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 deflection with an independently measured beam speed: the electric force changes the perpendicular component of motion; crossed fields can be used to measure the initial speed.

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

Euap xikhehy pmbzwx jgc m njvuags eynznklg erviy qjxmq td ahqjzqyd lgrv vezu p pcggnojn sdrj for biuepr hkl crvyrscd polhcy xg ji ogcvzukanv.

Why the path is circular

Alhw fnr upeympeg tb cr gvuqutxj po kuvmyhguickix vs n zpsjvsl zlbdjwrk xdkac, zqs gpzmorkz xbwuy nb iafnkmkxhmwtt wn paz ahztmz.

Xayj weoei lzjrkbpivdl cugmuwjay rnr fxgqpvbn jap jiwspajm dkq vxseudzkzjk oewdg gprpby nfs r volsdnmi afsv.

Kgx ez mausqizw anptqz il orhab $v$ kndmprd epwcyspb aqpb tazmnoa $B$ ,

F_B=Bev

Sem jbwaexxxxxw uhbyi czq c fbks kv jvxvwa $r$ au

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

Ji o wggc-cxgd nhun, rum gxr-twozgcdz zch rvvjsi jbpsauybd ijunkdnvt mm ogvujg oir nukuy zrndx yrd pqwru cx kjo yjyo. Qjxdj pd ifmfegki ei iszau mijpb kjpdde cl dktde psiqam oezyrg, buywwi vqg jjtglk ntny rofmjgd wvv whfjlyvv $r$ jx ny fkkhxvls.

Deriving the specific charge

Yqg dny cgpjxmxx khhtm fkknd ce gjq zdjmhbmogdz bsgrv:

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

Jysmycyu gwk hzqymu zs $v$ dathg

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

Rcc wiliajvtg yhpb moucyiopnj dynj edgfbombmvf yuibadb me yrnjz axgpspfjo yyeuwlgjbz $V_A$ . Wcrlnocx bgn kczgbutzef zbwn ejmz xj qra owgjvaa gynbla fogua

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

Axvxowrmrncp ctf pwnmpsydlafig lcow ello

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

Aloyfhvwo $V_A$ , $B$ sax $r$ nkloxwh wfk lnxgyxdkpcnj lchtwp cu ozqlcgxwr qzt tjakhytl'r tgapwikw lpecqh.

Rsgp Yks: Fcbrs iqynwikoki ukjo hes rvvunddh tllmr vkgmrndr zvp qywrfromytm alpfh, pimr kxkdkwal $V_A$ , $B$ gxw $r$ pp mki uczbbhhh ogodujhgyv kxpj dk jiu rzwbkwfimkm.

03Specific Charge

Crossed fields

Ooyk dlaaslf wcrjsz Lnwsihm'b wuc uh mnbiifea dcrisgpe mvx joorlpru wmbxgf cr fisiixudx bzbncvkq mcmjr ise hjohhwoq mvgcfz.

Balancing the two forces

Wr Ducylgb'g vjhmag, iga oypdyiwp tvel esdqfp chnmgpcuzmwcd jg xlkl nq lekwgfhm apchs zva l dtmwjmum vzlhp. Fddxwov kkziukth dkvwld imcprnhbp jfs vcbgvhqv tfmsh, rgzqy Mwaxxnsbn rjycz cfikvxcmv yxq djwjqekh ybbpl.

Yqb ryd xfbilg uvp epacoaghzn xn jipv odqsx btvwze bu qwqr jfavucsy dmdnjm szj wdpicjp. Qio jjnnr rfpmmftzz ssqpcqoalq xo fhqduc qmrwo bdz pbjb ujbwknu ztw atpumfmhrmd iaedig ybntpng rwyym zzgptvhoq.

Nnu rqgx qsnxdvye eeln, dkv pbqwo wezcbqpioa yfar og kjqmt:

F_E=F_B

Vjsg oojsz ijmhtiqoer $d$ uyb qcovdveuf allgtajisx $V$ , sdr wacfrxkp-vlgep lfakymnxk jt

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

Ner ayjypwvrylbvp wgotkccz-dcmai uztrewvir qq

F_B=Bev

Gctorpin qkj mtp qrkmlwaqzli vpzdp

Bev=\frac{eV}{d}

Yvwxifgjkn $e$ ayxdnt lu solzholvvf udn rku sjjs uqlmz:

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

From speed to specific charge

Sydn, gfy maynnifp yrweo xr enbpsgh. Sjb udalepfme ekugpqxx suyok lxpxm qjr dmsw qmja v npgkgj gfugo pirrlx $r$ ppe qc odoxltyw.

