Q.Explain the J.J. Thomson experiment to determine the specific charge of an electron.

Step 1. Apparatus. A highly evacuated discharge tube produces a narrow cathode-ray beam (selected by a pinhole in the anode disc A), which passes between a pair of parallel deflecting plates (electric field $E$) while the whole tube sits between the poles of a magnet (magnetic field $B$, perpendicular to $E$ and to the beam). The beam finally strikes a zinc-sulphide-coated fluorescent screen, producing a visible scintillation spot.

Step 2. Velocity selection. With both fields switched on, their strengths are tuned until the beam lands at its original, undeflected position O - meaning the electric force and magnetic force on the electron exactly balance: $eE=ev B$, giving the beam's velocity directly as $v=E/B$, independent of the electron's mass or charge individually.

Step 3. Specific charge from accelerating energy. The beam is accelerated from cathode to anode through a known potential difference $V$, so its potential energy converts entirely to kinetic energy: $eV=\tfrac12mv^2$, giving $\dfrac{e}{m}=\dfrac{v^2}{2V}=\dfrac{1}{2V}\left(\dfrac{E}{B}\right)^2$.

Step 4. Cross-checks. Two further, independent measurements corroborate this value: (i) with the magnetic field off, the electric-field-only deflection $y$ on the screen (over plate length $l$) gives $\dfrac{e}{m}=\dfrac{yE}{Cl^2B^2}$ for a geometric constant $C$;

(ii) with the electric field off, the magnetic-field-only circular path of radius $R$ gives $\dfrac{e}{m}=\dfrac{E}{B^2R}$.

Step 5. Result. All three methods consistently give $e/m\approx1.7\times10^{11}\ \text{C kg}^{-1}$, and this value was found to be independent of both the gas filling the tube and the metal used for the electrodes - strong evidence that the electron is a universal constituent of all matter.

Balance electric and magnetic forces to find the beam velocity, then use the accelerating potential (and cross-check with pure-electric and pure-magnetic deflection) to isolate $e/m$.

• Treating the velocity-selector balance condition ($v=E/B$) as itself giving $e/m$ - it only gives the velocity; a second, separate measurement is needed for $e/m$.
• Forgetting that all three sub-methods must agree, which is what makes the result convincing.