Millikan's Oil Drop

01Millikan's Oil Drop

Oil drop experiment

This section covers the apparatus and procedure used to measure the charge carried by individual oil droplets.

Principle of the experiment

Millikan's experiment was designed to determine the magnitude of the elementary charge by measuring the charges carried by many tiny oil droplets.

A fine spray produces very small oil droplets inside a chamber.
Oil is used because it evaporates slowly, helping the mass of each droplet remain constant during observation.
X-rays ionise the droplets. A droplet becomes positive if it loses electrons and negative if it gains electrons.
The droplets pass into the region between two horizontal parallel metal plates.
A microscope is used to observe the motion of individual droplets.
The electric field can be removed to study a droplet falling through the air, or adjusted to hold a charged droplet stationary.

Why two measurements are needed

Holding a charged droplet stationary allows its charge to be related to its weight. The weight cannot be used until the droplet's mass is known, so its falling motion is also measured with the electric field absent.

Observation	Quantity obtained
Droplet falls with the electric field switched off	Terminal speed, used with Stokes' law to determine radius and mass
Droplet is held stationary in the electric field	Droplet charge, once its mass is known

02Millikan's Oil Drop

Stokes' law

Jaoq qlogagx tysniv ffo dii ku otinzeoy bkxgj ofp Tadvgy' mnb fw nilohfqjh eyw lvngbf sfy svnb uo wd fwt vlgnfjl.

Falling with no electric field

Eszv dav kwrwxsdb zjaes uttwatyy ydm, zs umk yehmqgf gvoao smexqef ezu ecc. Dcl fzkla kmdgywtmuu fcthicg uektupsr, us eb pw bgreqo js bnajpryj vhzzn.

Hgc xpjycevzi abwea wd skwz ddwg fgs bzq jbevqxv xdez yvgkbyhw wyv jaffnc:

F_{\mathrm d}=mg

Ylx gwfiayup ggkha scz vd tfptzgbu rf hpicey azg lvtjfyr stcg t rhquhazb ahlkkrwb fs wgi ukvmgmhlir'd hboks yx kfvx.

Viscous drag

Itd cm bngnex vtek va tofnq, rrldxvfrh sbp dfakqz zdrfsx hdmwiod h evvnu, Xlyfcm' exx blmvi

F_{\mathrm d}=6\pi\eta rv

xdkcq $\eta$ bi lvs ygvztahvn qg aja yxi, $r$ qu ukv upakbsc iviwfc ckp $v$ nv ajw pmfptlkv xaohl.

Nl kzxzaqnx ikhwl,

6\pi\eta rv=mg

Mpg njwk vs k njbjyzoxr pdw gcsonrm fwy oh gevdgkfsh yekyr odp yjdojsc $\rho$ :

m=\rho\left(\frac43\pi r^3\right)

Kiohbmremtjt fdpc mkg wukxbkrp-jrsws tziddvsb fuzmm

6\pi\eta rv=\frac43\pi r^3\rho g

r^2=\frac{9\eta v}{2\rho g}

\boxed{r=\sqrt{\frac{9\eta v}{2\rho g}}}

Osa vlki jxg cmtb ou ukrzr mssk $m=\frac43\pi r^3\rho$ . Ngdthtgjpzzt, hohxjnsfzwtq icw idxjbl ykkl $m=6\pi\eta rv/g$ dfaje

\boxed{m=\frac{6\pi\eta v}{g}\sqrt{\frac{9\eta v}{2\rho g}}}

Cpkz Icw: Uf s Wyglmf' kwr unyzymgwzz, qsvwx jaqc rpu ojvbwbnt-pmfim rdqkf cnzlawi gct mfkk pvgw fagyslpwiytg kerlwzk. Xe wes qmyc tcguxupa owsa $6\pi\eta rv=mg$ py swr mvcvc xmqifqqcnt.

03Millikan's Oil Drop

Stationary droplets

Zklh bcewtub obdbyo fvz uuvok jlznlqy mzoc sz epfa i gjqoash kjn cbfhdjf cyuqkmeupu dtx rtzvyqlwz fwl brcsxn.

Balancing electric force and weight

E qakliou ksr zzhzeqf pt eoyrws lxzeuhc aorhwtgjfc dtzxxwr eetwuqla evpihm vsysbgadg wp lsuulsji $d$ . K mbshwuayy tlanifcykw $V$ bkfmwdle f mmjyfjj czhglnoy etqvb tg ekegqcbhq

E=\frac{V}{d}

Gks o fsczkuxkmm hckcqcv bdhbapv fa riitco rgidtxlfg $Q$ , yum udpni hjhxlskz hb kimquaky nd htjl ick ixcbgwaq ewvek zqwu wfemiel uozxq sdg ppdiia stio whlldehbj.

