= Proton exclusion by magnetic rigidity and range
{title2=$K=\sqrt{p^2c^2+m^2c^4}-mc^2$}
For a concrete independent example, stipulate perpendicular motion, $|q|=e$ with $e$ the <elementary charge>, uniform <magnetic field> $B=1.00\,\mathrm T$, incoming <gyroradius> $r_{\rm in}=0.100\,\mathrm m$ and outgoing <gyroradius> $r_{\rm out}=0.0500\,\mathrm m$. These are invented teaching inputs. Converting <SI> <momentum> to <electronvolt> units gives
$$
p[\mathrm{MeV}/c]=299.792458\,B[\mathrm T]r[\mathrm m]
\quad (|q|=e),
$$
so $p_{\rm in}=29.9792458\,\mathrm{MeV}/c$ and $p_{\rm out}=14.9896229\,\mathrm{MeV}/c$. Both a <Positron> and a <proton> with these <momenta> trace the same ideal arcs. Their <kinetic energies> differ because the <relativistic energy-momentum relation> gives
$$
K=\sqrt{(pc)^2+(mc^2)^2}-mc^2
=\frac{(pc)^2}{\sqrt{(pc)^2+(mc^2)^2}+mc^2}.
$$
The second expression avoids subtracting nearly equal quantities when $pc\ll mc^2$. For this calculation stipulate rounded <rest energies> $m_ec^2=0.511\,\mathrm{MeV}$ and $m_pc^2=938.272\,\mathrm{MeV}$. The <Positron> interpretation gives $K_{\rm in}\simeq29.473\,\mathrm{MeV}$ and $K_{\rm out}\simeq14.487\,\mathrm{MeV}$; the <proton> interpretation gives only $0.479\,\mathrm{MeV}$ and $0.120\,\mathrm{MeV}$. The corresponding proposed losses of <kinetic energy> are about $14.985\,\mathrm{MeV}$ and $0.359\,\mathrm{MeV}$.
\Image[/positron-proton-rigidity-energy.png]
{title=Equal <magnetic rigidity> permits very different <kinetic energies> for a <Positron> and a <proton>}
{description=Original calculation for the stipulated $1\,\mathrm T$ <magnetic field>, rounded <rest energies>, and radii used in the text. Lines use the exact <relativistic energy-momentum relation>; the <kinetic energy> axis is logarithmic. This is an energy comparison, not a measured spectrum or a historical <particle range> curve.}
The small <proton> <kinetic energy> follows from $K\simeq p^2/(2m_p)$, whereas here the <Positron> has $pc\gg m_ec^2$ and $K\simeq pc-m_ec^2$. Consequently magnetic bending alone cannot reject the <proton>. One needs an independent material constraint: could a <proton> at the proposed <kinetic energy> penetrate the plate and then travel at least the observed visible distance in the specified gas? A <particle range> shorter than that visible distance contradicts the hypothesis. The visible track need not include the stopping point: it already supplies a lower bound on <particle range>. No numerical <particle range> is assigned to this teaching example; that would require stopping data for the actual material and gas conditions. The experimental insight is to combine a geometric <momentum> measurement with an independent test of penetration.
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