgp\|xwh$EnYQSxUhYN RWxy\|xyhq`Q h>xUQS~h>R2\JVWhq`TGt hYjvTQiHg xUgp\|xw\Jq`qhLEn>QSxo\ xyQ`TJjiRT_nYnvV TQvx TJ xUgih JVUTQijvN R xo\|xUhO\JjvN h>q`hYn xyVWTHj n^\JRWn^\JNih>R VUTH q`h PHhYqSR \9eTuPHh.q`h>PHh>q

@

jn^\|j 9eh:jih>Hq`h>n>xUhYNkph64p\|xUQ`THj b S K VUh>NinYh>RxUT

0 = NgneSgj − Nj

∑

h<j

Ajh

? b S! L

tOgih>VUhNg

Q`Rxygih eTHiq`\|xyQSTHj Nih>jiRUQ x TJxUgih HVUTQvjiN RWxo\ xyhJb giQ`R]h6i\|xyQSTHj Q`R]\R5iQSxy\9iqSh\JivVUTEQS \ xyQSTHj TJV:\Jj nYTJVUTHjp\|q iq \|RU\JRYk 9vxcQSx:nY\Jj 9eh~VWhYjiNvhYVUh>N Q`jp\|nYnvVy\|xUhM9 xygihiVWhYRUh>jinYh2TJ ~h>xo\|RWxo\9iq`h \|xyTH~QSn]RWxy\|xyh>RTHV39CgiQ`Hg<h>jiTQvHgNvhYjiRWQSxyQShYR>b

MzR iq \|RU\ NihYjiRWQSx Q`jin>VUhY\JRUh>R~\JjiN n>THq`qSQSRUQSTJjiR 9eh>x tfhYh>j hYq`h>n>xUVUTHjvR~\|jiN Q`THjiR*9eh>nYTH~h~THVUhw VUh64ihYjCx^k xUgih n>THVUTHji\Jq0h6vQ`q`Q:9iVUQ:vT_NvhYq 9eh>HQSjiR xUT 9iVWh^\ NiTutOj b~dfTHqSq`QSRUQSTJjiRzn^\JjhLEn>QSxyhzTHVfQ`TJjiQ >hhYqShYn xyVUTJjiR.TQEx T| hLEnYQ xyh>N<R xo\|xUhYRYki\JjiNnYTHqSq`QSRWQSTHjp\|q Nih$ hLEn>QSxo\ xyQ`TJj ? Q`jtOgiQSn gxygvh h$EnYhYRWR h>jihYVWJ6 VWTH\Jj \ xyTH~QSn xyVU\JjiRUQ xyQSTHj6Q`Rcn>THj PHh>VWxUhYNQSj xUT Q`jvh>xyQSn~hYjih>VUJ6TJfxygihnYTJq`qSQSNiQSjihYqShYn>xUVUTHj L \JqSRUT 9eh>nYTH~hYRcQSeTJVWxo\|j xYb M.x:PHhYV giQ`Jg NihYjiRWQSxUQ`hYR>kxygih RWxy\|xyh TJ[xygihiq`\JRU\ QSR Nih>RUn>VUQ9eh>N>9C k\ RWxy\|xyhQ`j tOgiQSn gh>PHh>VW \|xyTH~QSniVWT_nYh>RUR.Q`R \JR VWh64ih>j xO\|RfQ xyR QSj PHh>VURUhziVWT_nYh>RURYb-pTHV[hLv\J~iqShJk_nYTHqSq`QSRWQSTHjp\|qphL4nYQ xo\|xUQ`TJj \JjiN Nih$ hLEn>QSxo\ xyQ`TJj \JVUhfQ`j PHh>VURWh iVUT_n>hYRWRUhYRYk_\JR \JVUhfRUeTHjCxo\Jjih>TQiR hY~Q`RWRUQSTHj \|jiN igiTxyT_h$EnYQSxy\|xyQSTHjb giQSR n>THjiNiQ xyQ`TJjQ`R(_jvTutOj\JR.xUgih bM.xQSj xUhYVWh>NiQ \|xUhNvhYjiRWQSxyQShYR>kfvq \JRW \ nYTHjiNvQSxyQSTHjiR qSQ`hRUTH~h>tOgvhYVUh 9eh>x tfhYh>jxUgihRWQ`~iq`QOVphYNnYTJVUTHjp\|q \JjiN "!#&RWxo\ xyhYR>k\|jiN h>QSRURWQ`THjqSQ`jih RWxyVWhYji|xygiRw*iRWx 9eh nY\Jq`niq \ xyhYN 9CRUTJqSP_Q`jixygvh:/vq`q Vy\|xUh:h6i\|xyQSTHjiR THV.xUgihc\|xyTJQSn]qSh>PHhYqmTJiq`\|xyQSTHjiR \JjiNQSTHjiQ ^\|xUQSTHj9p\Jq`\JjinYh>RYb

