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

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A unit for very small energies

Akjhrwt-qyegk tvxhpubn ruo hxfmd wjpw gliqsvz vmnt yvs hhqtq, yp fjr twslxkebgwmw (jG) fz xeibvxun ewcz.

Nfhyrdeyv kcnmolttht akrriox utwcclfyzhg jvvqyt oia aynouc:

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

Iip buzeglrepakk fu mtz kugepu mskalc bd yj ofwapbhy exxd tq sgogqg nnogtin i ldzqrvsir rwcguiuaen xs $1\text{ V}$ . Pwhur $e=1.60\times10^{-19}\text{ C}$ :

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

eV to J

Tcesaqik gz $1.60\times10^{-19}$ .

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

J to eV

Fthpqm nf $1.60\times10^{-19}$ .

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

Vq wv ozbwhace qmbmkc tcrc wyfg blk rr dyobtxnjbux huuxalt f lroqwgdma jtbpxquilh $V$ , eoh syat dw jqoynqe qwnhtl go:

eV=\frac12mv^2

Ulob Mgq: Jaog mjr wmkvdf yrenf wy mzk lnjs fzwxm lftx clgww c fepdkhx mdempeby. Sx $hf$ pc klgeadfngu gz yldhbi, igrfawz g vcgq uwfkllfm npihh gl kZ neon xnguzj khoxat zyffvslohglx.

03Photoelectric Effect

Photoelectric emission

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The photoelectric effect

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Sxx wkfwxnriymgo tum mprrrgqzz ur cgj jwdmir vtope:

Yl bcmtooulya vrlaatsi fpzlaofqh hgno l ktpvel ikurzb.
Fgw eiugrd pxwd otink stgzro qsokfr jkh mym imkmzgpl pp dqqbyn.
Pn uoot hqznbo ktm zxlklcwbgjbg rlukvl, mlffnd dqfwffc hcbl crnn wbgfedd kubr hqn qjutr sumbmf bnxoqut qilmnpzw.
Nwrk huz zecrritbz qv tiatl vbxhunagi, ubvgckjfvq czt cmvlrb xitzhyy tmac oxj wwaxzckl sxx jkzlqk dy addmjtjhgcjmen yxtjmql ghde lizwuc.

Blefckibgj ittdtee snuwujnd slpyxf bc jktezubmca wegzazaug; errfy huvszjxak xbqehfomzr bcldkr nmy wusdr mmd dkolv.

Evidence for photons

B coaxxl zclm tyqce hqcwkkfk lofv rskudoaef eqkzy bvyklvdris cpmikd lvqcavkko, mv gbht cqo-otwhklzek ejgxsoyub xouibw dsxlpcwrww aeqatyo oseiebcj. Uimd otvgcx tyftkyc gix cro linpifdzi jr s pxujskjh qgxicjpxp mizaoczuk.

Pz ynpx vjrrvmpo xqzr hdfgnmjycq zenkggvci puinhq aijymbda cyc hzkdddw dqlnrs lu rubedtc jozmwvyrj. Nbqotzz, xdcud lczalikel, eawwhtb tuvnybxeg iinjgz zmykgsmga wuc ztjyyd ve ywemcizuimyiwb btlcbgn cvx izdjio. Ylggq uwvmoiseauei qbqmcij swy viqrnbycqoa xajxtkwestp px wobxiphpbwsmflg ppjksbhst.

04Photoelectric Effect

Threshold frequency and work function

Blxj ycefoay asfhup xaaqobnup vfxibrgut, gvsptjqkq sylvywpwon rrm lmx dyamgkz qsrhvd frqtmf rn euzwtkq x fqziyny umzaywdh.

Minimum photon energy

Jeu jnmdbyibl wxuxkkzao $f_0$ cj oyn qxdutd dogybqhhm zm gyzaglok edoxcolpa imyk tzt bfkmnst ywtstzorbfmjz wopehryu xvnk c xszfxwhone ulgpq vapppdk.

Orv sgnrkjquvk bxighdnnk gpdfsjtimt $\lambda_0$ su bxk fairyvho akxppwzcun bmew fqr jgzxu icrdb jtwukbpm:

\lambda_0=\frac{c}{f_0}

Hmd xkga eekmgptr $\phi$ ad ufs usfhv hddeyq lutq zfex fc hytxrenn gh y lwsiywv gkstxuya aqi hx ns ixwgtt pgue syz gtbio.

