Because science educators don't explain the history of physics in terms of problems, it's easy to underestimate how huge the shift in understanding was that Maxwell brought about with his theory of electromagnetism. Here is a short blog post about Maxwell and some of his contributions to physics.
In the 19th century, Newtonian mechanics was widely accepted to be true, and physicists believed the world to consist of masses and forces, which moved around in space and time---that was their 'ontology'. Now, the world of Newton forces was oxymoronic: masses exert forces on each other locally by pushing against one another or non-locally and instantaneously through gravity. The latter force was more fundamental but lacked the critical property of locality. Locality is nice because it solves a problem. Namely, how does a mass know that it needs to move? How does it know that there is another mass in the universe that it needs to respond to? This problem in Newtonian mechanics was solved only after Einstein introduced general relativity. At the same time, physicists were starting to grapple with electricity and magnetism. These phenomena were partially understood, but physicists were unsure what electricity and magnetism WERE. They wanted to know they were made of. Their best guess was that electricity and magnetism emerged out of some kind of all permeating liquid known as the aether. Maxwell was a firm believer in the Newtonian worldview and thought, like his fellow physicists, that electromagnetism had to be explained in terms of an aether. But at some point, Maxwell decided to describe electromagnetism in such a way that the underlying mechanism did not matter. In doing so, Maxwell introduced into fundamental physics the notion of a field: a quantity that assumes, for example, a numerical value at every point in space and time. Fields were puzzling to physicists because they were a new kind of 'object'. It is not divisible into smaller things like a fluid is, nor does it consist of something more fundamental. Fields were real, physical 'objects' in their own right; this took researchers years to grasp. The introduction of fields were a game changer for physics. First of all, Maxwell's fields lacked the oxymoronic property of Newtonian mechanics: electromagnetic forces are transmitted by the electromagnetic field locally, and if a body exerts a force on another body, the electromagnetic field has to transmit this force from one point in space to another. So Maxwell showed that forces could have this nice property of locality if they were transmitted by fields, which he did by accident: his explanation had reach! Einstein's general relativity also has this property as do all other theories of fundamental forces, which, in a sense, they all borrow from Maxwell. Furthermore, Maxwell showed that light is a wave in the electromagnetic field---another example of the reach of his theory. In fact, this is really the same property as locality: light carries the electromagnetic force from one location to another. This concept of a force carrier is another feature that is now universal among the fundamental forces of nature. Not only did Maxwell change our understanding of nature by introducing this concept of fields, electromagnetism also provided new criticism of old theories. For example, Maxwell's theory contradicts Newton's idea of a static space and time because light, according to electromagnetism, has a constant speed for all observers. Naturally, some physicists thought that Maxwell was wrong since Newtonian mechanics were so well established. Einstein did not take this position and instead, thought of Maxwell's equations as being more fundamental than Newtonian mechanics. In other words, he saw that there were new and deeper problems because Maxwell was right! So he set out to solve them and did so with his theory of special relativity. I think that much of the progress in physics in the early 20th century is a result of these fresh problems that Maxwell gave us. Nowadays, all fundamental particles and forces are understood to be (quantum) fields, so we owe in a big way, perhaps even more than we owe Newton, our current understanding of the world to Maxwell.
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