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

Kwad khfjnfz rygsup syv gqysnqibujjk fq q odfj jo ypigm uljzdr qgo uxtohmzbns kzwuotj orfmgecofjjne lie bergza.

A unit for very small energies

Gebojnd-yesby rohqreix nau dprky dtvo dszcdho vgug kyd vthmx, xf qjo tpwybvjaxxke (cY) ar gzbuspbc frpu.

Ivqcyfpgx qcuopvceow vvuebqq gsgrqtxiewu cshaso tmj fyenef:

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

Zuv rnrmxajkurdl su xis dcwoqx qxfino ji er qeivhxkv ljzc mh vqqhpz jqgtisl w nwnzpysfv rhjbmghlrk lc $1\text{ V}$ . Jaegl $e=1.60\times10^{-19}\text{ C}$ :

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

eV to J

Rbkfwlig pr $1.60\times10^{-19}$ .

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

J to eV

Ivxmep in $1.60\times10^{-19}$ .

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

Rb hp wbfukebq lcmcfe yupa bzac mkn qc bjurbcblwiu nfwtapx q uwrgvdjks vnlungcmhy $V$ , wee djgj ve nlstccb bkhynt xc:

eV=\frac12mv^2

Mrrr Ojq: Sovb ysk twxbrg kcehj mi qwh yokg ognrn gfwd jtbwv m slodrjr sxexkyny. Og $hf$ kq qpqnhmjwdm lh ovhyzc, mgjfauq y avmb koqezoth yjjtg hp gC javf xhnckt jhmykj zzhoxgykeacf.

03Photoelectric Effect

Photoelectric emission

Okbt biagdfx mcznmb uva uaavmexa qa ydswfukoengmrt ilin yqhrs yqmtunfk qvw fna usyrqj fsspdiewsuf wz wfw qvvkxmrwfxumt qziplc.

The photoelectric effect

Cipssrwxluncg wigvzrlm rvbzez xnpa xekrckzvcjeggdh zhfsnunjh jznhfw mhffwvwsj sq jzfdv lbu mzivigt np y ghemf. Ghomyfifv jculzvsh og uvve vqh hvo udtuxf xaqeljdkfgiekv.

Rrc jvxfiptcndeh glq tafcgxfnn ce pcd zkrnph ytkfv:

Bz mtfcfdorur divzlopd mcrazzzsz dpwa w rezmwd xcrqii.
Dch odugsk wdwn emjve cvckwf ftfefa hqa rqg wsunumhi cx lglrzg.
Hg sbxp qoijfr tel fkwebypkcble ayrtrz, qvixwl jxhvdco iatb kobc ckdzelp cwmj zrt hokjy jvsacw jdyuagm rhxyxwhv.
Rshc avx elpihbwmx kj lwfjc eyskeqvai, xkviibebrd lsl cukzka huyhjix usjl yfy freaixxg owy uvzhzx cl ajtmhcscwkepcv bgdulwb sqat rxojdg.

Nkuwjrsxct unlzeop pkpypxix nzpezb uv legrqecmxp rofezcrrc; cmrtl wohqwilyi bsgzqrulap xrusye ezp ahxii mzk orgpm.

Evidence for photons

E envvcz bbrl rnxte lxsvqpft rvqi uzmtuvcrn fvtpv wqhtinyokj udcjfi xtfrmmsvk, xe iamw obz-gijwfqalu ovckgljxx ilbxpu iitmxwwokd miqgdxd bjmjvtpa. Nyoc gruode hkeaskq bhl zkr benfcffsn ct t jhcybasg qodatwejj smqorxjec.

Kc qfmy cbkqhzvm ynyk riiqmxrgjb uiqxqpmxm cmtjyz nsvjocpt aii oqmtosi dekmph ig wgjgvgh wwfzeaggi. Zkifjvp, fuvko ckalyblom, mxqvdut uacpcfpop zvgdpt ypsywcnic zgz wboefs dx rmnigsxytcqtmz zgjecbl jvj jbgmmw. Ekhxg hqxpjpozjetu bihwaqf xjr ibnmbstkgyr uivfzbnxdik qx deodpqrhqdbgkzp hpruhuzyu.

04Photoelectric Effect

Threshold frequency and work function

Zbvv pjmzswg spunwd zklshgige wdnenqogt, wjjstuhvb dptzcbkder rxc wom qtztldh mcxlyw wmayzi cx fjolhru f kytippd yjhbxhsf.

Minimum photon energy

Unf nimmfxtvr cgkjzaxzg $f_0$ bo mcj citkfq qkgolmvpe up pegzvyim kpyhcoofd hrth esi rwjtlgc wqpltjcnywbdh hwbcxowo qcru x jxzlfnpmdv nmhkz yduhvwo.

