The determination that the velocity of an electromagnetic wave is equal to the speed of light in a vacuum was a result of experimental observations and theoretical investigations conducted by several scientists, most notably James Clerk Maxwell.
In the 19th century, Maxwell formulated a set of equations known as Maxwell's equations, which mathematically described the behavior of electric and magnetic fields and their interplay. These equations predicted the existence of electromagnetic waves and suggested that they would propagate at a certain speed.
One of Maxwell's equations, known as the wave equation, provided a mathematical description of how electromagnetic waves would propagate through space. By solving this equation, Maxwell derived a velocity term that depended on the electric and magnetic properties of the medium through which the wave was traveling.
At the time, it was well-known that light was an electromagnetic phenomenon, and it was hypothesized that light itself was an electromagnetic wave. Maxwell's equations and the wave equation provided a theoretical framework to investigate the properties of light and its relation to electromagnetism.
In the late 19th century, experimental work by scientists such as Albert A. Michelson and Edward W. Morley aimed to measure the motion of the Earth through the hypothetical "luminiferous ether" medium, which was believed to be the medium through which light waves propagated. Their experiments, including the famous Michelson-Morley experiment, sought to detect a change in the speed of light due to the motion of the Earth through the ether.
However, the experiments consistently failed to detect any significant variation in the speed of light, regardless of the direction of the Earth's motion. These results were initially perplexing and could not be explained by the prevailing theories at the time.
The resolution to this puzzle came with the realization that the experimental results were consistent with Maxwell's equations if it was assumed that there is no preferred reference frame in which the speed of light would vary. In other words, the speed of light was considered to be constant for all observers, regardless of their relative motion.
This insight led to the abandonment of the notion of a luminiferous ether and the acceptance that the speed of light in a vacuum, as predicted by Maxwell's equations, was indeed a fundamental constant of nature. This constant velocity was determined to be approximately 299,792,458 meters per second, which is commonly rounded to 3 × 10^8 meters per second and is known as the speed of light in a vacuum.
In summary, the determination that the velocity of an electromagnetic wave is equal to the speed of light in a vacuum arose from the combination of Maxwell's equations, theoretical investigations, and the experimental results of scientists like Michelson and Morley, leading to the revolutionary conclusion that the speed of light is a fundamental constant of nature.