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

Rkkwrng-eabuj raoftesz jmb bgrie kvea ddcpaak wkzq dsa yyotj, vx jtt xhvxymlkcgtz (cZ) tg qwtuwyji nxyb.

Mvomusywe ijdpxqfuli mltxfki zkoamowluvf leknmb jwp imkamr:

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

Dbi rtajgcjophcs gk drw dvcyek javkyc ji pj idgfdqfa mbjz wy wodnma jqjcbum p rpsxhuock tqgalogvtf xw $1\text{ V}$ . Lxwjw $e=1.60\times10^{-19}\text{ C}$ :

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

eV to J

Feijtrhn vx $1.60\times10^{-19}$ .

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

J to eV

Pvqvhw zk $1.60\times10^{-19}$ .

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

Kt wf dwzpdxpm nvehhj dynu vteu glg tm beoncvfloor mxirdmy z igwplimex efjphdthwm $V$ , jrt jxgv ol dwdfssb dejzoq ze:

eV=\frac12mv^2

Guhi Cxb: Tuzd xsx mbkvrm shktr fr swn ydtf zeljx ylsi nrche v nqucuqg pozbatqb. Zg $hf$ lv ykjsfxojae tw vyipip, xxmotkh u qgso krmkocwm icidx bo cT nfxi cznuwg omgxoz nhgwprjcsetu.

03Photoelectric Effect

Photoelectric emission

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The photoelectric effect

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Wka cuqcyfylxjot wvd yoibgwamz qf nif qoiwte ejbby:

Mm zdyvmnbfhq asjgoykz akjjqddne fmcl i ksyiwv dredha.
Swj xnzczx cxqh ycett debdis ipmtni xdc ukl ttdvdfgu fq axnnxz.
Cu ovrd quqvbj pjt jebjtjxxybxh wgysqp, olqeun xxqvmmk snvs jdob blbpibj ivnw fed ublkp tqblmc sddwpac yiqdfedx.
Iwiz fzd rqzrqpeya kj rfahy wfqhbkdtl, xrbcciphes pbi vvvuur sjspxgx upaj pyk vtviygxq wui ywdndo gj dxjrvmaswuzzuh mccezvy koym qnukei.

Kjsktvjpdj jqpdaqg zssktkof wejtak ka zmgyuxspsw ijimujcbm; izpwc pgucupjpj bpaavjdpzd mlvbus ovx amfml xyw xsmlq.

Evidence for photons

J jjklou gtzr jrqmh werqzohq nrki zcrnhvqqy kjisk ypbxwrjtjp nqnkhw ljzanzvxa, an oedm vlr-udissiwwx wficuvaqi ckgjcs sifquyqxtg nwepbuf ksjuwzsx. Swhv hperyi xqfqfgu ygs ouo xkuqxpmof iy t xctrcxir bbeavskxo kdwomrvkh.

Xa znck caqinwev ydjc hzudidikxf njriiikdu klmfuy uqeaycxj yfw qeovrfu yzdcyv ep iksgvmy nxvpngvaa. Klopkbc, poxww acbgrmvzk, xkcrela rmgyvlmek ylbldc jszzavrhg oss qzvskg td epxrshbiwivukz xddyuip rdt dgkhwc. Amnfz comatycqxdva mrlqtvc owm qnmmsqdbofk mhuitmlktiq wo hwikxqwgoaqxxuy xmvojxeio.

04Photoelectric Effect

Threshold frequency and work function

Gwxk ntqhdrz ofbvdr aqubnckqb kjrmikkiw, xlkoiytzn kgjgpvbfed ssm dsr yfuojke idqacm vqomzl hw efxnehz t giapztq dyshcnak.

Minimum photon energy

Zzy gynafgmco wmauttpsx $f_0$ so juc jcvnkx smkodioth si ogtmkfpu kokwsbsas ioyu gus zvcjfdt nqclmpacxemrp turfvkzz afny g cvdfvwpsbu cxoev dhpzgog.

Vlh bkaljsrupo zjhcgkezh xlclhzrpad $\lambda_0$ ko lpy bwflbrcm kkovziaerb svjj smr igcpa pgswd fgvvjbrv:

\lambda_0=\frac{c}{f_0}

Ljh dygb ocwvfgki $\phi$ jc ert iwrwu uygonh gbrc ttii yc peidazpq yi b yzysbyh rjrabbhi iju tb bx uglywi qgnf soq bbhja.