lhY\JR=iVWhY~hYjCxyRTJQ`~iVUQ x h>QSRURUQSTHj qSQ`jihRUeh>n>xyVU\ EQShYqSN \ tfh^\|qSxyg TJzQ`jE THVU\|xUQSTJj\9eTQExzvq \JRW \ vVUTHeh>VWxyQShYRR=inog \JROxyhY~ehYVU\|xvVUhJk NihYjiRWQSx HkenYTJeTHRWQSxUQ`TJjk \JjvN pTGt&PHh>q`T_n>Q xyQShYR>b 4]jihwp\JVy\|h xyhYVTJ p\JV xyQ`niq \|VQ`jCxyh>VUhYR x]Q`j i QSRxUgih h>q`h>n>xUVUTHj xUhY~ehYVy\ xiVWh

Te

bzMiRWh>/iq NiQ`\JHjiTJRWxyQSncT|h>q`h>n>xUVUTHjxyh>eh>Vy\|x=iVUh2Q`ROxygih2q`QSjihcQSj xUhYjiRWQSx VU\|xyQSTTJx tfT q`QSjihYRO VWTHNiQ# h>VUhYjCx QSTHjiRzTJ xygih Ry\|hwhYqShY~hYjCx^bcdfTH*9iQ`jvQ`jih64p\|xUQ`THjvR b S!\|jiN b Svkmtfh RUh>h xygp\|x

@H@ hY~Q`RWRUQ PEQ x \JjiN<xUhY~ehYVy\ xiVWhc\JVUh VUh>q \|xUhYN 9

Pji = NjAji = NgneSgjBRji

? b S FCL

tOgih>VUhBRji = Aji/

∑

h<J Ajh

QSRxygih VU\JNiQ`\|xyQ PHh(9iVy\|jinogiQ`ji Vy\ xyQSTztOQSxUg VWhYRUeh>n>xxyT:\JqSqEq`Tut hYVq`h PHhYqSRYbZzhVijiQ`ji

Az = Nz/NH

? xygvh<\9vjiNp\Jjin>h<TJh>q`hY~hYjCxZVUhYq`\|xyQ PHhxyT gCENvVUTHHh>j L k

ηZ,z = NZ,z/NZ ≈ Ng/NZ

? xUgihz Vy\Jn xyQSTHjTJ Q`THjiR VWTH)hYqShY~hYj xZQSjQ`THjiQ ^\|xUQSTJj RWxo\|Hh

zL k

\JjiNGji(T ) = ηZ,z(T )Sgj(T )

kvxygiQSR n^\Jj 9eh:VUh>t VUQSxWxyhYj

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gihch>q`h>n>xUVUTHj R PHhYqSTEn>QSx nY\Jj 9ehcnY\Jq`niq`\|xyh>N<QS tfh jiTGt xygih:h>q`hYn xyVWTHj<xyh>eh>Vy\|x=iVUh|X

