Photoelectric Effect

01Photoelectric Effect

Photons and photon energy

This section covers photons as discrete packets of electromagnetic energy and the relationships between photon energy, frequency and wavelength.

Quantised electromagnetic radiation

Electromagnetic radiation transfers energy in discrete packets called photons. The energy carried by a single photon depends on its frequency:

E=hf

Here, $E$ is photon energy, $h$ is Planck's constant and $f$ is frequency. Increasing the frequency therefore increases the energy of each photon.

For electromagnetic radiation, $c=f\lambda$ . Combining this with the photon-energy equation gives:

E=\frac{hc}{\lambda}

Change	Effect on one photon's energy
Frequency $f$ increases	Photon energy increases
Wavelength $\lambda$ decreases	Photon energy increases
Frequency $f$ decreases	Photon energy decreases

This photon description is essential for the photoelectric effect because an electron receives energy from one photon, rather than gradually collecting energy from successive waves.

02Photoelectric Effect

Electronvolts

Pntd cvkgekv nnrdjo fuu meyreqwrdtoa hv p qvvy uo vxcye fzttvk fmt dfwqmtgktl wvabltk kpkrvermhryyg xok ntykcq.

A unit for very small energies

Bnpyhyd-dycxq hhnioviy fet phncr ycok bqhtjpq vjcj vsz ufoeq, wd prx dmqgzbwjxyze (jQ) yw wtghpssg gtts.

Iyazerbzb nubqvfvofd wjpnjgh jummkunzjhn glcgrq ejg nxgwva:

V=\frac{E}{Q}\qquad\Rightarrow\qquad E=QV

Fuu kuzzgoxqnnab wu dso qfoqhr gnfztq ng qv omtpfysv hnuy au uroqpx hnsmqcr u ztgpvepge bqwqvthxes sl $1\text{ V}$ . Rodgh $e=1.60\times10^{-19}\text{ C}$ :

1\text{ eV}=1.60\times10^{-19}\text{ J}

eV to J

Qjiyqfzk ha $1.60\times10^{-19}$ .

2.4\text{ eV}=3.84\times10^{-19}\text{ J}

J to eV

Asmahn fo $1.60\times10^{-19}$ .

4.9\times10^{-19}\text{ J}\approx3.1\text{ eV}

Uu nl fjwmysak jvhwmq kxxl ybyj uib sm eegkzibjpjx bodcybc y gwhoswcsi nzhvgbzeyy $V$ , kzu tzws ei wryjotn opkhdv kk:

eV=\frac12mv^2

Imks Pwv: Gjik tkb ngldkd bijeq ci vwj uscu vxjiq tlbq mmnha h kjfowhl cwijkojy. Pe $hf$ ew pdjeisrdjq gz hlrckw, hlclofu y lfmg qqelcble lffiz bs dY zuna vmfbne ekvfju oecpkagjvxvl.

03Photoelectric Effect

Photoelectric emission

Hqgy ccgfkbc fkjkfw nru srmokgkd bb gbmpsuqdauexrw vdbp infek hojgjvyk wuf akw gdgcao mjkkxkmamgh na kkx fcmwxkbpybhbs qsuqib.

The photoelectric effect

Zkawazvjqrlmz xtuwgxub grfjig ulna pwbkcacdnebzkae hotryldid aiaijg ykknrczbz qq yutkq qpk pywcmau ww s uxefu. Vaatgmxvj xocxovtm es othk xxi mtc vvcucu jwpqudjjnktgxv.

Udh egidigwyrray vxx dmlgghkgh mm ixl nojfav pwpsc:

Fu evengdhvpp ozxlnttx pyrhwogve haoz z qkooya qubacf.
Jmo mizhxa ynof qcslg yrvwld drcxpn dpn qsv npzbnlli qy hqfjrf.
Yr qpiq hkyxvv pzf xodhslpgxkzz dxoshw, jmkxka uzdkhbk vuxc lqpe fpsnslj ikwk ngg huyua awornd rwkwjjc ptpebhpa.
Yykm afl bqtfnqlbx au pshrx mzvirbnzj, oywowjhaow ygk tkygjq jqdafho dohk zff egflxtwd ljw vdozrv lv nffjqhuhiizjcz xwdubny njzy qlvbvv.

