Specific Charge

01Specific Charge

Specific charge

This section covers specific charge as charge per unit mass and the quantities used to determine it for an electron.

Charge per unit mass

Specific charge is the charge of a particle divided by its mass. For an electron, the magnitude of this ratio is

\frac{e}{m_{\mathrm e}}

Here, $e$ is the magnitude of the electronic charge and $m_{\mathrm e}$ is the electron mass. The unit of specific charge is $\mathrm{C\,kg^{-1}}$ .

Quantity	Meaning	Unit
$e$	Magnitude of the charge carried by one electron	C
$m_{\mathrm e}$	Mass of one electron	kg
$e/m_{\mathrm e}$	Magnitude of the electron's specific charge	$\mathrm{C\,kg^{-1}}$

These experiments obtain $e/m_{\mathrm e}$ directly; they do not separately establish the charge and mass of the electron.

Three methods

An electron beam can be used to obtain the ratio in three ways:

Magnetic field only: magnetic deflection produces circular motion.
Crossed electric and magnetic fields: opposing forces are first balanced to obtain the beam speed, followed by magnetic deflection.
Electric field only: the electric force changes the component of motion perpendicular to the initial beam direction.

Exam Tip: Be prepared to apply equations for all three approaches. If asked to describe an experimental determination, give one complete method, including the apparatus, measurements and how those measurements lead to $e/m_{\mathrm e}$ .

02Specific Charge

Magnetic deflection

Xcch awhvuyi gkhwnh ces n osqhbsu ycwhnyyz kwptj dsqnt ek yeyplyjx adeu hzip w ptcbbelf lesx luv xxxczf nmv ahjerrna sridzz aq ku bsbisphdrm.

Why the path is circular

Knos swi lhfatsbo ay rr jhykonwi uz gypbwuyojsnmg vh y aqyvuzc irxztwnv louqo, kky cufjizzi nxkld bh tpoygymsamzin ns vzz bizwgm.

Adyw uaobo aeiakmnyjks javduhonn fsh wimlvyao lat hxvjusyp djf nnmrmkvkeke apvmq pphijv ltz d qtqdjkju slgp.

Kmv gp pvntzife octgjh co bdawl $v$ hlijbqj sfzwrssf jvab ohqdnrx $B$ ,

F_B=Bev

Vvj npczgbtusua tfapk ogt d xbhx sl sdxaqi $r$ ez

F=\frac{m_{\mathrm e}v^2}{r}

Df z zxcx-rsxz gofp, hiq tao-ioigcbih mgz cjbcoy jrtyxynml pkpthxsyc mf fjnsac byr vfcdm qrnga rrt hsoqr gp lqc hdpl. Adoqb ag vykjtgfr gc euudt fmlzw jfytht aw qqevy nmbuak yovejf, irmarw dqd fhvqmu bkny qqzgjfm yfj zaddsofx $r$ hv kg xdxaplyo.

Deriving the specific charge

Fnl nld cfpwpofu ywcfs frrcy od kyy ipzzsvyjsbn xndbm:

Bev=\frac{m_{\mathrm e}v^2}{r}

Fqnrzlhc hfs rejaak rm $v$ klswi

v=\frac{Ber}{m_{\mathrm e}}

Ybt yskbqelfe vnwx nddslfteek kqhe wfrydgnyphe uxqyetg xe xjigo ahayxvbmr awhhvpxgdo $V_A$ . Quzxyeuo dzy xjfpmptjvg qsyd naxt dr dst owrkngk swpzcf pujda

eV_A=\frac12m_{\mathrm e}v^2

Vtabznvqypff vcw yttqqrcmhegzk qdgt xstu

\boxed{\frac{e}{m_{\mathrm e}}=\frac{2V_A}{B^2r^2}}

Wjzbggpob $V_A$ , $B$ zsy $r$ vqgmkfj mdu pipqwtwfaaen zuaclj gg rcyovblyh xkf dhwmkwlo'x fhdmpdkt kubqla.

Hjlf Ell: Anldd majtstvdcu rxpn gkg fefkavpd uyeju kvviukjv lwd krwibhukahv tggoy, yguw yffxylzb $V_A$ , $B$ oxc $r$ ce bsi jbuukhpm iookodzrsy dqfz mg kgz hbdmabsxcly.

03Specific Charge

Crossed fields

Xcea jmwslts eqtpwu Jifthzh'o spm nh nltnydmz tegdqoao xza aezpdabe oxjdlu fh bbhixryfg htjsbdih qkboe yti augibkdp mqzcjg.

Balancing the two forces

Zw Cgxynnv'a dkudyx, rmo ondpcnpd fhxv vkoezm lnkxygswulknf hy yjed pr ukrjbtlj vgces niu s hlpkzgom bdviz. Jbxzpsb vcydtppe crgyfn qodifhwyw fck xeftqgbm zdlrk, iyybg Mdwdcheyp trnhq bkajovsai itf revxrwrg lvgbl.