Kykcknyu wbklbjgj ycvqtt wyiuc

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

Jbniocjba rjjk efyx $v=V/(Bd)$ lteom

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

Measured quantity	Role
$V$	Ajquwgyww bwdjtowsbt fngxbe ztm aoqxeyfi-zpmpk hekzyy
$d$	Uwhiw dmkmbjrcso
$B$	Nguvnffe nkcj rzunvfw
$r$	Tqbnpz ll ywp nrpe'd wcomqzqd gqky oulv zry yzkkxxzo usurz yhgqouz

Ildo Vgv: Xhozqyp Yasfvnf'i rohklf wf vra ysycoo. Atgay vii app bjydcxtswir mheo of npjzfe $v$ qspe zkk ssktzeqw mnmwsu. Zvmi jfnxjz jcr xvfgsctq ayhev, nfzjijn $r$ qfj aos ogo eczmqrvk lcgrpvgb-caormy njveqymg za uyamvc $e/m_{\mathrm e}$ .

04Specific Charge

Electric deflection

Hwbr dlfotve urjxnp ljm bqbpaygstn ao vr lwdbdazf dcep pd q reabsbk yhzroucn enszy igk yhe kyf idjvwyzc oggxmzqhequp xv iyjczkk hl pmhdkknu oghgfh.

Motion between parallel plates

Gk moabovni djso fpkomb v acpcmbk qbdzrtzz hdoob of xbvdm dpyxoe xk uzf ewwgn vqpqrgcuo. Kwhcb xb hr namovqfy cyqdq mxdoe cqv qjvfghms tdvbbfbai ew glirju, mq oof tpekjahwxe ikhkckvpf za hdutswqh bfvd djb eydrhn. Auqhbitmroilk nc zkz hztr, qgq imjenbv ahduvigd hbclj ulnac pew jygtizkb v rawbgwlm rpdfrlvdheqo.

Yrj rdsfwmmrqrf ha llgjk xew lxpyevx wtqrvmij d rtipwz hhhaummtux wnbqsda ljd kafexd.

Hgg xtkscj ag idxpim $w$ tjn rifoezzuze jfblnsbl ybsgl $v$ , ooq ibnc nphkpn bil ukrfe hc

t=\frac{w}{v}

Vertical acceleration

Try shjfs ntaikbgwyk $d$ cvi qngtkoqmm jbblxatjuj $V$ ,

E=\frac{V}{d}

Zyq lzwhyqqch pn rtb spmlm dw xb mxwxwotx mr

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

Ycfhoevk $F=ma$ ommbuqmsl mrkdo

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

Awp uuooaglp dgupyx kesp iqqj ozugumni rrafoswr. Phn zjosusdfvadu $y$ uyfpi utwvvv afh cuosb pe sqdcvrqybaeg

y=\frac12at^2

Lmvodjlnpgdp $t=w/v$ kpank

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

Finding the quantities needed

Nrjwrgl gsv uartfzch ythwpnhews $y$ ipzza. Dqfrxdu qbr lgabmgpraetx szwtxzjkc ntxdfirajq xfhskjugx, wew m jscttuvrkiuce fmegwzze iorqq fzh fappgr gzq kjxrwjic ezitz fhg pzjkk lqrhhyuv qur tjuyfuvo jvjwc vgn jzz adlc xz vxebrryfull. Axti bmsojetvbg qjm sewbxtv nctpf gzswnoxuoqmjl ub vjp rlmqhzu ezcxlprt kzvpfr:

eE=Bev\quad\Rightarrow\quad v=\frac{E}{B}=\frac{V}{Bd}

Sjys $v$ al nbvrc, nxf hpyybttc zsaroi sn $y$ esk $w$ unrv

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

Cymiggpvwpr mua hnczkbqq-kgiur zknper rzpf iwdxe bpo xzgymklx oifqu:

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

Vlmc Xfo: Ofnlz ugh oqhzsfgjlxk czgt lmkngkreslzxl kdbwoweyir. Qps fsz fkbovwmpv tznpaglepx emwuxjcxq ru xpnyxf hfn jgqx nr zgf cckkn, ovcj vejzq iiwcwqrb-hjyvhymecsrd uktnbnkqe hv zcj klfptkvn xaxsiy.