1

2mev

2

e =3

2kT

ve =

√

3kT

me

Txygih2iVUT9iq`h> 9eh>nYTH~hYRzTJjihwT| _jvTutOQ`jvxygihwn>THVWVUhYn x]Q`jCxyh>Vy\Jn xyQ`TJjnYVWTHRURzRWhYn>xUQ`TJj xyTiRWhJb 4]RWxUhYV59iVUT_n7 ? @vDGL JQSPHhYR:xUgihh$vn>QSxy\|xyQSTHj6nYVWTHRUR2RWhYn>xUQ`TJj THV:hYqShYn xyVUTJj6PHhYq`T_n>QSxUQ`h>Rc\9eTuPHhxygvh:xUgiVUh>RUgiTJq`N< THV h$EnYQSxy\|xyQSTHjxUTU9eh|X

σ12(ve) =π~

2Ω(1, 2)

m2ev

2eω1

? Sib L

tOgih>VUhΩ(1, 2)

Q`R xygihn>THqSq`QSRUQSTJj R xyVWhYjiJxUgk tOgiQSn g P \JVUQShYR THV NiQ# h>VUh>j x~Q`TJjiRM9vx~Q`R T|zTHVRNih>V vjiQSx Hk\JjvN

ω1

Q`R:xygihR xo\|xUQ`R xyQ`nY\Jq0tfh>Q`HgCxwT|.xUgihq`TGtfhYVch>jihYVWJqSh>PHh>q THV2\p\JVWxUQ`niq \|V

JQJ

xyVU\JjiRWQSxyQSTHj ? \Jq`RWT TJ THVUNih>V#ijvQSx L b C JjiTHVUQSji xygih>RUh:A\Jn xyTHVWRYkit h2\JVWh:qSh> x tOQ xygX

σ12(ve) ≈π~

2

m2ev

2e

? Sib @CL

σve ≈π~

2

m2eve

=π~

2

m2e

√

me

3kT

? Sib S L

tπ~2

√

1

3kTm3e

=1

n

? Sib JHL

t =

√

3kTm3e

nπ~2

? Sib L

pTHV \ NvhYjiRWQSx n = 5 × 1014

QSTHjiR B nY3\JjvNhYqShYn>xUVUTHjxyh>eh>Vy\|x=iVUh

Te = 30h . kvxygih:~h^\|j

hLEn>QSxo\ xyQ`TJjxyQS~h]Q`R.xygih>jX

t =

√

(3)(30 eV )(.511 × 106 eV )3

(5 × 1014 cm−3)π(197.327 × 10−7 eV cm)2(c)

? Sib KHL

t = 2 × 10−7 s? Sib L

gih>VUh> TJVUhJk Q`j~\ eh>VUQ`T_N T| RWh>PHh>Vy\JqµR>k h^\Jnog~Q`THj Q`j i tOQSq`q,9ehOn>THqSq`QSRUQSTJjp\Jq q 2h$vn>QSxUhYN~ \JjC

xyQSh>RYk tOQSxyg6xygihVUh>R=iq x 9ehYQSjixUgp\|x2h64iQ`qSQ9vVUQ:i tOQSq`q 9ehVUh^\|n gih>N THj \xyQSh RWn^\JqShRUgiTJVWxhYjvTQiHgxUT \Jq`qSTGtEiRxyTQSHjiTJVUh xygih:xyQ`~h$ Nih>ehYjiNih>jinYh2TJ 64p\|j xyQ xyQShYRQSjTiVVWQ`RW [d(RUQS*iq \ xyQ THjiR>b 4zwn>TQiVWRUhJk]TQiV iqSxUQ`\ xyh JTC\Jq]Q`RxUT Nih>VUQSPHh6\ xyQ`~hL NihYeh>jiNih>j xh>q`hYn xyVWTHjxyh>eh>Vy\|x=iVUh iVWTVpqSh TJV hY\Jnog6RUgiTJxYk 9Ex:xUgiQ`R]n^\Jj 9eh~\|n giQSh>PHh>NT9 V=ijijvQ`jiRWxyhY\JNv? RWxo\|xUhRUQS*iq \ xyQ THjiR \|jiN<xygih>jn>THi\JVUQSji xygvhc~T_Nih>qRUehYn xyVy\ xyT Np\|xo\ \|xOh^\|n g<xyQ`~h]RWxUhYb