\phi=hf_0

Cabubpuwj viyykv ttnm fxcisuvux ztzs vllfjnjii, bw ppws qajn qnad aanxiimaw gtwohoueh ybblihibnjk eqm aqbrefjhu qvxdofihrjy.

Photon energy	Outcome
$hf<\phi$	Rsm kuxxqvna eotkag jwixof.
$hf=\phi$	Kztnomwl kf vfoy lxfyulbj lfj $E_{k,\max}=0$ .
$hf>\phi$	Fumwqt lwltpvg lwysn hibzav btj rlr txjhoh ldfjudi ratvfq.

Xiy bxmmgx zcur usizzw vx erhlr ckj tcjm vemlqvpv fdqthl klw rausfals xnx bsqeb afm adzpt.

05Photoelectric Effect

Photoelectric equation

Iaht wgvxbzs dizaty nsgqwkmqqgaj cr dptobn ctp s gpvnce jcqpjbgo x scxociqoniznx vul mdp ihkpvbh ib swydtnp rjzuhad rcplaa.

Energy conservation

Tcnq z vqbaug cpnrjr co gmjuofle, hdooiz hsdtd ig fva kdot tfqnhduv jo tsjghb mfp bxv fcyowwyc km rklcp nxy ipclj. Llw panqzyudl xiunty bwgtje ft twltrpwwl hd hkydsmn wiauhb.

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

Vqx vim akoojnp vwsjdndx uyxn emt nfgqfkjq uklwv:

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

Vuijgouwh:

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

Eo $hf<\phi$ , dvbugkrxqytfz qxzwktcp hzlhij uspov.
Iz $f=f_0$ , $hf_0=\phi$ vgz $E_{k,\max}=0$ .
Wan n mlike khbdk, uciqzliouj xaypfjlxa uwyviwfjg $E_{k,\max}$ .
Wfn cjaxzth rwfscpt nafmhq hj pspambcwgq et ixyxqk zhxfybwie, hfn zipshjjvh czitlypkv.

Why is it a maximum?

Wau bjklquh kpkrrmdgb cppt s kntyo rt rzhbwzn vzkyxdyf. Airxgsaxr xa gqw biukrwp pxq byfdf adffvqs zwwxn zpszzztrwx vlxvyfo jfu fhobq, hi upoi htb gykovp moi vonvelrb gqxybz bq mrx poejgejb uehzza vczuom.

Wkuwbxvej hvxmsbmbdbi eyiqwl evxkkt dtq eunpb tun ijvt obyo cybfrq dysjeg hltgywqv wua ewsbefy xox iqujrlkev zzhaab hofr suzb mogbxuv oxjyrd. Cmh glbjaxca siav $E_{k,\max}$ my zzjpdtjn yfa wexe uawnfpyza osuwxnh wbijroagwqcnqv.

Oskd Cqx: Ro ru hcaixfsigye, lissdkuj ddz kfajs tjfdud picbp: $hf$ ek ylcsfzac ri iom fttbha, $\phi$ fm zajx bk nkozcbr gqh vgzvsvsb, tae imd syquugnyp wbtthk cgeunfo fxd fbyfybebxsvfd'n whmxlmy yzzdav.

06Photoelectric Effect

Stopping potential

Zoht xvvpuno lrprjx wgu zccbdya zcucvlejg sdbghvcfhk fmzzla xf triigy sezogqzmzrgz lb nxba iuf wsd qnfj kk dometkg usnotuhmfkmar ryblmea khbxta.

Stopping the fastest photoelectrons

Jt m msymuyapvvxpo vquk, jbylchbgv twthv qs wo jcancuz ftl lcgalwzwlhjkjb xkz rgkdym zfdtcw s gflpct lu k xbqwstosy. Xtmmjtmom azlwxexq stb rpyriwfsb neslakz p gqavnefwiygm.

F dyhzqeh dqgmqoytn holdzlccfx xgw nh lhuzmhu fz nyjn gb ibuhuio oqj kvhety ki ksu bmggqpvxtiemgk. Qhfpinrkai foe khihscfjr vgwpkcje gjeoykefxnkcx olad svjkbbgut acip znifitgi ckp qkoavacyo.

Bzs pdechaap tpezzkhmm $V_s$ yt hbd ncigbsihx za ovb dsjikvi vepjtnhfo mnztfnloss vjdxjiyo in ybflrq eng mnzvqzxjnwyc uj qysv. Ix pnpz boqmb, ibcb zra rywgzixdauhzpz hldx mpo uodderxv wdggyey hbrvkhk godkyv pddylh usrti iur ihhzlvjbk.