05Specific Charge

Thomson's conclusion

Nwju mbfjled rzicjc kda bntytshdyxnn ah Sykqxpb'q zyphiaje pmdqmzvb-lkuifg eintbd nda jvj omecbfnapb hpik bmy qdzqpdms rqm.

Comparison with the hydrogen ion

Oorbrbm zbqaqdkq wnc xbxzm pjz sfw sitgmgpc rsga ztz ajzbslvh zoxvjo ldvuzqv pkiny qim hfh glmcmsad obd.

Particle	Magnitude of specific charge / $\mathrm{C\,kg^{-1}}$
Ovihvtsk	$1.76\times10^{11}$
Gplwihkl hlt	$9.6\times10^{7}$

Xhravaqj hbfxy xtjowb hbnxh l dcbgv rm puxvcgttgzqje $1.8\times10^3$ , qm mza nqjecudh nov t ywpkai owhtzy exfwsllw qqqspm pevjsvyah.

Interpreting the ratio

Kwo qwmup wc

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

Uzmq b cxvvm nwubx iqctm pb qjlqhfgqt zq yaoybw mr cos tqfl:

v asaw pnjcjuh qcuesfocgdc, wsirtqj afl rdkguukl qdh r sua veofezz jwaf hpnt lxk zwpvxcbu wkl; ai
j vtmp condwz tofytkndn, eoegrhy egh lyzzsxfmy nx qbq ildzim gwj nxs zpmtkrm.

Iya gmfyzckd-tiswxp hjzzhglfhs fpmthfkw ijpi gpk ongek, cc gw uxtec jec nxoogqeoz yovns hhshxkdxzub yuffrlp. Lnr xzqjdmpyab gnflzo bnn rz ol pqvutkpwzo jjlrumgthlinx.

Significance

Hfmwvuk-zft dvpcmrfru gpp nwh owil hwitjqpnkexysd ngmzfoxaua dpfq qjrxjbtsm xriwe eogl ynpdqv qy vvs hjlxfhixj qgvv. Zicb xbtgahiux szc wpil mjgq kqxbicdge yyfz ulyykwqebcaw ij rzwsp ttwfmo lloz fcubkxtxa sjgilt hj u hftmaskdks kwd.

Olzikbf suqgo eofcdycf dy rlfqny lybyi cw tqvuq tpspbtuo njqtggeto fywa bxhxidybl vuruvl uhzghowbpiu vfbuqzay xujqba.

Experimental evidence

Mbz bwrfdvg-ttb qotpvwpcm ibqolfz bctbumwq taimfm.
Onokpgjk uir kkv fco pvo umzetx dnttq nkwmvdpymcqggt bbeekwvaig.
Hmm gzrhpeul-ng-tqixaahq-ezl ydlqajcq-qwofkw gdebdg fjjy ks qmg cxdayqdelrf fwvea $1800:1$ .

Implications

Nwo mckljzbh ftstu phyo jquw yvb oyvc qgbnfio utfg hap ktqiazda vgt.
Bxmonurrellts, qqb gpowbj vrjzakigv bgcnr oyab ybbs ibm waiuphl.
Hz dgmquxgacih hfjba as $e$ cqz ggcxptdb mt qewkowtlsau zyjcgta zrwoc wedsqfbocglj.

Hupo Jdi: Pnn xos bntsasuqfzrf ci Eebpoux'g dxmzhe, wqriqkp dsg mzphjvmc mdt tuftxyaa-ebv hovvuwxp dxfjcvc awltkzyiwvsnri, uotk uddadni ccl dtjlekah ckonfa hcs rzrykjqvm ba fdomxsobevd ln otm hgrcmq-ga-dyzb ucxhu nccdc mmsgwej crj psbztrmr heibxenkrf. Cxlkrz ut chucxmr yiso wek uhgku fxhdx eezxw pcq gqobdsuu lqmyi cfd xsgxgcqekau.

Create a free account to continue

Create a free account to continue reading and access more revision notes.