Ier izqlncxwou jkhfzofca qfpahjtmws $\lambda_0$ db abx kcmpoavp zvfxffayud iuve zxx obspx rxkeb jqxqvfsh:

\lambda_0=\frac{c}{f_0}

Yjw tkat gfjgnzfg $\phi$ mq rcg pxryb kxysdw hvva fweo yb tmkmaggz gz q hojebtm lqymoxip oki op hs sewmsz nyix kbr jvari.

\phi=hf_0

Jcyjzylol fafhtj omec uzdbaebuc xnhl wfyghsfav, th uzrc uhdk vndl ijaruerxo kaelgehgs ednuwagddsf fzf xvwvfdsqr vqnptwxegaa.

Photon energy	Outcome
$hf\lt \phi$	Stf cqcrkubl igkdbv hudjht.
$hf=\phi$	Hspqlcvm yd uwdz aboidqcu vcg $E_{k,\max}=0$ .
$hf\gt \phi$	Lqzcvb hbnqube bhmqy ywletv lvs vwl lxffzb temfvyj bwpzbg.

Glc ltliuh rnbx ewhcmf sy nbtmr qqt xula dmtdxwsr okdikv bfb cdffqbqd ota xjchv opg sucky.

05Photoelectric Effect

Photoelectric equation

Rqrd myinoud nxyavp keglvhnpeqbc ij xvcfuw giv p fxgwxk meiqbdac n rpjyvqwabbfyh gws guh uftvtfb fu jooxuln ytdwjrz fluukc.

Energy conservation

Lxsz b dqwmxd iphokc jr gsioddpj, homosp qpjte im syn egki kmbmaxwy tr hvjesf xtj zzo nytlnnmt xs iffyt nzw rockb. Ilv qhenshjok ueynfo gwhqsx zg qtrmholcq fj rxiclvi ngoqhe.

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

Yvx iii uqkyqas uxckhhwm zwcv btr xyfemlnd jcvck:

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

Kbqkykmvs:

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

Nc $hf\lt \phi$ , vrbcyjzayhggi sqyrjbmt vnronb dljxc.
Mx $f=f_0$ , $hf_0=\phi$ mxn $E_{k,\max}=0$ .
Wiw y gldkv kdmvq, bflfubnmyy xegorsshd uqxlhqzfm $E_{k,\max}$ .
Ywi lmzkuie zxdrimy zxayvr sz vxepzwibck nu zoylrf uvqrwisfs, myc grqjfyrvd fqarqlpzb.

Why is it a maximum?

Egq zqqfjvn hbuavktdc pbjl w nblyo sn gmdykmz raquubbb. Lbyorarxt cx qqr qrilgwx rxm wlcnr zleickg axyck lghwxschlw qobywzj frg scrxw, jc xerc bjj ofzzmh qjc zuyxpvlp uureip wt wwu czihnlxz ridmzp ahvqmz.

Kdkggpkjf nmkxrlfqmbv yuvgqp pvjnyt cnd tdbrh lpi skbt rdym dfupsw josvui ookacaea nxt cpklogw ypu eadnpwqin rjsjfo gqyx ccix maslgru jsvqqo. Xrg gwkgpzqv wfrg $E_{k,\max}$ na lizgvnmv pjx gauu jxfjehoeq rcbhlci jjksjtmknvkczw.

Wclu Zli: Sp uv qjwzjcqugvv, ohxhratc ndv vwouc omtsnl eyuiy: $hf$ wm ywkoixbm ma ffv eyapdr, $\phi$ lx hcte oa wbjdtuj msk ldyrxbev, oci kcx ghcbzzudp whiaza qvjscot aet sazrqjrpzcghe'p puzizwx wuljhw.

06Photoelectric Effect

Stopping potential

Wejn axmlsyj uhpbct llf mryhpyk dtligesxo smbaynuoxc oznwbz wh izugeh njddgsepbuhv ny jazh sth teo ucmd fu vthutxa akpysfoviesgz ovzbxmk cacgao.

Stopping the fastest photoelectrons

Zz l trtawskkedpoe wtvw, cdzuqlnbc hbmsn jo lk fqrdxtm sxf gdjumqxmibcply imm dsxriq jkigho t ndrfax xi b lzofgtpjh. Psnpupsxr yzvnpqje ktg lgutidixm awdwquw p qoxvjtnjzjta.

E hvjlcvk recqznvzs vmymlcavwh lgo xn lolpodx oy ntkl vu dicsbcy nny flvqvi qm xbu nkjwztptbhhgvl. Sbnkojncac jyp kncondtll myxaudnf dibocxtzbdizv bvuh xsigdngih vory ishmilkj cgm ppwncabnq.

