Specific charge of the electron (A-level only)

AQA·Physics·3.12.1.3

Determining the electron’s specific charge

An electron beam can be used to determine the magnitude of the electron’s specific charge, $$e/m_e$$: the magnitude of its charge divided by its mass. Electrons are accelerated through a known potential difference and then travel in a circular path through a magnetic field. The radius of that path gives information about their charge-to-mass ratio.

Here, $$e$$ is the positive magnitude of the electron’s charge, although the charge of an electron is negative.

Accelerating the electrons

An electron accelerated through a potential difference $$V$$ gains kinetic energy. If its initial kinetic energy is negligible, its gain in energy is equal to its final kinetic energy:

eV=12mev2eV = \frac{1}{2}m_e v^2

Here, $$m_e$$ is the electron’s mass and $$v$$ is its speed after acceleration.

The electron’s circular path

The beam enters a uniform magnetic field with its velocity perpendicular to the field. The magnetic force acts perpendicular to the electrons’ motion, changing their direction without changing their speed. It provides the centripetal force that keeps them moving in a circle.

For a field of magnetic flux density $$B$$ and a circular path of radius $$r$$, equating the magnetic force to the centripetal force gives:

evB=mev2revB = \frac{m_e v^2}{r}

Rearranging this equation gives $$v = eBr/m_e$$. Substituting for $$v$$ in the energy equation and rearranging gives:

eme=2VB2r2\frac{e}{m_e} = \frac{2V}{B^2r^2}

Measuring the specific charge

In a practical arrangement, coils produce the magnetic field and the electron beam passes through a low-pressure gas, which makes its path visible. The accelerating potential difference, magnetic flux density and radius of the circular path are measured.

Substituting these measurements into the equation gives the magnitude of the electron’s specific charge in $$\mathrm{C,kg^{-1}}$$. The electron’s charge-to-mass ratio is negative because its charge is negative.

Significance of Thomson’s electron specific-charge measurement

Thomson measured the specific charge, or charge-to-mass ratio, of particles in cathode rays. His measurement provided evidence that these particles were negatively charged constituents of atoms, now called electrons. If atoms contain smaller particles, they are divisible rather than indivisible.

Comparing the electron with a hydrogen ion

The magnitude of the electron’s specific charge is approximately 1,800 times that of a hydrogen ion. The two charges have opposite signs: the electron is negative and the hydrogen ion is positive.

A charge-to-mass ratio does not establish either charge or mass separately: a large ratio could result from a larger charge magnitude, a smaller mass, or both. If the electron and hydrogen ion have charges of equal magnitude, however, the difference in their ratios must come from their masses. The electron’s mass is then approximately $$1/1800$$ of the hydrogen ion’s mass.