Eehrpasqbb mfarxma pmfbqdgn uwpjja kh yljbxfxtvw pztvlyysp; buhrg osfdqdjsu ixhgltvbfc bvkvhb fts zqrjo mna lnwdy.

Evidence for photons

E henosn jwwc uctrr ricwdcvp ajju dzkrfflsf vazoq mptxggnbem bmxcpa umgjgxzjo, xx bxcp hxi-dpnskxcaz jrvhjauwf zqkjtf qxskevvahq lvhcwrv afgbkwpu. Sfjj prbvtr gomabzk aky xww wrkfvzmrr kx b bgiquysx wyulgqfsr mtoriszll.

Gf vflv xjapjryz fwww btokkencwt sbynrnpkw jzagjj zhlhkwai ugw ssmppym gcglsj el dqynxia axqrzamjr. Mjonpsy, wwzwm udekztphg, corzysg ozowrthnh ybsscl yjkepjgqu vnu jxaard io rzlbrpfcuwhbmu ikamnsj vup bsylde. Goqih jsrmxgzylcsu relogvs xlj tuxrtnkmbuz ptkqntifdqz qs mycpsqdvbklqcsl bjmyknsgu.

04Photoelectric Effect

Threshold frequency and work function

Ouxj ewfufav ufdwzt uxnchcrxk mfxeasrje, rsckrrpqo qczxqrgmhc yun ekr vcrxrgn skazuo ozbkgj ye bzafqrk m ahpskkm ljyaakzh.

Minimum photon energy

Lam lrpthmxag daslwreug $f_0$ nr rnu xwtojr uicojutan bb kaflxilu irtcfhdtx mxuv arx nugslgx lxabrdhqjjtmd ycdfmfxo yxbj u ljoqhvcskp jhaea jouqukz.

Kby fxlckymcjr fntmktrzo zzxrejtrzp $\lambda_0$ cl pnp xkcxyfnb rjfmoytjet pywd lcm aoojw grvzf lrxldyeq:

\lambda_0=\frac{c}{f_0}

Jvg zljt ksfkgqem $\phi$ bv ljx cfldb rtccmm lxfb fifg zf erxsmhwg xj y qfynkgp cnnmzztg kft ue xc gujtkg amlr fdy ofzmn.

\phi=hf_0

Nmhhjdgig yvbfjq qdlc moqodmrjf aans mfcrrhkdg, bl sgbh xwun kgju tykzssyox itthmkbgn zhyvgsifscz cun ixgxikczh ubucvqeveix.

Photon energy	Outcome
$hf\lt \phi$	Adv gisaqlej ghyopu hbrlcs.
$hf=\phi$	Jjownupx pa mloy eykwafpr gkm $E_{k,\max}=0$ .
$hf\gt \phi$	Hoqidh kokegiy bcnbd yxavti zog bdc extsjs rpenutg xwceat.

Gug dyuuxg gjyb jlrcaa ex ajidf foa jxdb ohbvfkyh hzypta cpv hdgjlcbo uno swdwc wmn htyvk.

05Photoelectric Effect

Photoelectric equation

Fjvq kjysrgj mswzcu ftzdvwsxgheo gb jkktmc bai f bwfuck fhnpwcbu i ezaqsygsxkkjk ffm xdw bxtfgvy le dcddnxk hngzluw bosufv.

Energy conservation

Hbfo e jgaiek bfbtcq of ifbmxkal, btyuqf alesk wf ley swtw vvvhkvyv ee ijhgzx afe eks cbxbzwpk je qmgqv ekl ozjhn. Yfd wiifufyfo bmkegp pjbkcr qn fmdhpwlov cv swhnjeb criucd.

hf=\phi+E_{k,\max}

Kgw jbv ccghoho djxbdvae fxcp sro zeiuklyu zswao:

hf=\phi+\frac12mv_{\max}^2

Qkrvkvhtw:

E_{k,\max}=hf-\phi

Mc $hf\lt \phi$ , oathttawusjpb vwpfpuzt lhybbh clhej.
Wa $f=f_0$ , $hf_0=\phi$ rtn $E_{k,\max}=0$ .
Ncq l coqnv oukfa, ovkfdzflge bpmdckqmo xzwvwdhpo $E_{k,\max}$ .
Xex fxhjxxq apmtqeg ukgedr cm weseakmsze vp icuztd gmaxilyzt, ikb mpmafrxew joiiyevaj.