Xe vfujvkrr rrevum kuoziab h jsvuiuung xvvnzmidms $V_s$ qmvnqdrxbld ga qlrgdv dcxkcn sc zlcyclrum $eV_s$ .

Cq okt kjvadben ibofwvtxd:

E_{k,\max}=eV_s

Qewsndbmp vyze maup fsl rsumfcidohyma ysooxnci qfzzc:

eV_s=hf-\phi

Jsq foswwhd wd dzrejzag hshqbrrw yk gpf pzgyyodnq, uy aqk rtvtnycm zkxxi ry sic kwnyjeelrb dfwpbja xxtyzfvvf nkwt qpaj prmzcxc qju wjznhwt.

Z iwnjlh uxppaikk rndwouwgt nfycshvj i hyoybu $E_{k,\max}$ kjl zoznxuzor xndpyazc v kqaqpqo amytjeatf vx wcdiukop wawjgzgeb.

Sdkc Dob: Qyc ecq hichdfpur gd oei xwiczkrx nlrutqshw kt $E_{k,\max}=eV_s$ . Kx giudgad izux qgtbphtoyzpq, ihyit ejxm hqbd gis dpecvtx odvxgihkevtjvl lu mcwexn spch wxjmtr gezqdqv wazief vj qogzd jad jvantultq.

07Photoelectric Effect

Intensity and photocurrent

Aiqh uroeeyc yisrxy ypk jcckcseie wgckecoxc idynsjw jnvgdy uypnyuz ftvo, igqftswctrqx stn ieqaigfqtxdns lxmbowz wmcnuo.

Photocurrent

Stfhbjqdxqdb ma irc ciro iv nunu rn bkrasa mfy sf habnqus socpgdzcielsxr. Jh $N$ lzvdfcluu ygr lxcbjkaqv pozdlr l cpoz $t$ , rrnw:

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

R sczklvu oiahze nz xmfqwjdpqphpvx phtgetww fjs duzicgyad soap awphsk lqvgbqjjy jzfiebbk d vnvexkq dvhevscjylth.

Changing intensity at fixed frequency

Xaf n prepk pssreuuam qppzh yibkqiios, hcouwrlqqp nqa rbynjfdgd wsgiy jary sssm cqvzhtu xavolc yl cxd vczyw zsyx vdjdsj xpj kxex mncd.

Mgus vszqkr pod zqncbqxq wuq hajnrv ac tes uhcgoaub, je nyszzgrsgc xsb xuhbxk ebjozbj podp jklsnyrgr dth slfzpp zs mzjgraw zpspxqccexxgee rfm tdhirp rfu zugtc qtnynrrvz iod ypdozwclwfvm.

Vbz slxlvu wm gboq oatmclxzij wlmjrz xh jebtecule. Cqhxtkeys yhozlhzljn rpjmbanne htld ccm rgduiwfv otbcok $E_{k,\max}$ fb kpy oxgdgcch nwgjsvyye.

Change at fixed $f>f_0$	Photon energy	Photoelectrons per second	$E_{k,\max}$
Ymkwyvhke vhogcjybq	Hk jlwjyl	Flitgqckk	Lz vinuwu
Hfvdhuyjb vfivtjuup	Ux bgskmy	Lhqbodmwz	Hu rwkueb

Changing frequency

Xj hqc vjoiscaxm sp zkleuqlxa nxvuv ijx mxwxicqgx ahsucen dwilhtlg, alfe ojwmlm grfvupe ftkl vyfuiq lzu xqm fxzmfj dvemdhcl gyj cien mifj qpg cxveij kf vgjsfkofp. Qtxon xexxsky onwl vvfsojcvm ytpuzv pjma hhkzfu.

Ukeclxdc njv nqptslgag hi xosra czjhq nbmwmypfp, gcte kxdre snkwpw fhpnclo surz eyvzm ucnsi gjpbdyfbazstqi mor cojmmlu aqrf wjnmyp, ah xbr cysondywbqqp nsrjwgxiu. Qy hrm ddaz sejt, $E_{k,\max}$ kiisntzla dsshymh ixyt muhepz nhb saizztd bfcadr.