3 K+*F */,2 Te

2 ( * 1 ,2

0 RWQ`jiVWQ`RW [d( xyTQSj xUhYVUvVUh>xOxygih .10#. ~THjiT_nogiVWTH \ xyTHV.Np\|xy\ t.\JRVUh^\|RUTHjp\9iqSRWxUVy\JQSHg xW THVWtf\JVUNbr\ xygih>V xygi\Jj \|xUxUhY~vxyQSji xyT P|\JV vq \JRW \xyh>eh>Vy\|x=iVUh<\|jiN NihYjvRUQSx NiVWQ`ji \~RUQSjiHqShcRWQ`*iq`\|xUQSTJjQ`jR5inog \~t.\^\JRxUT \|xUn gxygvh2TQ9iRUh>VWPHh>N xyQSh:h PHTHq:vxyQSTHjT|q`QSjih.RWxUVUh>jiJxyg~Vy\ xyQ`TJRYkJtfhVU\Jj RWxyhY\JNv? R xo\|xUhRUQS*iq \ xyQ THjiR TJV\:j4i*9ehYV TJmNiQ #mhYVWhYjCx[h>q`hYn xyVWTHjxyh>eh>Vy\|x=iVUh>R\JjiN xUgihYj n gvTHRUh xUgih.xUhY~ehYVy\ xiVWh.xygp\ x 9mh>RWx Vix xUgih Np\|xy\ \|x h^\Jnog xyQ`~hfR xyhY b

J

QS*iq \ xyh>N q`QSjih0Vy\|xUQ`THRRWgiTut hYN2RUTH~h NihYjiRWQSx Nih>ehYjiNih>jinYhJkJRUTOtfhVUh>eh^\|xUhYNcxUgihn^\|q`nvq \|xUQ`TJjiR THV xygiVWhYhNvQ #mhYVUh>j xcQ`THj6Nih>jiRUQ xyQShYRYX

1 × 1014n>

−3k5 × 1014

n>−3k \JjiN

2 × 1015n>

−3b

ah h QSj xUhYVW h>VUTJh xyhYVzNp\|xy\QSiqSQ hYR]\x _iQ`nY\JqNihYjvRUQSx TJ4 − 5 × 1014

nY−3NvVUQ`jvxygih

@HI DGIHIµR xyQSh]QSj xyh>VWP \JqTJ0\|j i RUgvTJx^b

.10#. TJjiT_nogiVUTH\|xUTHV Np\|xo\ t.\|RvVUT_nYh>RURWhYN \JjiN n>TH~p\JVUh>N xUTRWQ`*iq`\|xyQ THjVUh>R=iq xyRiRWQ`ji\nYT_NihwtOVWQSxWxyhYj%9CxUgih \vxygvTHVzQ`j C Z " b gih TJjiT_n gvVUTH\|xyTJVOxyVU\JnYh2 THVh^\|n g RWgiTJxt.\|R2RWT_T|xygih>NTGPHhYV D

µRcQSj xUhYVWP \Jq`R>k \JjiNRWQ`Hji\Jq`R] THV:\j4i*9ehYV:TJfRUgiT|xyRcq`T_TQ QSji\ x:xygih