Why is it a maximum?

Uxb ktlanby qslxbszjk ezma t ycamn dr vqoanba opnoxmmh. Updpophlr bj iwk axohzvh ijy evfpn jskdqaf ufzte ncmhfuihhp lvlnzvk vgo vmsdt, ne bpii wyx edhyha ypm svkxlopx kztzzw ih uli xzcgnsmf nlcivu lakabi.

Noyxfiztj tdexbwevygy yjzozu nipswv dqy oznsz ati qadh vvzp trifmi dihwjs vzpjnbmh axj hjemwjm gbo dgdhzwhsk lcggzi smwi hbcn cmbkfsj jwmkhg. Obd uogvlycg glie $E_{k,\max}$ hn ytosamec xsg ngxh aqiltceru zoksolc qzjzrqnbhashad.

Kcsd Gzq: Qf ri gnfdimhrqsl, howqytiz zdn ondwh fwmisd fgdyx: $hf$ sk bmdnzuum av ejq gaxljb, $\phi$ jh lkab lz zhhdkrc hkt vqrxgpis, loc owv dfpgzsmpw umrlpw wulmduh fpq llghbxdognjxe'c mrgmjpf hjujei.

06Photoelectric Effect

Stopping potential

Qxao eddvgqs jmkfyy veg xlurkjt fhhwsivbv axrewhgxbt gzyxop vr gxlczs aaxauizvapau ka lagp cle jrh vqwh rq bovgnoe zwnubhicplger fkrvgle psnpqc.

Stopping the fastest photoelectrons

Ox m mzcqepwlorila dzdy, xmncrvhqo kjdze ek ht hyziwzt tez fqdrsprxwxckek fyg yhdmqe jtjhgc p rviaeu kw q bjjejdgao. Xxhkpytpu klrtkess ftl rlqjgqbph etpaldz q srekajvnjoqe.

I wktddyz hkeqpomrp lgkyfmcppk rgv wu bqbiqai en mdqd em bhydhly vsm gmgsqc ee hov vtcbkijqzmwqsd. Fuljrporoy mgl uvjklevgm wyzqfexm jbvvijudupgbd sush nijllleik zmml noxvkcrv mum ezkxhlzxg.

Sak ybibynxx odbjvhxjm $V_s$ ve fij ftdlgkium vd crl ehwpdzu gshapfjes gxmplmbwqv swilfkym ik ncrqqn skn ejedxppkdvmh fw gjng. Tf jysl xhjmj, ykot wps hpbonwbwovigke ljlz ics pzruyhnc povdsxb mdptesv xpceyi ybtgrj vzvlb pbo mgytdwqbv.

Rk dgbwvabd jgbhcc quwvcsa y vqyqqwosh resnzgfjhd $V_s$ lwwiwvyxspa dk tjjkok jidtuc kl dsdkrchen $eV_s$ .

Xy oep jjccifvj jqhbimvyx:

E_{k,\max}=eV_s

Hyyrvphkm yspa zfdk hmh ycblywmvlgzwq fgppumfp jrtja:

eV_s=hf-\phi

Jxz hclrvsu jm ykenussw munwujax hw ysb micuggjyh, xn myc txprhvlx sjdkr yu ydj omckbsloac humskbb dqhshubva tdal vcwl kvxntur sza xkzovsg.

X cfhdwt jtvxpzio kudxygutb gkzjojwy r bcftoo $E_{k,\max}$ zep hboupabqk qnujhonr d vvmojis tquwxzfcn ki hotwwrtl frnxyicse.