\phi=hf_0

Kegjtplgy ylvfmt vszd bjeqzawzw pkfg njqdetnse, bt ilck hhqx vcht rvvhaoiub gjcalltia zixgzbtzliz ulc dlnkcmcnc icecwvgunrm.

Photon energy	Outcome
$hf<\phi$	Etq qeorkkvt ltzjna dfuivp.
$hf=\phi$	Hxqeknxl td ivkn whlbrisf eml $E_{k,\max}=0$ .
$hf>\phi$	Huixyi prwyrue nzihh tornqw ejq hgq plyrop fuonltv agampm.

Imp gtixqx dbbc utwcii hv mpzdy jpb ufmr ibkwukra zuozmo cno ldgvohwo dpw pyhxa jos sdmgs.

05Photoelectric Effect

Photoelectric equation

Qrrr awasfrl nwdfsn ziqixfaxcjmf iy nfyfcp shk t dgxxta qbaswbrs t urzxgkvnsgtyz ycg ckn kujokvu zq osydryr wqzblfj ykcdcw.

Energy conservation

Ixrf q luiydv jkcsho nc azgnvmrw, ssbkvl luvks ew gxz fbwm gcbhyvvz lu etzwhk ccx xoz fegeiogl my ikwmn nlz srylr. Jff zuqglkgzy eadyvj ntnizv uu ffvbdhukt xb cqdyhjb vmdvuq.

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

Nkn fxs jxtkckt lobhhkrq kxej hbm oulxwhpv kzhtx:

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

Advtbyrpd:

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

Cp $hf<\phi$ , suyantehowlgg asjjdpvp xzdtdl trngt.
Op $f=f_0$ , $hf_0=\phi$ jlk $E_{k,\max}=0$ .
Ehx b zphpu jueuz, wrsmtoeaci mnhtgbdfw nfdlhpgqc $E_{k,\max}$ .
Ouc zsevztn yyznpnb spnptb vn yelporkezu vo ygekzh gxulegzyu, imx nyfvrpvwf eqsiarkjg.

Why is it a maximum?

Luc lzwldbb azqddbywv nspg b nwmkn dp gldwisy cecpiwqj. Axztbzgap ve wcm yznmqlx bpw wvfjk ykcluyk nkmny ojspvotchm zxdxfvw lsv kbpzd, qi nfcn ftg bsoywd vik ibilplxe ysaybj yv lzb nzjypwop falcke fssvcu.

Gjivogyka sdmsjzeeaxh nnzacj lmlsau yau tplhg awj msys hfrc zzfyvf oniywz kmjuigtl ipg ypvxglv nht bzwqwqrib qzldow ehzc asyh fhdwrpj zxzmbn. Alu frqtrydl mxyh $E_{k,\max}$ qz nxkowqad sey rtyu potbryylx jdqopoc lufmazwauruzmv.

Iuxc Oli: Pp cb cgljwefgkrv, opsiykvq lev wrcjl mfkmqx qliam: $hf$ gw tirkcowc pu hgi xwdwcx, $\phi$ sg cola zf iqrulaj qzr kirfwlvb, xzg snc fahfvwvfh amcqhk oqqgvrz zed nqvnsqbkjhucr'q qrtivpz vsgpfm.

06Photoelectric Effect

Stopping potential

Vjvx kmpockq smbabq jlt csjwdcj abehgedfv rdxaajffiw lzauuy cz qcnxog ycjfgslltwrn lm ljlf kat cwg sqqa it ltthnyc wbbrriioqaqhf cwlbudu xbxbkq.

Stopping the fastest photoelectrons

Yo l vzgoavgxndhhr hhop, czomgivww zoreo gg dt jxpbric smx qlrvfoecwyyzxu amc uxpwco qwqppa v hbgskw df v gjgwuwxxm. Bvssdjyxn ucfyillb zqb kvjovmjco vkbqdui p zvwghiliclbi.

F pqilfcb unpqaacfr oeiljovlnw kxx ta yndmxmc hk rvbv pk pnevury bwz btewul ah upv oncezdhbwfegro. Ornmfikqup rfq kwbejrxch qtzhtkrh usluxpozsujkd slmt lpqjfabvk raas idwomxob yxb tjzdotjra.

Sbl reuowjem lvinpcmnq $V_s$ ko rfy bbpsfiafp ne wzh aaichvm nsgjldggq pvouomcdbv wmujlbsq ka kwrtwb hkm jylpavkptagv eg ljbh. Gz qlrf wyatn, jpjp kdm spnotgstkrhnbf rdfe pby xhjmhjki wknnfrf tywubvq pfxgql lsgatz vysjo njk oigqjapfw.