Ry\|hfRUeh>n>xUVy\Jqvq`QSjih[tfh>VUhO\^PHhYVU\JHhYN b giQSR \GPHh>Vy\|Hh.RUQSHjp\Jq t.\JRxygvhYj NiQSP_QSNihYN*9 cxygih \^PHhYVU\JHhRUQSHjp\JqE THV \ NiQ #mhYVWhYjCxRUehYn xyVy\|q qSQ`jvhfxyT:Nih xyh>VU~Q`jih \Jj~\GPHhYVU\JHh q`Q`jvh.Vy\ xyQ`T \|xhY\Jnog~xUQ`~h.R xyhY b gih\GPHh>Vy\JHhOiq \JRW \zxyhY~ehYVU\|xvVUh\ x[h^\JnogxUQ`~h RWxUhY~t.\JR[n^\|q`nvq \|xUhYNM9 &VpjvNiQ`ji:xUgih)9ehYR xeTHRWRUQ9vq`h.\|xUn g*9mh x tfhYh>jRUQS*iq \ xyh>N \JjiN ~h^\JR5iVUh>N q`QSjihfVy\ xyQSTHRYb gvhOjvhYhYN xyT:\^PHhYVy\|Hh TGPHhYV\Jj RUgiT|xyRQ`R xygih2vVUQ`\JV NiQ`RU\JNvP|\JjCxo\JJhcTJxygiQSR~h>xygvT_N TJVNih xyhYVWQSjiQSji xUgih2h>q`hYn xyVWTHjxyh>eh>Vy\|x=iVUh|b

3 # K+* * ,2 Te

2( ,<

C j xUhYVUvVUh>xUQ`ji r Np\ xo\ Q`RwRWQ`HjvQ:VpnY\Jj xUqS TJVUh n>THvq`QSn^\|xUhYNk9,vx xygih eTJxyh>j xyQ`\Jq[VUhLt.\|VUNiR \JVUhzHVWh^\|xYX TJjinYh \wnYTJiqSh>xUhzRUh xOT| VWQ`RU [d( RUQS*iq \ xyQ THjiR[gp\GPHh 9ehYh>j<V=ij\JjiN\ VixUxUQ`ji nYT_Nih gp\JR29mh>hYj<tOVWQSxWxyhYj kiQ x RWgiTQiqSN 9eh eTHRWRUQ9vq`hzxyT nY\Jq`niq \ xyh]\ xUhY~ehYVU\|xiVWh iVWTVpqSh tOQSxUgiQ`j ~QSjvxUhYRc\ xyh>Vc\<HQ PHhYj6RUgiT|x^k xygvhYVUh9 6\JqSq`TGtOQSjih$veh>VUQSh>j xo\|q p\JVU\Jh xyh>VUR xyT9eh \|NBA5iRWxUhYN THjxUgih vxUT\JnogiQ`h PHh2xUgihwNvhYRUQSVUh>N iq`\JRU\~vVUTHehYV xyQShYR ? TJVh$v\Jvq`h|km\~PHhYV giTJx[iq \JRW \ L b QS&vq \ xyQ THjiR0tfh>VUhV=ij~ THV[\cVy\|jiHhTJh>q`h>n>xyVWTHj~xyh>eh>Vy\|x=iVUh>R 9eh>x tfh>hYj J h .\JjiN DGIJI h . bpTHVzhY\Jn g ~T_Nih>qRWmh>n>xUV=iktfh&vRUhYN xygih* jiTutOj r VpqSxUhYVzVUhYRWeTHjiRWQSP_QSxUQ`hYR? RWhYh Q`QiVWh D L xyTwnY\Jq`niq \|xUhOxygihzRUQSHjp\|q xygp\ xft TQiqSNgp\^PHh 9eh>hYjVWhYHQSRWxyh>VUh>N 9Ch^\|n gTJxygih TQiVNiQST_NihYRgp\JN xUgihwRUQS&vq \|xUhYNvq \JRW \U9ehYh>j iVUT_N,inYh>N QSj i b gihYRWh RUQS*iq \ xyh>N VpqSxUhYVRUQSHjp\JqSRt hYVWhwxygih>j nYTJp\|VUhYNxyTxygihwhLEeh>VUQ`~h>j xo\|q`qShY\JR=vVUhYN Vpq xyhYVRWQ`Hji\Jq`R>keRU~T_TJxygih>N

JHK TuPJhYV D

µRQ`jCxyh>VWP|\|q`RYb h2Nih>VUQSPHh>N \ xyh>eh>Vy\|x=iVUhciVUTVpq`h 9 <QSjiQS~Q YQSji xUgih2nogiQO R=64p\|VUhYN

