Investigate, assess and model the experimental evidence supporting the existence and properties of the electron, including cathode ray tube experiments and Thomson's determination of the charge-to-mass ratio of the electron

Thomson accelerated cathode rays through a potential difference and passed them between two parallel plates in a region with both an electric field $E$ (between the plates) and a magnetic field $B$ (perpendicular to $E$ and to the beam direction).

Step 1: balance the two forces so that the beam goes through undeflected. The electric force on an electron is $qE$; the magnetic force is $qvB$. Setting them equal gives the beam velocity:

$v = E/B$.

This velocity selector lets him measure $v$ from the field strengths alone.

Step 2: switch off the magnetic field. The beam is now deflected by $E$ only. The vertical deflection $y$ over a horizontal length $L$ inside the field, with no force after, gives:

$y = \frac{1}{2} a t^2 = \frac{1}{2} \frac{qE}{m} \left( \frac{L}{v} \right)^2$.

Solving for the charge-to-mass ratio:

$\frac{q}{m} = \frac{2 y v^2}{E L^2} = \frac{2 y E}{B^2 L^2}$.

Thomson obtained $q/m \approx 1.76 \times 10^{11}$ C/kg, far larger than the corresponding value for hydrogen ions. He concluded cathode rays are charged particles much lighter than the lightest atom.

Markers reward both fields and their roles, the velocity selector, the deflection equation, and the conclusion (small mass relative to atoms).

Three observations supporting particle nature:

1. Cathode rays are deflected by electric fields, in a direction consistent with them carrying negative charge. Electromagnetic waves are uncharged and would not deflect.

2. Cathode rays are deflected by magnetic fields. They follow a curved path through the field, as expected for moving charged particles experiencing $qv \times B$.

3. A small object placed in the beam casts a sharp shadow, consistent with particles travelling in straight lines from the cathode. Cathode rays also turn a small paddle wheel placed in their path, transferring momentum as particles do.

Observation against electromagnetic waves: the rays are deflected by electric fields. Electromagnetic waves carry no net charge and are not deflected by static electric fields. (Equivalently, the $q/m$ measurement made cathode rays definite particles.)

Markers reward three particle-nature observations and one wave-incompatible observation, with clear reasoning.

Two observations. (1) Cathode rays are deflected towards the positive plate in an electric field, the direction expected for negative charge. (2) In a magnetic field, cathode rays curve in the direction given by $\vec{F} = q\vec{v} \times \vec{B}$ for a negative charge (opposite to the direction a positive charge would curve).

Unit. $q/m$ is measured in coulombs per kilogram, $\text{C}\,\text{kg}^{-1}$.

Marks: one for any two correct deflection-based observations naming the negative-charge direction, one for the correct unit $\text{C}\,\text{kg}^{-1}$.

At balance the electric and magnetic forces are equal: $qE = qvB$, so $v = \dfrac{E}{B}$.

$v = \dfrac{1.8 \times 10^4}{6.0 \times 10^{-4}} = 3.0 \times 10^7\ \text{m/s}$.

The charge $q$ cancels from both sides of $qE = qvB$, so the selected speed depends only on the field strengths $E$ and $B$, not on the particle's charge or mass. Any particle (of any charge or mass) at this exact speed passes through undeflected.

Marks: one for the force-balance equation $qE = qvB$, one for $v = 3.0 \times 10^7\ \text{m/s}$ with the unit, one for explaining that $q$ cancels so $v = E/B$ is independent of charge and mass.

The model. Thomson proposed that the atom is a diffuse sphere of positive charge with negatively charged electrons embedded inside it, like plums in a pudding.

Correct feature. It explained the overall electrical neutrality of the atom (the negative electrons cancel the positive charge spread through the sphere).

Marks: one for a correct description of the model (diffuse positive sphere, embedded electrons), one for a correct explained feature (neutrality, or the small mass of the electron relative to the atom).

(a) Acceleration. $a = \dfrac{eE}{m} = \left(\dfrac{e}{m}\right)E = (1.759 \times 10^{11})(1.2 \times 10^4) = 2.11 \times 10^{15}\ \text{m/s}^2$.

(b) Deflection. Time in the field: $t = \dfrac{L}{v} = \dfrac{0.045}{2.5 \times 10^7} = 1.80 \times 10^{-9}\ \text{s}$.

$y = \tfrac{1}{2}at^2 = 0.5 \times (2.11 \times 10^{15}) \times (1.80 \times 10^{-9})^2 = 3.4 \times 10^{-3}\ \text{m} = 3.4\ \text{mm}$.

Marks: one for $a = (e/m)E$ with substitution, one for $a = 2.11 \times 10^{15}\ \text{m/s}^2$, one for $t = L/v$ and $y = \tfrac{1}{2}at^2$ set up, one for $y = 3.4\ \text{mm}$ with correct unit.

(a) The graph is a straight line through the origin, so $y$ is directly proportional to $E$ ($y \propto E$), consistent with $y = \dfrac{qL^2}{2mv^2}E$ at fixed $L$ and $v$.

(b) Gradient $= \dfrac{\Delta y}{\Delta E} = \dfrac{(14.1 - 3.5) \times 10^{-3}\ \text{m}}{(4.0 - 1.0) \times 10^{4}\ \text{V/m}} = \dfrac{1.06 \times 10^{-2}}{3.0 \times 10^{4}} = 3.53 \times 10^{-7}\ \text{m per V/m}$.

(c) The gradient equals $\dfrac{qL^2}{2mv^2}$, so $\dfrac{q}{m} = \dfrac{2v^2 \times \text{gradient}}{L^2} = \dfrac{2(2.0 \times 10^7)^2 (3.53 \times 10^{-7})}{(0.040)^2} = 1.77 \times 10^{11}\ \text{C/kg}$.

