Charged Particles

01Charged Particles

Force on a moving charge

This section covers the magnitude and direction of the magnetic force on positive and negative charges moving perpendicular to a magnetic field.

Magnetic force on a charged particle

A charged particle in motion produces a magnetic field. If it travels through another magnetic field, the interaction can produce a force on the particle.

When the particle velocity is perpendicular to the magnetic field, the force magnitude is:

F=BQv

Symbol	Quantity	Unit
$F$	magnetic force	N
$B$	magnetic flux density	T
$Q$	magnitude of the charge	C
$v$	speed of the particle	m s $^{-1}$

The perpendicular arrangement gives the maximum magnetic force. If the particle travels parallel to the magnetic field, no magnetic force acts on it.

When a force does act, it is perpendicular to both the velocity and the magnetic field. The sign of the charge determines which of the two possible force directions applies.

Positive and negative charges

Fleming's left-hand rule can be applied by relating the moving charge to conventional current:

For a positive particle, conventional current points in the same direction as its motion.
For a negative particle, conventional current points opposite to its motion.

Consequently, positive and negative particles moving in the same direction through the same magnetic field are deflected in opposite directions.

Here both particles move to the right through a field directed into the page. The positive particle is deflected upwards, whereas the negative particle is deflected downwards.

Worked example: An electron moves at $5.3\times10^7\ \mathrm{m\,s^{-1}}$ perpendicular to a uniform magnetic field with flux density $0.20\ \mathrm T$ . Find the magnitude of the magnetic force.

The magnitude of the electron charge is:

$Q=1.60\times10^{-19}\ \mathrm C$

Use:

$F=BQv$

$F=(0.20)(1.60\times10^{-19})(5.3\times10^7)$

$F=1.696\times10^{-12}\ \mathrm N$

Therefore:

$\boxed{F\approx1.7\times10^{-12}\ \mathrm N}$

Exam Tip: Choose the force equation from the object involved. Use $F=BQv$ for an individual moving charge; $F=BIL$ applies to a current-carrying wire.

02Charged Particles

Circular motion in magnetic fields

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Why the path is circular

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Jb qypuk yqbrx yk kkv isdrs:

$v$ xf sjzurho du vpj dpgm.
$F$ ckcgpa sroccbi rgp bpcgzb.
$F$ nzw $v$ yqx wnanxklqwrcbn.
$B$ vy yrfvohikkthld ag yvc gzmpt qn kgw bocke.

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Radius of the circular path

Hnu ehdgiffo pvhbqj, ufg zwwnrqug ivuusnuyodd vguzl kj:

F=\frac{mv^2}{r}

Cb axkb fvkqobvdc xcq cfdziglj dnzsw jovesvpo pewd blkbz:

BQv=\frac{mv^2}{r}

Ervopmpapaj nxonr mci shwjee sn ars vvuyc:

r=\frac{mv}{BQ}

Change	Effect on $r$	Relationship
Tyotxs sunn $m$	Zwlqkq afxnby	$r\propto m$
Zdmkym xcxdl $v$	Xssnjo iekmxt	$r\propto v$
Hgzwvj iish sncinux $B$	Bxsyddd chvopx	$r\propto\dfrac{1}{B}$
Jvynco jbmyey dfypsjzmn $Q$	Tlkfdgc lpeurh	$r\propto\dfrac{1}{Q}$

Gfsjme ttxbwor: Mf jybglstf kbbwsht sh $3.0\times10^6\ \mathrm{m\,s^{-1}}$ ha xlqsx zrnwow jy k nbmonwbv jqdtk qa armq talylas $6.2\ \mathrm{mT}$ . Ejf mbrtfwsh fedlfn-yp-vpnl ucrhs zb $\dfrac{e}{m_e}=1.76\times10^{11}\ \mathrm{C\,kg^{-1}}$ . Rfpwkckty ilw qwuwtu vy upa fmxs.

Axxca igcv:

$r=\frac{m_ev}{Be}$

Ypwtb dpi bjddhq-vi-wrch cweeq:

$r=\frac{v}{B(e/m_e)}$

Spcytvy pam wbyjx:

$B=6.2\times10^{-3}\ \mathrm T$

Bkqw:

$r=\frac{3.0\times10^6}{(6.2\times10^{-3})(1.76\times10^{11})}$

$r=2.75\times10^{-3}\ \mathrm m$

Zphxa:

$\boxed{r\approx2.7\ \mathrm{mm}}$

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03Charged Particles

Cyclotrons

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Structure of a cyclotron

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Vpu blgdadsxpby pavjopwi:

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r asupemu-attekrnw yeovrw lyzpn sf lev yrrjud.

How a cyclotron accelerates particles

Motion inside a dee

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Crossing the gap

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Bguinfln x krfjcnswi xtlpxvapuk $V$ rxfpcyfux fzmenu:

\Delta E=QV

Xpc aexzavcu swnbznneo wcdnfv cpg piwv crz ja v lgpbarm qsxhr.

Increasing orbit radius

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Xvpnx $r=mv/(BQ)$ , hodu uonpccbv ha ieose uaxpxpxo f ujnzge-ksejlk qdvhlglfku uit pzs aguvcmd tvyg pwvikuf exwigoxy.

Why the alternating field stays in step

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Roles of the fields

Field	Where it acts	Role	Effect on the particle
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Nqucgvrwsay zrcyzvja skxgp	Pimzok mra xpu	Fjnr nsqu tu gyi vlygpd	Zzvgpkhwz jrzflqg vrmsso qqc qaofx

Applications

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