D2 notes: Fundamental charge
Thomson and Millikan
In 1897, whilst investigating the then mysterious cathode rays in discharge tubes, J.J. Thomson essentially discovered the electron, going on shortly after to determine that particle's charge-to-mass ratio, or specific charge. A few years later, in 1909, Robert Millikan, continuing with what might be called “a tale of two experiments” built on Thomson's discovery by determining the fundamental charge.
A summary of the relevant particle physics
The electron is a type of lepton, a fundamental particle, not composed of anything 'smaller', as opposed to hadrons, for example, which are built from quarks. Like the much heavier proton, it has a relative elementary charge, negative in the electron’s case and positive for the proton, though of course the elementary positive charge on the proton is due to the sum of the charges of its constituent quarks.
Thomson's experiment
In Thomson's experiment, conducted in Cambridge University's famous Cavendish laboratory, he applied opposing electrical and magnetic forces on a stream of electrons (accelerated from a hot cathode), finely balancing them so that the particles experience the same magnitude of parabolic deviation towards a positively charged plate, as the circular deviation in the opposite direction due to the centripetal force provided by the magnetic field. With the two fields suitably tuned, and the charged particle beam undeviatingly reaching the detection screen:
\(q_\text{e}vB = q_\text{e}E\)
so,
\(v = {E\over B}\)
where \(q_\text{e}\) is the charge on the electron, \(v\) is its velocity, and \(B\) and \(E\) are the magnetic and electric field strengths.
Equating the kinetic energy of each electron with the energy transferred by the accelerating voltage \(V\) allows the specific charge to be determined as follows:
\({1\over 2} m_\text{e}v^2 = {1\over 2} m_\text{e}^2 ({E \over B})^2 = q_\text{e}V\)
where \( m_\text{e}\) is the electron unit mass. And so:
\({q_\text{e}\over m_\text{e}} = {E^2\over2VB^2}\)
It's worth noting that Thomson, who postulated the so-called plum-pudding model of an atom, whilst investigating the positively charged particles that we now know to be protons, assumed that they carry the same magnitude of charge as electrons. Noting that the charge-to-mass ratio was so much less for the positive ions, he predicted the mass of the proton to be 1836.15 times that of the electron: a figure which is remarkably close to our current value!
Millikan's experiment
It was through his famous “oil-drop” experiment that Robert Millikan was finally able to determine the absolute charge of the electron. This procedure involved allowing tiny droplets of oil to fall through an electric field created by the potential difference between two circular plates. Unlike the low-pressure gas filling the fine beam tube as used by J.J. Thomson, Millikan's plates were separated by an air-filled space, so the mist of oil droplets upon entering the apparatus through a hole in the upper plate was essentially descending under gravity through a fluid. This meant that by measuring by observation the terminal velocity of the falling drops, resulting from viscous drag force (opposing gravity) without the electric field enabled, Millikan could use Stokes’ law (\(F = 6πηrv\)) to determine the average radius \(r\) of the drops, where \(η\) is the coefficient of viscosity of the air.
Now, by switching on the electric field and finely adjusting the potential difference \(V\) across the gap of height \(d\), any oil droplets which were negatively charged (by virtue of friction, x-ray ionisation or radioactive decay), could be made to hover, at a standstill, so to speak. Since they are now stationary, viscous drag can be left out of the equation, the two pertinent balanced forces being the downwards weight of the drops versus the upwards force due to the electric field. Using the following formula (which you can derive by equating weight and electric force for uniform fields):
\(Q = {{4\over 3}πr^3ρgd\over V}\)
where \(ρ\) is the density of the oil. After many measurements, the total charge \(Q\), on a drop, always turned out to be some integer multiple of -1.6 ×10-19 C.
As a caveat to the above, if investigating further, you will probably discover an abundance of subtle variations on this experiment (and accompanying equations) in different sources; some texts even incorporating buoyancy forces on the oil drops as they ascend or descend; however they all basically amount to the same result.
A special number?
To finish, some of you may have had occasion to wonder 'why' it happens that 1.6 × 10-19 C should be the fundamental magnitude of charge for both protons and electrons. How is it that a bunch of quarks should 'know' to arrange themselves in a baryon to have exactly the same charge as a lepton? Well, if you have ever reflected on this particular phenomenon you're not alone - and a journey into deeper, more profound particle physics awaits!
Images:
- J.J. Thomson (1856-1940) y R. Millikan (1868-1953) by Jorge Meinguer
- Fundamental particles from https://pages.uoregon.edu/jschombe/cosmo/lectures/lec07.html
- Cathode ray tube 2 by Sharon Bewick, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons
- Simplified scheme of Millikan’s oil-drop experiment by Theresa Knott, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/>, via Wikimedia Commons