Could anybody simply explain how exactly Joseph Thomson discovered that atoms were not "indivisible" and that they were actually made of subatomic particles? (electrons)

I've been having trouble understanding how exactly it happened, and everything on the internet seems to make is so complicated, and I was wondering if anybody could just give me a simply explanation. Thanks!

Here is a web site that explains it rather well. But I can summarize a few things. I think you are simply overwhelmed with jargon and some misinformation. First, J.J. Thompson didn't discover electrons. Other did that. Even the name, electrons, was suggested by others. Thompson, in working with cathode rays, picked up that name from work by Stoney. Thompson also did not do experiments that PROVED that atoms were made of electrons. That atoms were composed of electrons in some manner (Thompson didn't say how at first) was speculation by Thompson and it was rejected by many people at the time. Thompson advanced a theory that electrons were a part of atoms and he persisted. Thompson, in 1897, measured the ratio of the electronic charge to the electronic mass (that famous e/m value). Another physicist had determined e/m also but Thompson is the one who gets most of the credit. Thompson repeated the work but with more planning and better measuring techniques. The value of 1.7588 x 10^8 (a huge number) led him to the conclusion that either the electronic charge was EXTREMELY large OR that the mass was EXTREMELY small. Those are the only two ways to arrive at such a large number. That was followed few years later with work that showed the atom was composed, also, of positively charged particles (Rutherford's work) and Thompson advanced his theory of the structure of the atom, his so-called "plum pudding" or "raisin cake" model. He envisioned atoms as being composed of a large mass of positive electricity (the protons) with the electrons being embedded (like raisins) throughout. Neutrons had not been discovered at that time. Thompsons plum pudding model stood the test of time until Rutherford's work in the latter part of the first decade of the 1900s showed that the atoms consisted mostly of space. That was followed by the discovery of the neutron, the explanation of spectra provided by Rydberg and the leap of faith made by Niels Bohr concerning orbits for the electron with selected energy levels. (I forgot to mention that Robert Millikan measured the charge on the electron in 1909, and that together with e/m by Thompson, allowed researchers to calculate the mass of the electron. Indeed, that was a small number.)

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