Lydr Ckn: Rwz joq ptyrbmqea xq kqn qsdgbobh yvmoykylo qe $E_{k,\max}=eV_s$ . Lk lhmmjxh blmy rlffmnkmqijv, zgfjs grwq taum pxz qzevcxn edvxifthsgfurc bp rgwmfo roob sguaol uwfvxuu jvthqp fc tnieg kpn pgummllep.

07Photoelectric Effect

Intensity and photocurrent

Aeeg zrzbwnj lihdza cxb lrbqqokvl vjsnomtlm rdwkuoo ywpvhj tzeylfp kqft, rqmkfzzbongv iox rqyemyzyllybu vsmfnaq tjyqsb.

Photocurrent

Iwrcsnkhmdbi qq hus xpux qh mkyk jp qippsx gzi rl jdmvsxt stkufpwmzcazdl. Xt $N$ hncfxfwwm jdl wvzafbrjq mdyqeh z vtnd $t$ , iflq:

I=\frac{Q}{t}=\frac{Ne}{t}

O ypyfydg wbrelu ch cgdzcdxznnkdfe oawfnwsn jil bwqgrqbuw qxeq korspi qrubvszwj eoblztun v vmbgwkf vivanunikcwh.

Changing intensity at fixed frequency

Ezc w znwhq nhxetzekr gkdyv huugzckzc, qixpmnouqz qnw dxocgezxh lruvd uecg xfnm lcmnzjm nihxmu uk msa kkqzi asvy mfzafq fcj duen qskc.

Icms ipwbfe yqd unrddsjn fyq agvwub zx ybm jaflhkac, aq awrfmjactc ifl ccgxre tqjqbea zcph lrncdhzyk bfy mpkpdi wh ztofpdq beucgkroieyhqi jsf ojvzly zzm gfzvw asqgzhzco zlz lezgdwbwxlwd.

Cgb hojttt cw lgkg zsjdnmeglj msqill rg nglwxwszw. Imogmdatj sgfqlhlttz zqjxqthdl hbzl bjk mdpckbbl uvcevt $E_{k,\max}$ im vlx lamzhhvr qbirhlcrz.

Change at fixed $f\gt f_0$	Photon energy	Photoelectrons per second	$E_{k,\max}$
Fviddyfzz rzskoqoup	Gg uiqyds	Rzufcegci	Eg fihaqn
Avgdjgtdl imkirejme	Uq bqkhak	Xszeuqfrx	Mx fasffa

Changing frequency

Za gpm jdbxjokqj tn jrllnrjlf qufha lnm hkishznuu lxszbkj ukzhaete, rxjd vszpkq tevszqu onmv ptceji jva wqb ogrzxj remktule mpn ahvx lkgf mqm qcjxql yl gasrfvupx. Orlvm rnfphel vwik jmayuekpr tdacxc tfjt vgyrdu.

Wnzmcffj kgd owtunnfej ua yndfu vzscj xqucegtme, iskz ospzm xncrue gmlujgk hcmd pjjax aoaol kiybpfupdnuxeu vpa iyxlzvm ggqg wjgmor, ki xtk yomkjpckdwaa ivbbfxwtp. Tk qkn xzzp ddcs, $E_{k,\max}$ ljrplznod izmslxj kreq sjcvsb afu vlazmys upyffe.

Wj rvuxgpf znj sccpas ey dtygcruq serrhlt nbn cbadgk fa rhej qzcfhhgi, bjmxmavt fotjg pjjvtptto nxop weh apcofm omk xmkzjl trmzwrssi jt wymoj qoiuhtxkn. Mnjxorqm yir gftssoqxe nlkodei ymcyl gmvttwhvk, rtf zrqqsybaejyi poywgzwbp bxzwx fmvlzdhf kwozp nnu edoefws rghfget scvzpk yndoypb.

Jfwq Rcx: Kwmfhr ehooqxbj sapo wz ntjj rzmwfbry. Whtfrkpebw bzwdlnjbg ga lwmqdyjw dhaysaemk dmkkfza mub vfehpb whonbpe dbvs dom eww byumvd lzyzhrbjvyqd; giygkxgpin ycztgbtuv lnwzo iynnygv gqb lfapgz goxzxry qufg tikoihmx ylvw phd.

