A HUNDRED AND EIGHTY SIX MILES A SECOND

Before Maxwell's electromagnetic field equations and Einstein's theory of relativity, it was not known with absolute certainty whether light had a finite speed or if it travelled instantaneously. Maxwell's equations united the electric and magnetic phenomena and hinted at the presence of a so-called ❝electromagnetic❞ field across the alleged all-pervading luminiferous ❝ether❞  through which the electric- and magnetic-field vibrations travelled at a hundred and eighty-six miles a second. 

This quantity, this c, is one of the most important parameters of our universe. Speaking along the lines of the Ancient One who almost blew the sanity out of Stephen Strange, it is part of the source code that shapes reality. So to understand the universe, we have to understand light, a seemingly inconceivable task that demands more than a single lifetime. But since we have only one lifetime, let us focus on whatever we can learn in these fleeting moments. Three questions matter the most; what is space, what is time and what is light. To avoid ending up with the age-old debauchery, we are going to stay far away from explicitly defining the individual terms and instead focus on some of the wonderful experiments, novel ideas, inconclusive reports, off-the-mark estimates and how it took more than two-thousand years of various efforts before we could pin down one of the most beautiful string of integers one can ever imagine. 

The speed of light in the SI system of units. Light moves precisely at 299,792,458 m/s, or, approximately 300,000 km/s. In the CGS system of units, the value is approximately 186,000 miles/s. 
Image Credit: Author @Nature&Science

In our previous article Not as it is but as it was we have seen that Galileo made the first attempt to measure the speed of light by recording the time delay on a two-way path over a known distance. Unfortunately, light moves so awfully fast that the time delay in Galileo's experiment would have been about 11 microseconds. The pendulum clocks of Galileo's time could, in no way, measure such a small interval. In his Discourses and Mathematical Demonstrations Relating to Two New Sciences, he cleverly concluded that light travels incredibly faster than sound. The reason why he favoured the finite speed of light can be understood from his Discourses. Galileo argued that during a lightning flash in the clouds eight to ten miles distant, the head or source of the lightning starts at some place within the clouds and gradually spreads towards the surrounding regions. Had the speed of light been instantaneous/infinite, there would have been a sudden uniform illumination with no distinct head or centre and no noticeable spread of light into the surroundings. 

Galileo's null result did not cast away the possibility of light travelling instantaneously. But for centuries, the instantaneous propagation of light across finite distances has raised some serious logical inconsistencies. There were two schools of thought. Empedocles considered light to be a thing in motion and hence believed that it must travel with a measurably finite speed. The Aristotelians, as always, preferred to rhyme with the contrary. They did not suppose light to be a thing in motion but a disturbance, a sort of transparency in a transparent medium like air or water. The Greeks were not that much concerned with having a theory of light or determining its speed. They were more interested in understanding human/animal vision. When one school of thought supported the finite speed of light, the other argued for an infinite/instantaneous speed. This picture remained more or less unchanged until the modern age. 

Roemer's calculation of the speed of light was based on the orbital geometries of the Sun (A), the Earth at different positions (E, G, G, H, L, K) as it revolves around the Sun, Jupiter (B), and Io as eclipsed in the shadow of Jupiter (DC). The portion of Earth's orbit from L to K denotes the planet receding away from Jupiter, and the portion from F to G denotes Earth approaching Jupiter.
Image Credits: Wikimedia Commons

Towards the end of the 17th century, one of the important problems in astronomy and cartography was to figure out an efficient method of determining the longitude of a particular geographical location, specifically at sea, since the standard method used on land were impractical at sea. This problem was to be solved by precisely timing the eclipse of one of Jupiter's four moons, Io, from a place of known longitude and that place whose longitude was to be determined. In the years following 1671, a group of astronomers, Ole Christensen Roemer, Giovanni Cassini and Jean Picard, tasked themselves with providing accurate astronomical tables of the eclipse of Io by Jupiter. Roemer and Picard, accompanied by Erasmus Bartholin, stationed at the island of Hveen, made simultaneous observations with Cassini in Paris. After initial observations, the group discovered an anomaly in the time period of Io. Roemer observed that Io's mean time period, when calculated from the satellite's emersions (when it came out of Jupiter's shadow), was always greater than the same calculated from the satellite's immersions (when it slid into Jupiter's shadow). He also observed that the eclipse times differed significantly depending on the relative positions of Earth and Jupiter (as shown in the above figure). From the ephemeris of Io, Roemer understood that the eclipse would be delayed by some amount of time when Earth is at its farthest (denoted by E in the above figure) from Jupiter. In 1676, relying upon his calculations, Roemer predicted that the upcoming eclipse on the 9th of November would occur almost 10 minutes late from the actual time estimated from previous observations. He reasoned that the delay comes from the simple fact that light travels with a finite speed and is not instantaneous. He also estimated that the speed of light must be of such magnitude to cover the diameter of Earth's annual orbit in about 22 minutes. However, Roemer himself did not calculate the exact speed of light. Based on Roemer's original data, Christiaan Huygens, with some refinements, arrived at an approximate value of 220,000 km/s for the speed of light. When Newton performed his own calculations, he found that light from the Sun must take about seven to eight minutes to get to Earth, where the actual value is about 8½ minutes on average. 

