Physics in the early 20th century is fascinating because by this point physicists had pushed classical mechanics right to the limit of where classical physics could go and were starting to grapple with the contradictions that popped up when they went any further.
One of the big problems at the time was the problem of the self-force of electric fields and the structure of the electron. In classical mechanics we approximate an electron as a point charge --- an infinitesimally small source of electric charge. This works for a lot of practical problems, but by the 20th century it was known that this couldn't be the real story because if you integrate up the energy in the electric field around a point source you find that it's infinite! And with Einstein's discovery of relativity, that meant that it would be infinitely massive.
To avoid this infinity physicists generally assumed that the electron had to have some finite radius. (Estimates of this size are how we get the notion of the "classical radius of the electron.") But now there was a new problem. If you assume that the electron is a little sphere packed with charge, what keeps it together? The self-repulsion from the electric field will try to make it explode apart. That means there has to be a new force acting like rubber bands to hold the thing together.
To make matters worse, if you accelerate an electron, the electromagnetic waves generated by one part of the electron will act on the other part. This will introduce new forces. Moreover, when the electron radiates an electromagnetic wave, that wave will in turn impart a force to the electron, resulting in a "self-force." Calculating these self-forces and internal stresses occupied many of the brightest physicists of the early 1900s, including Poincaré. But the results they got were paradoxical. They imply that due to its self-force an electron can spontaneously accelerate exponentially --- or, even worse, act with retrocausality and start moving before you even touch it!
In the end these problems were "resolved" because the classical picture doesn't hold on these small length scales --- you need a quantum mechanical description. But interestingly, the fundamental problems that those physicists were grappling with didn't exactly disappear in the quantum picture, they just got transmuted into new problems in quantum electrodynamics. The problem of the diverging energy of an electric field around a point source has a mirror in the issue around renormalization in QFT.
One of the big problems at the time was the problem of the self-force of electric fields and the structure of the electron. In classical mechanics we approximate an electron as a point charge --- an infinitesimally small source of electric charge. This works for a lot of practical problems, but by the 20th century it was known that this couldn't be the real story because if you integrate up the energy in the electric field around a point source you find that it's infinite! And with Einstein's discovery of relativity, that meant that it would be infinitely massive.
To avoid this infinity physicists generally assumed that the electron had to have some finite radius. (Estimates of this size are how we get the notion of the "classical radius of the electron.") But now there was a new problem. If you assume that the electron is a little sphere packed with charge, what keeps it together? The self-repulsion from the electric field will try to make it explode apart. That means there has to be a new force acting like rubber bands to hold the thing together.
To make matters worse, if you accelerate an electron, the electromagnetic waves generated by one part of the electron will act on the other part. This will introduce new forces. Moreover, when the electron radiates an electromagnetic wave, that wave will in turn impart a force to the electron, resulting in a "self-force." Calculating these self-forces and internal stresses occupied many of the brightest physicists of the early 1900s, including Poincaré. But the results they got were paradoxical. They imply that due to its self-force an electron can spontaneously accelerate exponentially --- or, even worse, act with retrocausality and start moving before you even touch it!
In the end these problems were "resolved" because the classical picture doesn't hold on these small length scales --- you need a quantum mechanical description. But interestingly, the fundamental problems that those physicists were grappling with didn't exactly disappear in the quantum picture, they just got transmuted into new problems in quantum electrodynamics. The problem of the diverging energy of an electric field around a point source has a mirror in the issue around renormalization in QFT.