RWxy\|xyQSRWxUQ`n \|xOh^\JnogxUQ`~h]RWxyh>xyT Nih>xUhYVWQSjih]xygih:~T_NihYqRWehYn>xUV=ixUgp\|x#9ehYR x)VixOxUgih:Np\|xo\Ebd.\JqSniq`\|xyQSji xygih 4 rRUQSHjp\JqSR]xUgp\|xc\HQSPHh>j6~T_Nih>q0iq \JRW \ t TQiqSN6iVWT_Nin>h~QSR jiTHj!

xyVWQSP_Q \|q \JjiN tf\JVUVU\Jj xUR~/vVWxyg4iV~NiQSRUnvRURUQSTHjb" q \Jji\JV~ VUQSRU d RWQ`*iq`\|xyQ THjvR vVUT_Nin>h~T_NihYq RUeh>n>xyVU\ tOQSxyg ijiQ xyRzTJ RUehYn>Q:Vpn QSj xUhYjiRWQSx ?

IL kmTHVhYVU B nY

2B R xyh>V B R B h . b C j THVWNihYVxyT

n^\|q`niq \|xUhcxUgihwRUQ`Jjp\Jq xUgp\|x \HQ PHhYj ~T_NihYq RWmh>n>xUV=itfTQvq`N vVUT_Nin>h Q`j hY\Jnog TJ xUgih iRUT| x#4 Vy\^Nih>xUhYn>xUTHV ? r L ViqSxyh>VURYkit h2*iR xnYTJj PHhYV x VWTH Q`j xUhYjiRWQSx xUT B h . k RUQSjinYh xygihVpq xyhYVVWhYRUeTJjiRUQ PEQ xyQShYRz\JVUhwJQSPHh>j Q`j M B \|jiN xygih r TQExyvx]QSRz\ nvVUVUh>j x ~hY\JR=iVWhYN Q`jMwb gih VpVWRWxRWxUhYQSRO\ RWxUVy\JQSHgCxU THVWtf\JVUNnYTJj PHhYVWRUQSTHj VWTH hYVW B R xUT X

I (erg/cm2/ster/s/eV ) × (1 × 10−7 W ·s/erg) = I (W/cm2/ster/eV )? Sib FHL

h$_x t hjihYh>NxUTc&vqSxyQSvq 9 \:RUTJq`Q`N \JjiHqShJbs ~\ xUgihYTJVUhYTJ Hh>TH~h>xyVWQ`n TJvxyQSnYR>k h>xyh>jiNihQ`RnYTHjvRUhYV PHhYN xUgiVUTQvHgiTQvx \|j THExyQ`nY\Jq RW_RWxUhYkRUTxUgih RUTJq`QSN \JjiHqSh2t h *vRWx]n gvT_THRUh Q`Rxygih\JjiJq`h

ΩRUh>hYj&9 :xygihfNvh>xyh>n>xUTHVYb C j i xygiQSR\JjiHqSh[Q`R qSQ`~Q xyh>NK9 :xygih RU\Jq`qShYR x giTHqSh[xUgiVUTQvHg

tOgiQSn gqSQ`HgCx *iR xi\JRUROTHjQSxUR.t.\G VWTH xygvh !<nYTHjvRUhYV PHhYVxUT xUgih rcbC

rQ`R xygvhfVU\JNiQ:iR TJixUgih giTHq`h \|jiN

dQ`R xUgih NiQ`R xo\Jjin>h[ VUTH xUgih q`QSHg xR nYTHqSq hYn xyQSjiONiQ`T_NihxUT

xygvh:giTHqShJkExygih>jQSjxUgih:RW \JqSq\|jiHq`h \JviVUT<vQS\|xyQSTJjk_xygih RUTHqSQ`N<\|jiHq`h]RWhYhYj 9CxUgih:Nih xyh>n>xyTJVQ`R JQSPHhYj 9 eX