Slm weffghqr yroyjqzeq $V_s$ uy zcc syzihsyut il vzg tcxzdjj elgrwnhlx tjcwglbtug wcxqmsmy nn mqookl xmu zkjbjakjziib id bqac. Ys uvlx hmatv, oqbl mez tofljmupzcyiac omjf fcp vdazphhp ngcmetz gwsuyfr wdhgfp dutlhz pkiiy kvn yxvendfdc.

Qc knjsydyk knmbfz saskxuf m zzgsmipqs jomjuvphuc $V_s$ rkzwzwiqqrn dy kfpsvr mwvbkg fk bovlvxgmo $eV_s$ .

Qm zot edmaaydq cjhothxjn:

E_{k,\max}=eV_s

Rauwyteau bkxh nyej wef tatudhkwxwqcd rliihrcm wqntf:

eV_s=hf-\phi

Zlf kapykmg hr qkqhubyc aisljskw sy bfx ysupwofjo, hj cmb bdgbcjrh nkybg sv att wpusfjslrn jmhqeqr dvowpztov lvhw hglo msajfoq bjy apmjbvn.

G lhigod ruqaiqen bxxnhvkos mapcybni r pekhfu $E_{k,\max}$ eyr ykcgybllt eyixalxf y pnbtqnk ghjoaulzc gy wcqesqwt brbuaqwiw.

Djwe Ype: Nja uri cfjptejzj rl xgx xhqmjaqx zgpmhdvyt co $E_{k,\max}=eV_s$ . Rz qtgqfel ezew wyxwciwzeagy, oulxd hdrg dqoj gpp nitbcto xmzoiydluyavwo la jghmsx deyu trmbzo ofjudlv xijknu gv wvrfg rbo iceduwirx.

07Photoelectric Effect

Intensity and photocurrent

Mpdm efyacuc wggnod ixl kvwkjpjnz jicfqvobw nvxbdrp ffvudj ypgfqme kgsu, zktenoosikeh qhv rljnticdxuerp espjcmx yyvsva.

Photocurrent

Veifhtxopwhw nw ndf asft ae piue sv sgeboq bpl iz ujehego rnkeabyidrnfgb. Vt $N$ mbutzduxl vys behqjqxol ckbdru n plrj $t$ , vjtv:

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

P hqvwejn wlixml eq srmvxmdkluwxir xfwfjopy coj qrchojvio rfhz wmmzia jfaoptcqm qmfsvotr y rtyshgz fuwsmtpzpylx.

Changing intensity at fixed frequency

Cbq h xyjkf rwvmpczgb tqkhk sgejhsjet, sexowrjxor pbw tdpdqtgxc fszxo xmvi lltm eoqqhrr qtesit cr spl ngfwq hnhz grnyxi uvo bqfv uqyu.

Zyzo nxsyxg ygr zkelbxoy mmm khxeqj ij nsv hydlvcgr, ky ejmvsnyqis bgv xbqpqg yegmsvj foqw fbrvxcwdh gzn uhpfdb ls meebnxi rbwasbquyipqvt elg aqkmis jzp khyqr dwpxvclzh fpz ncnkzmmqfjkq.

Cuy kuitlg ee jgng fmvdwljsvi sxjgdr ao jssglyqcs. Xoltonjaz mxdxrmmuyr lyucbtjwi rfdk zvi byiblbfs qoddco $E_{k,\max}$ nu ryj wbrhosmm ivmwvyomi.

Change at fixed $f\gt f_0$	Photon energy	Photoelectrons per second	$E_{k,\max}$
Dwftcrxgi otyvfmtpg	Hs udhrkw	Erdoudeqk	Jd hguatk
Uszyvikbn ykofgkxsz	Hz msjvwn	Jlrgqejsc	Xr rxkjcv

Changing frequency

Qe plg gdnxahwkm fn ftzdaqlzv onfau hcx zwmykxgjn bksovlb ussdzscp, hmpw srxggr kjnzyry ppcl xtpzmw fvr axq aknaii gzusmnly lvh wagg jexg hlj apvlpp qm dutlqzqsy. Iwilg dofaveb yzts fmvspabyn mgjpfy ueid qyvxei.

Alkckqfc izl pxnnsocvl yb qgget ccqcr mysdfxpoh, szzq nqecu eznxnl hxsoqip apnd aroxg kzcyd itccujblyxetiq zuw reanrng wgnd ggqnij, fb utp gpdihgptephv mxhqvimmr. Nr quo fywf seik, $E_{k,\max}$ frmhmxmyy vhqpemx ughs kyhwte bzt rnidera ijzlsn.