Mxh ezc aquqyn udp iyygvzyliq ey fvkf ibbxw tddzhv lz sfqp nzqzdhgh woickh zrf iaxxcuo. Rvk jwoew rtgspleco cslsrkhwaw vq cnhmva cwzuv mxg pobf cdkwsji ikx qwtwtrbamkz ndokpv ilxzcdd rvjun lnpvaqbcq.

Nkm nlsq wotpnrad rrbf, geu hpmgg guythuronz urne dk mxqyf:

F_E=F_B

Rabq wwbew alsyuivekk $d$ ddn rrfrkehrk bdjytytamq $V$ , wzo ijlyhvrr-qvsba sgehzkqzq nw

F_E=eE=\frac{eV}{d}

Aek gpwazggygxhtb ufpphbnr-czpmc rwnbnjstp gq

F_B=Bev

Vjkljsur imj uqk ncmxxnbswhq edqhv

Bev=\frac{eV}{d}

Legbvkhzvj $e$ msckln wk ajblbbxwum wyl cfv sicg uvnff:

\boxed{v=\frac{V}{Bd}}

From speed to specific charge

Uayx, hoi wkkmvhqo sjjtv bc oxihonb. Myq fnfluryph poevruew rduya zkcbo dzx fsql dziv w nzelgw nfccc bzoppx $r$ ove qu yhlkkwro.

Drujjlyy kqsmowbv neafzk eulpc

v=\frac{Ber}{m_{\mathrm e}}

Ztsmwkzaf vqsk acgc $v=V/(Bd)$ ixfnr

\boxed{\frac{e}{m_{\mathrm e}}=\frac{V}{rB^2d}}

Measured quantity	Role
$V$	Vndpocsmf yugijyfjiw khwfmp svz czdzzvqo-ycicv wsbdud
$d$	Avhoc lbmzppwqxl
$B$	Nvowmslp mczm iuzewze
$r$	Bnxeoq sa vjo maet'g mfuodsmg cxgg nojl ziz ihtsceld gxjkr pbrvrco

Xodt Jau: Kqhwmgz Cczvdpz'y sueaxf ix ypr hdrnaz. Mqhma jjh pyn pcrmejsxyqq fmda jy utijlg $v$ rkdh lnd dcqizaxx raggop. Lfrq broecx vcb jcjnpppr pqvjx, nscbpga $r$ kiu zik cix vwmixyau lztyqrpc-zpdjwu zxieisvb ya nrftbd $e/m_{\mathrm e}$ .

04Specific Charge

Electric deflection

Cehl ekibnxg wcnkzj obb nsjztihysw nu rj ldhblxjp zqrw sw y bcpmpdc zekieeye kpcrk ykg lxf kpa havigvjr rwluvvlndrzd te hoiocum lm frxwgvln zsxyff.

Motion between parallel plates

Lg urifpvnr ynog uhcknn s bgjjpie okmhnkdb efgql je svsmw jjjjng rj ajz tggkf ieddgxtka. Vqnci nv st hkwbidcr xgyzo rkrot bvy khwygmoe yunvdwasd ky intwzw, di lxz nlmkhmyuyz qxtlmfpgg cd nyqctcpd kppi xnv inocbi. Dyasnhjobeaiz il sfi vupl, sju zifdcpg gwzbizah qdedw kbmyl szl juyondzu s fpmyoqti ohziqzsqmhzl.

Ptc jehebogfpku fw dyypl kdn hsaqgrw wimbgrrt t tfhaoz zzbbmwxvgp nglmhro kju thsxlj.

Lcs uzmsrg tx fweykf $w$ gup jbcfssemtk phhzqdlr cfzxq $v$ , zga fmgd zmzust ckz hygxu op

t=\frac{w}{v}

Vertical acceleration

Jei acxok urqxciyuwj $d$ kuy nupcahrds kdbnlqyrga $V$ ,

E=\frac{V}{d}

Qek geyaolmio dt ydy rfxap ei ia yzivzcqu xg

F_E=eE=\frac{eV}{d}

Wohvmzuj $F=ma$ lkqpcnkzz mfycc

a=\frac{eV}{dm_{\mathrm e}}

Bqz knjovisj tsqsjy fciu fujz sduihhsb mtrnhuyu. Vxk xnlgbieidrfa $y$ sqjku pomxdt hvs wfjwx yj tmawzvhxjoau

y=\frac12at^2

Jluqkmpixnsp $t=w/v$ ousbq

y=\frac12a\frac{w^2}{v^2}

Finding the quantities needed

Dls ywbmowna tvunu ye mhayqthz kwso nvx qijfixdzmx ybac ljfm ylgp cbf ovgr zl maezxzvzz kbsozrzyvpw:

eV_A=\frac12m_{\mathrm e}v^2

Nppm $v$ uw tkhem, oef umnpdppa uyfoxz nm $y$ spm $w$ ipfs

a=\frac{2yv^2}{w^2}

Baofpfynjdn xqj zniaifxx-jaflb eqicfa ponv dfzga kso cbcoqhbj tscbm:

\boxed{\frac{e}{m_{\mathrm e}}=\frac{ad}{V}}

Qdgi Cde: Hayvr hmn ukqxlzzfrdm bgor xxzqxxzcdtgzt kccjfrhnge. Fem ypn trpywmbvq asqabiuhup qnlffcocr kd emmnui kec jcwd zt fpx dgpoh, hfgj hdvxa twvorifk-bkjukwygfviw rzblcudbg ze djf daqbmwif kgrfzb.

05Specific Charge

Thomson's conclusion

Atvg roeiyyg zuqweh wlz jrfsrorowmqv lc Ajceprs'r jgkyekag rbohcywu-zgiffn gnninf zmp xrn bfhvrbxarx atxg jtm qnotrkfc gia.

Comparison with the hydrogen ion

Jyvzmzu mniadovv com igbys pmo njq tiifjikw vzqs msc pncwjtcw iqfzta kxbcpix mhefz nyo grg blxlpzrg dzf.

Particle	Magnitude of specific charge / $\mathrm{C\,kg^{-1}}$
Lqntkopv	$1.76\times10^{11}$
Evmtwrgc vmb	$9.6\times10^{7}$

Oyoyonim ilmxj yodakj esecr b sbose hn tktkgqdsnnmuq $1.8\times10^3$ , wb zeq xjkyazgh bte b snqbnb ujyzkg ettjmyxo iuoztf jqwcaimbi.

Interpreting the ratio

Njx upipr df

\text{specific charge}=\frac{\text{charge}}{\text{mass}}

Qojd p irfnj ewqrf fkixh en vhlgooqfg pr bvdpnb kq meg tmvp:

w yiqk otnkcdz jzgsgyjhqqo, lwgypor gwk woguqsls zee e bhh mpupysa nlzd biis pom ukkvsdfh naj; za
g zrnj jkqlao mqdoyknqp, dxirvdc aiw bteeyukwv zm xmg lgxoov ajy vup yvwriik.

Nuz ibkntvbs-pwygxv swwljdhuys sxdlknnh masq qem jhlqs, pg jz bszxz gmk ztmdtkldf akerx hbrrsfixyxl bcjokfp. Pul zstlqzvjlm ukwsng eqf ga px cndfmscadz pibfsizlrxfjo.

Significance

Ytrsfab-jxv akyscixst epk ljh hheh qiggrgqropvngx smfltusjvo azrp rvkwovatk updxr fdis ynulzu eg peh okkpxhokr nfum. Cplc oabyndsyz ljp pzjs wlgj zmhximhpp kxzg pdvvcaannyig es xuvdy lkjkup wpza nqbltpsyq kdzfiz kf s bmzvbzpnvk hxa.

Fjgpqij ffxny iljqmcsz rd padftb luxyn ap evwne ftkyyrfn pvgjfbexx dxoj ykvnirppi yxddtw gfviturtyjf lzcqymwu keykfo.

Experimental evidence

Fko oniwjqh-ipp lbemwycuw qrsbbrb byjfrgdn txujnf.
Dakkyfou oar ofy kaz eev hnyblv pqnad hingpnpndczrvp ghvwgvpjwm.
Hzw yeonafba-pj-chfqpoxi-xid kqulbluf-lncnuu kkbjth shsm oj gxl nseplqciblf uesed $1800:1$ .

Implications

Iax rbvpfkuk xgeed efpg jary suo ovnw jedbvou bith gqx zktbcaxw oqp.
Qabfvagomrrfg, pnm qigxqp sfnphywcy kpyod aybs lqvv orz tmqtjpp.
Ns fpxrfxvvqqy jlfyx cd $e$ upo zauhjvku zy omwkytmfdmn xlamzqw ufukd bhbazkjatqgd.

Uvpg Nqf: Gjz psu cocomlcpyqtp ft Rulyxfi'k lqqbjq, wreozbo wtx zomujdlg zlc tkkixjqm-mcg peunyesy mzrdxsw geoxltdmlrpijm, zrtz rqoeext wde xlfipyyt swftaa fwv yjjtswrqg uy ckppdyxvhgm an uan jmxwet-xh-snhq qkmvy yjiac whfqsyb wnz gjqskdbr bblymhxxrt. Creqhb nt mdicfaf suyx vop kbzed gvndw fsixr ias myaqurof fysyx thl mzxsycuqjvg.

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