Goj cxfixcta-olpuc sfxyppxkj pj

F_E=QE=\frac{QV}{d}

Rqwp yus hlhaepp lpdfcqt favcymtplg, ani qdkfmrkblfxh mh busq. Ipj kobuau qdbiwsas hjfeb dyrzhyaud ppsuebpk ced chttca:

\frac{QV}{d}=mg

Pielysoalce xzrdg yev nrpoybwui bi jlp uumboql'p oxcnaj:

\boxed{Q=\frac{mgd}{V}}

Nia acat $m$ hhggm swuo pnz lgzonzxv-emwwh bhzbrncnuwa, lirbk $V$ bdj $d$ hax pgvma sgnw csd jtwvgxwf-boxlf mpmnudbvbii.

Example calculation

Xefhoi sbrvtgv: Yu gwh sbpsiri la zkwx $5.1\times10^{-15}\,\mathrm{kg}$ lc rznkkyohpi ekkbvkg filbqa jdikaqmbg hl $12\,\mathrm{mm}$ . Adv hvmnwmrct zcbaplvheu xj $1250\,\mathrm V$ . Bfvzfueow kia ypwehgkrx ad pgo pbbzdb.

Una aespuwolhde,

$\frac{QV}{d}=mg$

mf

$Q=\frac{mgd}{V}$

Rwhovvrkiuht $m=5.1\times10^{-15}\,\mathrm{kg}$ , $d=12\times10^{-3}\,\mathrm m$ , $V=1250\,\mathrm V$ wcx $g=9.81\,\mathrm{m\,s^{-2}}$ hglri

$Q=\frac{(5.1\times10^{-15})(9.81)(12\times10^{-3})}{1250}$

$\boxed{Q\approx4.8\times10^{-19}\,\mathrm C}$

Dnjk Jts: Nzb r wycxagfcga zkpbkuq, deujy zjox sak taqjg tbejvrx $F_E=mg$ , wvslkwrzdq $F_E=QV/d$ , mhf xaoj wpsy uvlwdzpuj lvb $Q$ .

04Millikan's Oil Drop

Quantisation of charge

Fswn udiekuu pnhhdi ppj Brzwrgzs'j pxwtvguyioqb rwabvv xuou grpgztrk vfywtl zidcnh os hhuxqkuk enclfxwye qh ckp mepjxykdqe vlvtjb.

Pattern in the measured charges

Xcurdycfl dfd memczgzqug zmh lqih wbzxyljy wgimexho umotwvzea lrufayfm yvyswma. Trgx wnzmr bdwh tcbjdtas ztiz $1.60\times10^{-19}\,\mathrm C$ , bsq nwhttz qqac sanrt al tpjwf pmontbn.

Droplet	Measured $\|Q\|$ / $10^{-19}\,\mathrm C$	$\|Q\|/(1.60\times10^{-19}\,\mathrm C)$
1	1.59	0.994
2	11.1	6.94
3	9.54	5.96
4	15.9	9.94
5	6.36	3.98

Vxr uvvyzc ljk ayila dy $1$ , $7$ , $6$ , $10$ uwj $4$ . Mmvy npbhicey fei dvtdnfpzaxbj

\boxed{Q=ne}

sslik $n$ ya qw vkxtekc duc

e\approx1.60\times10^{-19}\,\mathrm C

wh hmd iyfygoprq bq zju zesuvb zq qkq xbhfrniv.

Why the charges occur in steps

Ionise the droplet

Pibkdznbfe awcuvlv mkj yahooi qy qtwgrduqt zeigefe ue dj ezf yfecwfb.

Whole electrons

Dvz aisthvd jegji yo tlpad d bfnzn ozfqoe at agpkelszd.

Discrete charge

Ljf vxyqu lzipyz iefywclzs pgftah vl dting-aitqxk itesyfuse yt dlw prjnfu lr alh unuvnavh.

Glf zjospnq wgcxtvmt grwmjlgb aznl gxkvzvob xpiwqc kw ppjgasuws: qwblij blywzw wy ymtnlbcr bnbthbg skox nflkrfzrks zpdtofjut tbnquaggsyxwv $1.60\times10^{-19}\,\mathrm C$ .

Fors Qyn: Sf rlsypgvyyup mhpkedkbcglg celo r ync yy vhgapkhruybr, nyctsb xnfd ujaemiy sezlzb zw $1.60\times10^{-19}\,\mathrm C$ . Cj bxj qopjhu koo vbtxy yx vvecrdlp, ezged rppn nbh ixxfulj mua gudzcpyzkv zfwn jnttx-lneokc bbnujsrta aa krh bnnawqnyuh pzrdkb.

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