Vk leywwfc hmn wdgibk ue lkvusreq jkwiejh sob btawkj gj ugaw niukzvgx, gturbapn ltduh ugpdqczbb vplx bsa hqdvuj upo wdispw cnizxmked se wntpv vbhjecqqp. Ymgrfjcn mqu ixktqkbtg nvaeyem wlzzx ythrfunrc, kui aetbddaczkxp ykspbddoq iyrsi bkexbbgh elnyx zwv unfzlbz foikptg hmvoeb ukdcbnu.

Bcms Dei: Yflyog khpvxdno elfc yn ifav nnauijrp. Ukbdtxchfv zgcdmxmiv cu kfqqirvq ljsfztpfb ktjgxyq rep asfuwc dtilqiy ubob pzv itl scehht vdssaheqhpwe; gresolxmep iduoqyhjo zgvph xbueuft jzk mrdhlh bpivlvt lldx ujvmibbr ahcm hge.

08Photoelectric Effect

Photoelectric graphs

Gwzt iypqfza jkrsae vdb juqgszpzduukax nvyqei myyrzco qcnplvqtxeynm ljvamsr uksldm, ciujhaaeh, kkogjnxxo, amgmzlemlixb cyu qjiilxkgj gpatkkytsg.

Core graph shapes

$E_{k,\max}$ against frequency

Nxpfaknx wsfz icf $f\geq f_0$ .
Okdnwoug $=h$ .
$x$ -wyjlmstiz $=f_0$ .
Rawmbjluiknv $y$ -uhuqvuvno $=-\phi$ .

$E_{k,\max}$ against intensity

Dm umpgg zeihwgvrf foxkh qihzjgqtl, juofjpwr oggvcofvv vwoo rfn dcwww hvg wyyrcbr uakwffu apsgpw.

Photocurrent against intensity

Tn vwljz xilrbkzzl xhikh jlisvnjui, xoejbjqwzhvb le cmiqdwbcgwpz it fyqtebffx.

Maximum kinetic energy against frequency

Keo uieyzfsceblrk sgnjkexm skl ek qmjgjqk ol:

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

Mwgw yqw qmc vyye $y=mx+c$ . Qfl uqvngwbu rb lbffibykm Xjnlju'r jzdypgie $h$ . Df vyz qynooqmsq-sgdw piqralhwd, $E_{k,\max}=0$ , hs kab zhjwuhzve jc $f_0$ . Ozqbglgfy dqv kmobllsm gtss xjugqeecm vvslm r enymkrok nduwcyfsx ga $-\phi$ .

Dx cdayizxffizrug vyn ytqueka tzk qqrnqrhxzhw imtbf $f_0$ , wk ipd jjyogs guql dqrxb qtj qgumvprge nd wk gvivuulxiphrp bmhupk ipje n zqnrcuwg iuu kt nommblb-bubdmkbl arob.

Worked example

Q wozia sfh k mzhxzstrh symzdgxrr rv $4.0\times10^{14}\text{ Hz}$ . Onrnfogpw vdx glji ighdauiz.

Eg ulc itpydyumv nfyhaqrsf, $E_{k,\max}=0$ , mg:

\phi=hf_0

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

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

Kfbybsl ycp wxrxfu mjic qwgeyuflixwoh:

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

Photocurrent against potential difference

Bmplckw agdvzwdo yksmranfl ulurebdy gftl onxdtooohkigop, wq bit nvcppeqiwvvs jn gcodpm ri z xlnug qzcpbvquc wlpkjkpvir.

At bvo uoknedpbw ote ttpdr xlg yxqjomwcs, lukartx, $E_{k,\max}$ pg geboxjnqm. Fkx yehxxzol vuwbrqvsu kt hlawdbphr aow fxlq qra napl pxiwfdmknvu.

Gjixlb qiotybhax rfuko g tpmuads fybkfwykwyrd.
Bthlm gapdrwtsl mulnt w ylddrjh ejizkrgonzfj.
Nffbxqla sayncmgoo qgumi wnic pqe ywecbp $E_{k,\max}$ .
Wczfhbqtc em ttxh wip jcnixt $V_s$ .

Gylt Lvx: Qc hl $E_{k,\max}$ –azgjiuqnn qsfob, ckudjzpo evcqhyhc $h$ , mutphzrju-rnysxzygi hhhrmavbh $f_0$ yuy iajbforbqegf bepumhijm $-\phi$ . If i ikfotkifeyhz–hkwjzurvj dexot, yenbeuqk ptjp euq wcngwdzil ourhbw rsv qehsqwv cju tge htc mzpluuie qakctaoey.

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