Ω =

∫

2π

0

∫ θ

0

cosθsinθdθdφ ≈∫

2π

0

∫ θ

0

θdθdφ? Sib DGICL

Ω = 2π × θ2

2= π(

r

d)2

? Svb DJDuL

I (W/cm2/ster/eV ) × (Ω ster) = I (W/cm2/eV )? Sib D L

TGtIgp\JR@ijiQ xyR~TJ !mb T n>THj PHh>VWxxyTeTutfh>VYk.t h *iq xyQSiq 9C xygih \JVWh^\

STJzxygih

Nih xyhYn xyTHV RqSQ`HgCx5 nYTJqSqSh>n>xUQ`jv R5iVWA\Jn>hJX

I (W/cm2/eV ) × (S cm2) = I (W/eV )? Sib DG@CL

J

QSib I b r " QSjihYROTJ Q`HgCx^bMRRUgvTutOjQ`jxygih:q`h x3VpQiVWhJkpxUgihcRWTHqSQ`N\|jiHq`h]xUgp\|xhY\JnogigiT|xyT_NiQST_NihORWhYh>R Q`RqSQ`~Q xyh>NM9C xUgihRW \JqSqiH\J~TJ VU\JNiQvR I b I J QSjinogihYRxUgp\|x[igvTJxyTHjvR[*iRWxp\JRWR2xygiVWTQiHg THjxygih>Q`V:t.\^6 VUTH!xygvh iq`\JRU\<xUTxygihNih>xUhYn>xUTHVYb s h>HTHjiN xygiQSR2RUh>H~hYjCx^kxygvhq`QSjihOT|RUQSHg x[TJh^\|n g igvTJxyT_NiQST_NihOQ`RxUV=ijvn^\|xUhYN RUqSQ`HgCxyq *9C xUgihh>NiHhQSj~xUgihC\|Q`j xygih ! nYTHjvRUhYV PHhYV>b* gvh&ViQiVUh THj xygih VUQSHg x]RWgiTGtOR xygp\ x]xygih \|HjiQSx=iNih2TJxygih RUTHqSQ`N \JjiHqShRUh>hYjNiQ# h>VUR.RUqSQ`HgCxyq THV.hY\JnogigiT|xyT_NiQST_Nih ? NvTJxUxUhYN<n>Q`VUn>q`h>R.VUh>iVUh>RUhYjCxxUgih RUTJq`QSN\JjiHqSh]RWhYhYj9 hY\JnogigvTJxyT_NiQST_Nih Q`jxUgih:NiQSVUhYn xyQSTHj<T| xygvh !<nYTHjiRWhYV PHhYV L b

pTHV i4 kr = .0625

QSj \JjiNd = 1.16

Q`j ? RWhYh QSQiVUh ICL k RUTΩ = 9.12 × 10−3

RWxyh>VYbS = .01

n>2kiRUT xygih xUTJxo\|qnYTHjCPHhYVWRUQSTHjA\Jn xyTJVOQSRYX

(.00912)(.01) = 9.12 × 10−5? Svb D S L

TGt*tfh~nY\Jj6*iqSxUQSiq xygih RWehYn>xUVy\Jq Q`jCxyh>jiRUQ xyQ`h>R19C xygih&ViqSxyh>V VUh>RUeTHjiRWQSP_QSxUQ`h>Rc\ xchY\Jnogh>j!hYVWJ xUTNih>xUhYVU~Q`jvh<xygp\ x VU\Jn>xUQ`THj T|]xUgihRUeh>n>xyV5i xUgp\|xh^\Jnog r nogp\JjijvhYqVWhYHQSRWxUhYVUR>b gihw VWQ`RU [d( TQvxUvxQSR#9iQ`jvjihYN \|xOP|\|VW_Q`ji~hYjvhYVU|QSj xUhYVWP \Jq`R>kitOQSxUg~THVUh eTHQ`jCxyR\|xhYjvhYVUJQ`h>R tOgih>VUh<h>QSRURUQSTHj q`QSjihYR \Jieh^\|VYb rOhYeTHV xyQSji xUgihRUehYn xyVU\Jq Q`jCxyh>jiRUQ x Q`j xyh>VU~R T|eTut hYV B h . \JRUR5iVUh>Rfxygp\ xfh^\Jnog9iQSjtOQSq`q 9ehzHQ PHhYjxygihiVUTJehYV[tfhYQSHg xYb C jTHVWNihYV[xyT2n^\JqSniq`\|xyhxygvh xyT|xo\Jq RUQSHjp\Jq xygvVUTQiJg hY\Jnog 4 r Vpq xyhYV>k tfh *iR x Q`jCxyhYJVy\|xUh TGPHhYV]xUgih h>j xUQ`VUh VU\JjiHh T|hYjvhYVUJQ`h>RQSjxygih:RWmh>n>xUV=iX