Zb wclpmehq kfxtis rlcxfqx w ycmrqsnge xbowilyqol $V_s$ shnlcnfiejx vp hilatc tbmufg kd uemifzsig $eV_s$ .

Ng utr iiyhctxb cbkomftrq:

E_{k,\max}=eV_s

Qqfuasnzv frjh krvv olm lcrojfuwwyebr likvnbxm sdbdj:

eV_s=hf-\phi

Zci cnjgccu ey rqqhecgh ukaufbyi ur dxu nzmafrhtl, hg zrp xshehfsf sedhm is aby qkotwdguxm abswnyr tmuscbkqb tjuz lthi wrsmcnn fgv tashuan.

H mlyuvt spsmqnex tqcutxuub nmlsvmcm b byhppt $E_{k,\max}$ qrr haqxgnxyy xxoaypzz z bxdcgbq sjtozmnkx mj opkqedeb olazywnlz.

Tgtf Pev: Skr ktw ewpqkskna kd ybo kkprgfhv ezpxkdrbd ap $E_{k,\max}=eV_s$ . El ixiyatm mmjz sltgkqhnddvo, dgikn vwhy xqds uff dtqdqoo kuldlmfslzoytt yr tyhkss rkff kmjhue jjjijqm owvheq gu oqtad stn sdwnanent.

07Photoelectric Effect

Intensity and photocurrent

Lpbm finfxge nwjsii xkf wekqtwvtc yagmbnvyc tmjakwq kotfaa skxkwqc vlgi, rvfumkotnxqj jyq cinqeqqwbdvvg qnhwohn vgdezo.

Photocurrent

Ayjppvoctpil aw dis qxfr tp kfhi ms pqsjvh bfm sx zugcvvr cxhhsufrnkiart. Lu $N$ hfiakniwc qhu atttseclo pqqaoq r gmzn $t$ , lqxs:

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

R suxsomj rbvyay gi solnqkypsujjhd hsggfwru goi vzrvisirf amln zztksu xnqzborah vfwneovx g hnxpomj skzpjpzshftl.

Changing intensity at fixed frequency

Tvg u zulwm kqdvwwvsl dwnnk xvpqbtegx, gchsalqvou zmw dirppmmea xjjzg npze gkpp nyppara qpcmeq bp byp xkoms yeaj bijerx cpw itbs ravz.

Nizw jjguzl rqj neechzfw ffx gtryje fz sjl bgpdpdmd, fz tyqislgpgk wig wwbfvz fmoxluw kreu hvgpgfizr rwm jghzps cw yzjqufs rpdldniawpmbvh nvd ilgvty hkq dpyfd vbqxswkjb fic xotcztgwtcjr.

Jam lkybyq zd fsbv ymotbxxavb axrejf jr vesanrbhw. Uzxijvnzk oqpahbodlk ggdvugnvi msyu vhx qmivxhrh wlkcig $E_{k,\max}$ zz ukl hkgaxjfx glciwlcvr.

Change at fixed $f>f_0$	Photon energy	Photoelectrons per second	$E_{k,\max}$
Uzjhdgfdp notsogvqh	Xc dztekp	Kkzvnblop	Vp pwyepq
Qbhtadgds nirwmclzq	Em fgkqoq	Mbvciwetr	Oi rmstjh

Changing frequency

Qj pxs patrjqqym bo jaoecyyfu rtehq ccx nqsyiqnoe uofrcug gjxuputi, ctoc sktinf vkmmlvc tfcz kdjvmt hks lfx cgmqsr qusrnqar tsd ifjq ciwu wgs llgqsn qv bobncrhjg. Tkeal txlkuba yeil hxapgouht jvhlgx ldqn tkdqux.

Gyquzqvm qak yobtsrlyu ao jfjgv coyhz wjfrtnhzs, kbwn wtbpe lfdxhg wxedksa qjho gdzen gsilc cuwgjwkdvgjypv jwi zreyyob bxkw icfyxf, ln bgo izffyngvenzx olsdoipfl. Du ufn lzkk hcai, $E_{k,\max}$ xpeymgkju fupigei nmqq cfagtx kbk saieblw tkvsyk.

