3.12.1.3 - Specific Charge of the Electron

Meaning of specific charge

The specific charge of a particle is its charge divided by its mass, with SI unit $\mathrm{C\,kg^{-1}}$. The signed ratio for an electron is negative; in electron-beam calculations $e/m_e$ conventionally denotes its positive magnitude.

A large specific charge means a large electric acceleration in a given field, because $a=(Q/m)E$. It does not, by itself, tell us whether the charge is large, the mass is small, or both. An independent charge measurement or a justified equal-charge comparison is needed to separate these possibilities.

Measuring specific charge with a fine-beam tube

A fine-beam tube contains an electron gun and a trace of low-pressure gas. A pair of coils produces a magnetic field approximately uniform across the beam path. The coil axes must be perpendicular to the plane of the circle. Electron collisions excite the gas and reveal the path; the gas pressure is low enough for energy loss over the measured path to be small.

The gun accelerates electrons through a measured potential difference $V$. The beam enters the magnetic field at right angles to it. Magnetic force is perpendicular to velocity, so it changes direction without doing work: the speed remains approximately constant and the beam follows a circle.

[DIAGRAM: asset_name: D.1.3 - Specific Charge of the Electron - Diagram 1; asset_slug: D.1.3 - Specific Charge of the Electron - Diagram 1; caption: A calculated electron orbit with perpendicular magnetic force; description: Plan view of trajectory in a uniform field into the page. Electron velocity is tangent at the top of the circle, pointing right; negative charge makes force downward there. Coils and supply wiring are omitted. Radius is calculated from V=200 V and B=1.00 mT.]

Deriving the measured ratio

With negligible initial kinetic energy, $\frac12m_ev^2=eV$. The magnetic force supplies the centripetal force:

Writing $k=e/m_e$ makes elimination clear: $v^2=2kV$ and $v^2=k^2B^2r^2$. Equate them and cancel the nonzero $k$:

Measure $V$ with a voltmeter across the accelerating electrodes, $B$ with a calibrated field probe or from a calibrated coil-current relationship, and the path diameter with the tube scale; halve the diameter to obtain $r$. We are determining the ratio experimentally, so do not insert accepted $e$ and $m_e$ in place of the measured $V$, $B$ and $r$.

For $V=200\,\mathrm{V}$, $B=1.00\,\mathrm{mT}$ and $r=47.7\,\mathrm{mm}$,

This gives $e/m_e=1.76\times10^{11}\,\mathrm{C\,kg^{-1}}$.

Checking a measurement

A finite beam width limits where its centre can be read. Measure a full diameter to reduce the fractional effect of a fixed reading uncertainty; view an external scale normally to reduce parallax if the scale and beam are in different planes. Repeating readings can reveal random spread, but does not remove a calibration or alignment error. Keep the beam perpendicular to the field: a parallel velocity component produces a helix rather than the intended circle.