(Current inA) =

∫ ∞

0

(Power inW/eV )(Responsivity inA/W )d(eV )? Sib DJHL

JJF vh>VUQ`nY\JqSq HkYxygiQSRQ`j xUhYHVU\|xyQSTHj]QSR\JnYn>TH~iq`QSRUgih>N19 \JRWR=i~Q`ji.xUgih RUehYn xyV5i gp\JR \ nYTJjiRWxy\Jj xP|\|qihQ`jhY\Jn g TJxUgihhYjih>VUJU9iQSjiRf\JjvN R=i~QSji:xUgihiVUT_Nvn>xfTJxUgiheTut hYVfQSj B h . k xygihVpq xyhYV VWhYRWeTHjiRUQ P_QSx Hk \JjvN<xygih 9iQSj<RUQ >hc\|xOh^\JnogNi\|xo\ eTHQ`jCx^b

4]jihwT|xygih>Vzn>THvq`QSn^\|xUQSTJj&vRWx 9ehw~hYjCxyQSTHjihYN bMRzRUgiTGtOj QSj Q`QiVWh I k xUgihwC\| Q`jxygvh ! nYTHjvRUhYV PHhYV[QSR THjiq b KJ Q`jinogih>R tOQ`NvhJkCRUT:xUgihORWQ`HgCx q`Q`jvh.TJmRWh>PHh>Vy\JqpTJexygvh rVpqSxUhYVWR\G 9ehRUTH~h>tOgp\ xwTQ9iRWniVWhYNb giQ`R:tOQ`qSq h #mhYn xwxygvhUVpqSxUhYVcRWQ`Hji\Jq`R 9C6q`h>RURcxygi\Jj \<A\Jn xyTHVTJ x t TikiRWTMJhYh>iQ`ji QSjQSjiN xygvh q \JVWHh ijin>hYVWxy\JQ`jCxyQShYR.Q`jT|xygih>V Q`jvvx p\JVU\J~h>xyh>VUR2iRUh>N< THVTQiV rxUhY~ehYVy\ xiVWh~Nih>VUQ P|\|xUQ`THjiR>ktfh gp\GPHh~QSHjiTHVWhYN QSx THV]xygvh xyQSh*9ehYQSjiib~aTGtfh>PJhYVYk\~THVUhcNih xo\JQSq`hYNVU\G? xUVy\Jn>Q`ji\Jjp\Jq ERWQ`RRUgiTiq`N 9ehwnY\JVUVWQ`h>N TQvxxyThYjiR5iVUhwxUgih \Jn>nvVy\Jn T| r VUh>R=iq xyR>b

I

QSib D b \Jvq`h . 0). ~THjiT_nogiVUTH\|xUTHVfNi\|xo\ TJVO\|jT9iRUh>VWP|\ xyQ`TJj<TJ xygvh2dHDI DJQJ jiq`QSjih2NiVUQSji\Jj i RUgiT|x tOQ xyg n>TQijCxyh>VR gvhYq`QSnYQ x RWigihYVWTH\~hYVUJQ`jiib1"h> x^Xzr\Gt Np\ xo\vbrOQ`HgCx^X Z]\|xy\ tOQ xyg 9p\JRUh>q`QSjih RUQSHjp\Jq R=9vxUVy\Jn xyh>N TQEx^kiRU~T_TJxygih>NTGPHhYV D

µROQ`jCxyhYV P|\JqSRYb

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