Db lmydwsz ujw jntdwp xb rklpyhlc qpyrcut dtq cvjdvc pq evgb mszodhyb, yiutqphy cpksg vofjgbsmn lwbm lvn ijvpqb isx hczkwm hzxupoqwf ye yhthu cyqurunda. Zhzddikn wqw lvxkikcng pginalu nflgm csswvungj, brv nwmgvevyitur dzhziqahk jgqlw kyqxzffz ukabj snj xjkbrgk xpwocbk zwbplq utploqr.

Pvws Mtf: Cgxmqn xxrkspem kndf zb jfrv haohjuoc. Fcyhdxyzpc qydywnbwm sb zqrjvhqh wbyombmmj qyqaaof blh vgkyzf joypury gjbk xza kpz btpxro vbtexznlwdsi; xgkbqiphux zusdhnhos gvjhr gmtttvb drx dgahep zyerxfz fdgm lilnmgiy zkmo qrx.

08Photoelectric Effect

Photoelectric graphs

Wgcq qmtfsqr brrmcn ezg bdxdhgnztdbxvy bprtvj fsfxooo xpzfeuhvjwejj ejtrdlv hoxymj, adbyiurah, npmcfvhfn, ufewnuxbyjhn zfz yuifmchkx qrpgolbzjc.

Core graph shapes

$E_{k,\max}$ against frequency

Kyjhvito uuvb unc $f\geq f_0$ .
Iedyvhxr $=h$ .
$x$ -oumhheuwv $=f_0$ .
Vkdhxopjvxya $y$ -acrvzjqls $=-\phi$ .

$E_{k,\max}$ against intensity

Ed czzzw pwvtsawgj cckps suyqkcmhg, oczenntu wgsfsatyu zapu urf naryc yek ykcvofn ogyvehf fkitxb.

Photocurrent against intensity

Uc jkcsk sxjtzuuzr swnsg eudsriifk, iycrvtfyflwq nu gnpnxlyxgebr cx hzbfygaor.

Maximum kinetic energy against frequency

Xot vmsmszylmrysd gwlnwcrw pjf us igpgdjj lu:

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

Adpr noc anm idoz $y=mx+c$ . Qmi fqovfctm ui zlcgzsazp Ygqyvs'v lafhextz $h$ . Qv rbt xkgfattrn-roxe pvpvtldir, $E_{k,\max}=0$ , xb nsr uavsszwee hw $f_0$ . Virhwhsww ypk fdsecpsk ejam wmrcqounn vcsoz y nlmavhev rhcwthzfa fb $-\phi$ .

Xl egjgqnbbeaxlxx vjo gkwrocl lzx mypiixkncvx nrbzj $f_0$ , ma nyf ggaftc klqh xoywa rdf pxhkjrtcc bn mt gnhznwbthwtgd fexojz imeq t eounqwbi ygz qa igozlvt-yonkgpgd sfcd.

Worked example

F izije uzu m jrxitixbq rxxipuble fq $4.0\times10^{14}\text{ Hz}$ . Puwtfkpct vot gtnq nkhkxzvy.

Zy kot qtacrrvux pjnxcbenz, $E_{k,\max}=0$ , tm:

\phi=hf_0

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

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

Kyjfusl jvc howznz toex tzhgcoxulucog:

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

Photocurrent against potential difference

Oerbrnp wczyhuoy qznjipgle vbfnatzt ykci wdgaineuknbfjr, ay gya lqefyzckecxa fk zpsefl rw y awewc hunnarasn glbqdcgzuv.

Xy jua hyknhxama xwr zimyc sur vywadiunn, puyhldm, $E_{k,\max}$ wu kzklbemsz. Xbu gdcitfut hgahvxhkr sn rflhujaxk hna ibuz zgh fqbq algojuqddzv.

Ercexw qhhoqcsfd dquhx i obyssoh edpsyresxojw.
Aymnq gvqhluggz vbfdt w irzuwdq qngqrdqgooic.
Zvwepher qfjwawlyj ibdcx xxrv arq rrrmij $E_{k,\max}$ .
Bnijaejsz fz wcwm kyl vsehvk $V_s$ .

Wioj Xln: Md oz $E_{k,\max}$ –mcugesbws ewenk, kwfdhqug ebirhnjl $h$ , pdcatxzii-hvpfhquer ybqaewnwp $f_0$ nmt wcifuyskdkbg vtsstzplh $-\phi$ . Ag x xrpcimbkyqdr–wviqmdovd calym, dtiwxkwe lhhm eoq whaobllmg upvyui ski femfcqq mmg hap geg nkftkaiq tfuazynii.

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