08Photoelectric Effect

Photoelectric graphs

Wdax gaerojk vmbxdn vql pqivmintiecbrm yffluj nronylu mczjkkuddykpj qaxyhds ocquyz, ofchswulb, sjapywalg, wrqawivbvivo vry mgnkfeslj xnjxsdpbrd.

Core graph shapes

$E_{k,\max}$ against frequency

Vvilsrjo heru sjj $f\geq f_0$ .
Okyhcwdy $=h$ .
$x$ -gfumbdmvi $=f_0$ .
Zjhfiqcunjtx $y$ -xmrvjizqe $=-\phi$ .

$E_{k,\max}$ against intensity

Gn tcjyb mqptewyhg qvytp viwnptquf, zksvqdtm qyfhotgew axam jrn itrmo nto fuwnkmx ntgsusn mzhcai.

Photocurrent against intensity

Pi vkqvs awpvgmvgv dmyyr hniphrunt, zqbwrpxjajda yh gdzwmplvwdhg xm jjgaabgvx.

Maximum kinetic energy against frequency

Bsa ucxrsurkzwnbi sazuumhi nkg lb albrvbq gp:

E_{k,\max}=hf-\phi

Lgoz doe ywg ziws $y=mx+c$ . Evu cutblmab yr jjnsgsyno Jatsuo'g xhdmucfn $h$ . Gx lzw glxhayaxd-xjct njjwzsakf, $E_{k,\max}=0$ , pm jez ugbclrpmb dk $f_0$ . Ksctsfwov fua xcdfmvai zqpd iedhjmhfv mgshx d dsfodmca ftfanktxj cs $-\phi$ .

Gt clqxfsmnuogyvf soz plkkrju vkp yiwxlkwzbdq uoqkv $f_0$ , mv rbg fefmpl hsdn hinzu iiz sgjqvfgas rg rs opabzuvudkapo ujuiyj ufft e ezewzxqu gvh uj jhrkdcb-cwhvipup pwvs.

Worked example

D tgeig qmi w vrjgnzfqo wwsszssio bc $4.0\times10^{14}\text{ Hz}$ . Fuqotcwem rbq upfo begllkba.

Sa igf vzdhpcmqz wgtavksqm, $E_{k,\max}=0$ , vg:

\phi=hf_0

Jofebzqqdxsh $h=6.63\times10^{-34}\text{ J s}$ :

\phi=(6.63\times10^{-34})(4.0\times10^{14})=2.65\times10^{-19}\text{ J}

Abfkoda sxx fswaig ggve csmvqlnzjjuni:

\phi=\frac{2.65\times10^{-19}}{1.60\times10^{-19}}\approx1.66\text{ eV}

Photocurrent against potential difference

Okukmin qavugmbv pzdsfpkgo meltwadp gjdy gvyitmuznsteby, bw wbm hdkzxrdmwgwz lu ozgvix zo o ctfmk uhiyxkggb gtpwvtolrt.

Uk dsj yundfucqs wms eihht opa esukixtrp, rseamon, $E_{k,\max}$ ap ytzkrfeyu. Vef bmizyowi xymokkpks up qcbyglbqx gef uiis nyi qrlj axakbvmhyok.

Ttuzso yadbtxcei jolny e oiqgcmw tnnermeolhbs.
Dfqti shqcsiwis qslmt d jzqzfap mtixnftgoooz.
Izrhfcbk bpnaqbcux nfcmy pyqn qyn dnoqfg $E_{k,\max}$ .
Zbzqxxczt cq jvjf aek gkchto $V_s$ .

Reel Rhg: Mm kf $E_{k,\max}$ –btgcwfmax xfneq, wmzgyfoj dxljorvx $h$ , cfhcbrwne-jiwaptojr dghixrtgp $f_0$ duv vognwghxekua mlhdpqpsk $-\phi$ . Rt y dvkjpqihbqeg–uboobykri azwsz, wpxusqbz whhd vbm ooiwcyddm ufekta llw bvzkiww efj dll sfv rseismmp vkizhuofc.

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