The 17th-century astronomical tables and ephemerides were not as accurate as we have today. With the best estimates available in their days, Roemer, Cassini, Huygens, Halley, Newton and Flamsteed, among others, did a great job determining the speed of light, which is about 27% less than the actual value. In 1729, when James Bradley discovered stellar aberration assuming a finite speed of light, he calculated that light would take about 8 minutes and 12 seconds to travel from the Sun to the Earth, which misses the actual value by a hair's breadth. Stellar aberration is the apparent motion of celestial objects about their true positions due to the motion of the observer. This stellar aberration causes stars to appear displaced from their true position (when the observer is stationary) by a small angle, which is of the order of v/c, v being the velocity of the observer (in this case, Earth as it revolves around the Sun), and c is the (finite) speed of light. Stellar aberration has its own historical significance. Stellar aberration was discovered in connection to stellar parallax, which is the apparent change in position of a relatively nearby star against the distant (unmoving) background stars, occurring due to the motion of the Earth around the Sun. The parallax shift of a nearby star meant that the Earth must be moving around the Sun and the determination of this parallax was the only way of validating the Copernican Heliocentric Model. Even though stellar parallax and stellar aberration were already known for a time, the credit goes to James Bradley, who saw them as two distinct phenomena and provided a thorough explanation on the basis of the Earth's motion around the Sun and the finite speed of light. 

Fizeau's Toothed-Wheel Experiment
Image Credit: Author @Nature&Science

Armand Hippolyte Louis Fizeau, in 1849, developed a novel method for determining the speed of light. Commonly known as the toothed-wheel experiment, Fizeau obtained a value of 315,000 km/s for the speed of light, which was only 5% more than the actual value. Fizeau's apparatus involved an intense source of light and a highly polished mirror placed 8 km away from the source, and in between, there was a toothed cogwheel with 720 notches. The cogwheel was such that it could rotate for up to hundreds of times a second. When the cogwheel was stationary, the incident and the reflected beam could pass through a notch and reach the eyepiece upon reflection. When the cogwheel was rotated, the path of the incident beam would be obstructed by the teeth, and the observer will see no light at the eyepiece. Now, if the cogwheel is rotated in such a way that by the time the reflected beam returns to the wheel, the adjacent tooth would move into the position of the notch and block the reflected beam. Fizeau wanted to determine the rotation speed necessary to block off the returning beam. Had light travelled infinitely fast, then the observer would see the reflected beam whenever there was an opening in front of the source and no light when there was a tooth, irrespective of the rotation of the cogwheel and at any speed. If light had an infinite speed then the cogwheel had to be rotated infinitely fast to block off the reflected beam. But if light had a finite speed, then during the time it takes for the reflected beam to return to the cogwheel, the wheel can be given a rotation rate so as to block the initial notch by the adjacent tooth in that same interval of time. Using these data, i.e., the rotation speed of the cogwheel, the distance between the source and the mirror and the width of the cogs, Fizeau calculated the speed of light to be 315,000 km/s. 

Foucault's Rotating Mirror Apparatus 
Image Credits: via Wikimedia Commons

In 1862, Fizeau's good friend, Leon Foucault, independently performed the same experiment by replacing the cogwheel with a rotating mirror. The Foucault apparatus comprised of a beam of light incident on a mirror R (figure above) which forms the image at M and is then reflected back through the same path. R is stationary at first. When R is rotated through 𝜃, the reflected image from M seen at the eyepiece will shift by an angle of 2𝜃. Foucault's experiment can be understood as follows. During the time the reflected beam from M returns to R, the latter would have moved by an angle 𝜃, and hence the reflected image seen at the eyepiece would shift by 2𝜃. Knowing the distance between the mirrors, the time between the first and second reflections on the rotating mirror, the angular shift of the reflected image and the rate of rotation of the mirror, Foucault obtained a speed of 298,000 km/s, which is only 0.57% shy of the actual value. 

The actual connection between light and electromagnetism was not made until Maxwell. Maxwell began with four equations; Gauss's law in electrostatics (1), Gauss's law in magnetostatics (2), the differential form of Faraday's law of electromagnetic induction (3), and Ampere's circuital law (4) with the displacement-current term that was added by Maxwell himself. E and D are the electric field and the displacement current vector, respectively. B and H are the magnetic flux density and the magnetic field strength, respectively. ρ is the free charge density, and J is the free current density. The electric field vectors, i.e., E and D and the magnetic field vectors, i.e., B and H, are further connected by the absolute electrical permittivity 𝞊₀ and the absolute magnetic permeability 𝝁₀. These six relations can be represented as follows:

Equations (3) and (4) represent a pair of coupled partial differential equations linking the electric field vectors with the magnetic field vectors. Decoupling the two equations by standard mathematical techniques yields two separate relations for the E-field and the H-field. In a charge-free region, where ρ and J are both zero, these two equations take the following form. 

We see that equations, (7) and (8), satisfy the classical wave equation given by (9). These three equations show that the electric and the magnetic field vibrations can propagate through space in the form of classical waves with a velocity 𝒗. The numerical value of this 𝒗 was already determined through an experiment performed in 1856 by Wilhelm Edward Weber and Rudolf Kohlrausch. Although they found that their experimental values brilliantly matched with that of the speed of light determined by Fizeau, the actual connection between light and electromagnetism was still due. In 1857, Gustav Kirchhoff found that electricity itself travels at the rate of 𝒗 in a resistanceless wire. Maxwell's genius lies in the fact that by hearing of the Weber-Kohlrausch result, he immediately realised that light must be an electromagnetic wave.  

Between 1877 and 1879, Albert Abraham Michelson perfected upon Foucault's measure and obtained a value of 299,910 ± 50 km/s for the speed of light which was the most accurate measurement of the day.

References:

1. Galilei, Galileo. 1954. Dialogues Concerning Two New Sciences. Translated by Henry Crew and Alfonso de Salvio. Dover Publications Inc., New York, p.42
2. Cohen, I.B (1940). Roemer and the first determination of the velocity of light (1676). 
3. https://en.wikipedia.org/wiki/Speed_of_light
4. https://en.wikipedia.org/wiki/Fizeau%E2%80%93Foucault_apparatus