This is within about $1\%$ of the accepted value $1.759 \times 10^{11}\ \text{C/kg}$, well within the precision expected of a field-and-ruler measurement.

Marks: one for identifying direct proportionality (line through the origin, $y \propto E$), one for a correct gradient with unit ($\text{m per V/m}$), one for equating the gradient to $qL^2/(2mv^2)$ and rearranging for $q/m$, one for $q/m \approx 1.77 \times 10^{11}\ \text{C/kg}$ with a sound comparison to the accepted value.

Band-6 plan. (1) State what the crossed-field method measures and how (velocity selector, then deflection). (2) State the numerical result and what made it significant (universal, ~1800x larger than for hydrogen ions). (3) Explain the two interpretations (large charge vs small mass) and how the same-ratio-for-every-metal result supported "new particle" over "special ion". (4) Link directly to why a diffuse positive sphere with embedded electrons was the simplest model consistent with neutrality and a light, universal negative constituent.

Model answer. Thomson first balanced the electric force $qE$ against the magnetic force $qvB$ on the beam so that it passed through undeflected, giving the beam speed $v = E/B$ independent of the particle's charge or mass. He then switched off the magnetic field and measured the deflection $y$ produced by $E$ alone over the plate length $L$, giving $q/m = 2yE/(B^2L^2)$. This procedure measures a ratio without needing to know $q$ or $m$ separately, which is what makes it possible without first knowing what the particle is.

The measured value, $q/m \approx 1.76 \times 10^{11}\ \text{C/kg}$, was the same regardless of the cathode metal used (aluminium, platinum, iron), which was strong evidence that cathode rays are a single, universal constituent of matter rather than metal-specific ions. Comparing this ratio with the already-known $q/m$ for the hydrogen ion showed it was roughly 1800 times larger. Since later experiments (Millikan) showed the charge was comparable in magnitude to the elementary charge on an ion, the huge ratio meant the particle's mass, not its charge, was the unusual quantity: cathode-ray particles (electrons) are far lighter than any atom.

This directly motivated the plum-pudding model. If atoms are neutral overall but contain very light, negatively charged electrons as a universal constituent, the remaining positive charge and most of the mass had to be accounted for by something else, and with no evidence yet of a concentrated nucleus, the simplest consistent picture was a diffuse sphere of positive charge (carrying the mass and positive charge) with the light electrons embedded inside it, so the atom's total charge cancelled. The model was a direct, minimal extension of what the crossed-field result had actually shown, not a separate speculative leap.

Marker's note: the top band explains the physics of the crossed-field method (why the velocity selector works, why deflection then gives $q/m$), correctly interprets the large ratio as "small mass" rather than "large charge" (using the universality across metals as the key evidence), and shows the logical chain from that conclusion to the plum-pudding model rather than stating the model in isolation.

Band-6 plan. Thesis: the combined evidence decisively resolved the particle-vs-wave debate but left the internal structure of the atom only partly resolved. Structure: (1) summarise the wave-vs-particle debate and the key deciding evidence (electric-field deflection). (2) explain what the crossed-field $q/m$ measurement added beyond simple deflection (a precise, reproducible, universal quantity). (3) evaluate the limits: no measurement of charge or mass separately, no model of the positive charge, no explanation of atomic stability. (4) reach a judgement.

Model answer. Before 1897 there were two camps: continental physicists who treated cathode rays as a wave phenomenon in the aether, and British physicists who argued for charged particles. The decisive discriminating evidence was deflection by a static electric field. An electromagnetic wave carries no net charge and is not deflected by a static $\vec{E}$, whereas cathode rays deflected consistently towards the positive plate. Combined with magnetic deflection, straight-line shadows and the turning of a paddle wheel (showing momentum transfer), this evidence made the particle interpretation essentially unavoidable, resolving the debate.

Thomson's crossed-field measurement went further than qualitative deflection: by balancing $qE$ against $qvB$ to select a speed, then measuring the deflection due to $E$ alone, he obtained a precise, reproducible number, $q/m \approx 1.76 \times 10^{11}\ \text{C/kg}$, independent of the assumed values of $q$ and $m$ individually. Crucially, this ratio was the same for every cathode material tested, which is strong evidence that cathode rays are a single universal particle (the electron) rather than an ionised form of whatever gas or metal was used, a claim that simple deflection observations alone could not establish.

However, the resolution was incomplete. The crossed-field method measures only the ratio $q/m$; Thomson could not separate charge from mass without an independent measurement, which had to wait for Millikan's oil-drop experiment. Nor did the experiment say anything about the positive charge needed to keep atoms neutral: Thomson's plum-pudding model, a diffuse positive sphere with embedded electrons, was a reasonable but essentially unconstrained guess, since no experiment had yet probed how the positive charge was distributed. It also gave no account of atomic stability or of what would later turn out to be a concentrated nucleus, gaps only closed by Geiger, Marsden and Rutherford in 1909-1911.

On balance, cathode ray tube experiments and Thomson's measurement fully resolved the particle-versus-wave question and established the electron as a real, universal, very light charged particle, but they left the arrangement of positive charge in the atom essentially unknown, a limitation the plum-pudding model exposed rather than solved.

Marker's note: the top band separates what was resolved (particle nature, universality, the $q/m$ value) from what was not (separate $q$ and $m$, the structure of the positive charge, atomic stability), and reaches an explicit, weighed judgement rather than simply narrating the experiments. Naming Millikan and Rutherford/Geiger-Marsden as the experiments that closed the remaining gaps is rewarded.