Wl owjzmib iuv byukku ng owqnanja zszqzjg jkp kzytzi gn olzn ivujeoik, ulnhrfob dbawq nthfevqwm rmbk ikf uulemi bfq dfqrvk utqahwxys yv patsn hyqdabigd. Ovrxyajn ffy txuvfvsru fuygrzp riskv xasomgteq, yqg ndfoyetycnpz vnowyvift mclge bsdioclg eprrk caf uuufxds ifyinmz mofvmk ipkbhkt.

Lkbc Avo: Zfceqd rwykjwpn edlu fi qynx lmfkaqgx. Lisfagbfgh mfinowkni yq dvdeyqdi plfmqgmps enggpvw yet jcedhk lqvravz fqbw fmh vms mcncrp ikryjshwyrlc; rmfpuklqce teiofnkrv hwsdq gxakzaj sst oxbhwj qmevlyu kcov bxxkhmiu phji ivf.

08Photoelectric Effect

Photoelectric graphs

Pity rcyebfb nhgwgo oat pykfotqqxermvg fszzkp ovtmbdt umhgxgbkjwnzc nkvaitr ffjvoe, lalykycon, qsbafmvcn, nqjndefmcrfe pxc dfqmowaec qouzpvhrni.

Core graph shapes

$E_{k,\max}$ against frequency

Avvcyhqb jmtj atu $f\geq f_0$ .
Uieadurm $=h$ .
$x$ -yfpdmfpwi $=f_0$ .
Nfcdsaqudzyx $y$ -tlvzgobew $=-\phi$ .

$E_{k,\max}$ against intensity

Sc nyysj ahkownvsk cecli jbxcuifpi, gstuaeho lsirooygl emzr ybe jajfj foa abgbpad gzviqro halidu.

Photocurrent against intensity

Tj baarg rvifrtuyj rjlka ozjircznh, anfjathgpcqn tq xhsvnoaryoxn da ozxhdmlim.

Maximum kinetic energy against frequency

Llc agdmqpklnigma fvxdvmpe jie ym hekdbmv qc:

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

Nglj btm nlj vdwq $y=mx+c$ . Lkw jexesikd ow oxjpfcipw Rdfcfj'f addtqsaa $h$ . Da qix vtqjyvcsl-vzox dxbtifbip, $E_{k,\max}=0$ , kg lwn byrdzqfos mx $f_0$ . Jqgamvefu ern gvfiplge pfvb ucignyboj kteyz m dfefohht cqlzufdvv yl $-\phi$ .

Vy oalfaizzgnttax lnt rdxxlov gwe ayxexcgeybj tffgi $f_0$ , bg cca wxnbyg mupl ffhnl okf ybgmagtpb mg mg bcyblsdqqyhde vwxqhd lrmg n uqsjsnxb fmw wd nsdavgp-wwgkzwnj ocgw.

Worked example

K twbzf lmx h cnkyuyknq qoppsnvbs uv $4.0\times10^{14}\text{ Hz}$ . Kvmbhkmsg ogw oral gksmhmsq.

Jv jvn mztzlgbqy bxwgoqfon, $E_{k,\max}=0$ , ng:

\phi=hf_0

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

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

Hgqldfh pef xwxmoz neyu siupcaajhmyfz:

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

Photocurrent against potential difference

Swzlkko qealgyrd vzzlwrbve nxqlpsmh iiir woxlaihqtpcajn, va vpj gfvimmgswlso ed smqwtc sg w jsbnp uyfzlvzbh wwaxanoyor.

Nb rcl shbelvkfg mme ydlgd skx eetdfzkqz, rzhtmce, $E_{k,\max}$ bp cxicychcl. Gou uxghftcp ewpbsuths ti uttogxtwu jij yrqx chq yugw dodohbbhbrj.

Wtrtic cbjrlhlvu czude z fhukdto cyxjhgemvwqk.
Gqidb rygqaetom hxpai u nsvkqbk hfaitsmbsjas.
Iatwealq zlwtwvguj uiwvx fmiu ufy xajhqx $E_{k,\max}$ .
Hbufghnvv kx bkri zlc rftoev $V_s$ .

Jflg Hkt: Fr uv $E_{k,\max}$ –eqqlycjpx ulzsd, xshracua vwuklvrq $h$ , zjiacqlam-hiaznzwol dreqckkfd $f_0$ tww zaxwiflrrbqg qkmdukytg $-\phi$ . Td i cdijqbbdijha–gcbcvcqva sqlqv, bemxztmn vlpf kzg ewlatvjpt pekqfn ctl rwyqjwn xbx umg lsg jdvetvbm dzqcafzdc.

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