= Thomson charge-to-mass experiment
{c}
{title2=$e/m=v/(B_a r)$}
= Electron charge-to-mass experiment
{synonym}
The <Thomson charge-to-mass experiment> infers the magnitude of an <Electron>'s <charge-to-mass ratio> by combining electric and magnetic deflection of <cathode rays>. The central idea is to determine <speed> separately from magnetic curvature: a known <speed> turns the observed <gyroradius> into a measurement of $e/m$.
In 1897, <J. J. Thomson> used two grounded slits in an evacuated glass tube, then <aluminium> plates $5\,\mathrm{cm}$ long, $2\,\mathrm{cm}$ wide and $1.5\,\mathrm{cm}$ apart. A <battery> supplied their <electric potential difference>. Deflected <cathode rays> produced a <phosphorescence> patch on the glass; an external scale and movable luminous pointer recorded its position in darkness. Greater evacuation reduced gas <electrical conductivity> that screened the <electric field>; reversing plate polarity reversed deflection. External coils supplied the <magnetic field>, calibrated by induction against larger coils. He adjusted equal electric and magnetic deflections. <Cathode Rays by Joseph John Thomson>[Cathode Rays (1897), Fig. 2; second m/e method, apparatus and calibration]{ref}
His comparison included air, <hydrogen>, <carbon dioxide>, and <aluminium> versus <platinum> <cathodes>, supporting a common carrier. He explicitly noted uncorrected magnetic bending outside the plates. His tables give $m/e$, the reciprocal of the magnitude used here. The inferred ratio alone could not separate <electric charge> from <mass>. <Cathode Rays by Joseph John Thomson>[Cathode Rays (1897), second m/e method: table, corrections and interpretation]{ref}
To connect this readout with the <Lorentz force>, derive a simple nonrelativistic model in modern <SI> notation. Let the uniform deflecting region have <length> $\ell$, followed by a field-free drift of <length> $L$. Assume a narrow <speed> distribution, small deflection <angles>, negligible collisions, and matching electric and magnetic field profiles. For an <Electron> with <speed> $v$, the electric <acceleration> has magnitude $eE/m$, the transit <time> is approximately $\ell/v$, and the subsequent drift adds exit transverse <velocity> times $L/v$. Thus the displacement magnitudes measured separately with only <electric field> or only <magnetic field> are
$$
|y_E|=\frac{eE}{mv^2}\ell\left(L+\frac{\ell}{2}\right),\qquad
|y_B|\simeq\frac{eB}{mv}\ell\left(L+\frac{\ell}{2}\right).
$$
The electric expression follows from constant transverse <acceleration>; the magnetic expression uses its small-deflection approximation. Equal displacement magnitudes then give
$$
v\simeq\frac{E}{B},\qquad
\frac{e}{m}\simeq\frac{E|y_E|}{B^2\ell(L+\ell/2)}.
$$
This is an ideal derivation of the comparison principle, not a numerical reanalysis of the historical tube. Different exterior <magnetic field> profiles invalidate cancellation of the common geometrical factor. The comparison determines $e/m$ through deflection and the known apparatus dimensions; it does not require a downstream semicircle.
For a second, deliberately different teaching geometry, combine a <velocity selector> with a separate region containing only a uniform <magnetic field> $B_a$ perpendicular to the selected <velocity>. Here $B_s$ belongs to the <velocity selector>, while $B_a$ belongs to the analyzer: they need not be equal. The magnetic <Lorentz force> supplies the <centripetal acceleration>, so
$$
\frac{mv^2}{r}=evB_a,\qquad
\frac{e}{m}=\frac{v}{B_a r}=\frac{E}{B_s B_a r}.
$$
If the analyzer turns the beam through a semicircle before it hits a screen in the entry plane, the separation $D$ between entrance and impact is the diameter: $r=D/2$. For a shorter observed arc, the screen displacement is not generally a <radius>; fit the appropriate <circle> geometry. This two-stage <velocity selector> and curvature apparatus is an original ideal teaching model, not a claim that Thomson's Fig. 2 used a separate selector and semicircular analyzer.
\Image[/electron-charge-to-mass-readout.png]
{title=Inferring an <Electron>'s <charge-to-mass ratio> from selected <speed> and magnetic curvature}
{description=Original ideal analyzer with zero <electric field>, <magnetic field> out of the page, and negative <electric charge> bending upward. The semicircle has screen separation $D=2r$. All numerical inputs are stipulated teaching values, not historical measurements. The <magnetic field> $B_a$ differs from the upstream <velocity selector> <magnetic field> $B_s$.}
For a worked example, stipulate $E=2.00\times10^3\,\mathrm{V\,m^{-1}}$, $B_s=1.00\times10^{-3}\,\mathrm T$, $B_a=2.00\times10^{-4}\,\mathrm T$, and $D=0.1136\,\mathrm m$. The selected <speed> and inferred <charge-to-mass ratio> magnitude are
$$
v=\frac{2.00\times10^3}{1.00\times10^{-3}}=2.00\times10^6\,\mathrm{m\,s^{-1}},\qquad
r=0.0568\,\mathrm m,
$$
$$
\frac{e}{m}=\frac{2.00\times10^6}{(2.00\times10^{-4})(0.0568)}
\simeq1.76\times10^{11}\,\mathrm{C\,kg^{-1}}.
$$
The <SI> units provide an independent check: $(\mathrm{V/m})/\mathrm T=\mathrm{m/s}$ and $(\mathrm{m/s})/(\mathrm{T\,m})=\mathrm{C/kg}$, because $\mathrm T=\mathrm{kg/(C\,s)}$. The <charge-to-mass ratio> of the <Electron> is negative; this calculation gives its magnitude. The bending direction gives the sign of the <electric charge> once the <magnetic field> direction is known. Scaling both $e$ and $m$ by the same positive factor preserves every displayed trajectory, so an independent <electric charge> measurement is required to determine <mass> separately.
The curvature formula assumes nonrelativistic <momentum> $p=mv$. At higher <speeds>, the <relativistic cyclotron orbit> gives $p=\gamma mv=eB_a r$, hence $e/m=\gamma v/(B_a r)$, where $\gamma$ is the <Lorentz factor>; the ideal straight-through selection $v=E/B_s$ still follows from vanishing <Lorentz force>. The stipulated <speed> is about $0.0067$ times the <speed of light>, making $\gamma-1$ about $2.2\times10^{-5}$. For this example the nonrelativistic approximation is adequate, but no massive particle can be selected at $E/B_s\geq c$. Finite slit widths, collisions with residual <gas>, uncertain <magnetic field> calibration, and fields outside the ideal boundaries limit a real inference; repeat at several selected <speeds> and analyzer strengths to test consistency of $e/m$ rather than trusting one spot position.
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