Invisible forces

Measuring motion

With Newton’s laws, physicists could treat a planet as a single point with a speed and a force acting on it, and use calculus to work out its path. The idea of force had already grown far beyond a hand on a cart. The sun pulls the earth across 150 million kilometers of empty space. Yet physics never stops at the surface. What exactly is a force, and how do you measure what it does?

Here are three puzzles that troubled the best minds of the late 1600s:

The puzzles have no single answer, because there are two ways to measure motion, and both matter. One is momentum: mass times velocity. A force acting for a certain time changes an object’s momentum. The other, which the German philosopher Gottfried Leibniz championed in the 1680s under the name vis viva, “living force,” grows with mass times velocity squared. Today we call it kinetic energy: one half of mass times velocity squared. A force acting over a certain distance changes it. Followers of Descartes and of Leibniz argued for decades over which was the true measure of motion. Both were right; they measure different things.

Notice that each step made the ideas more abstract. First came forces you can see and feel. Then came gravity, which you can’t see but can feel. Then came momentum and energy, which you can’t see or feel at all. Energy had entered physics, and Course 5 follows where it led.

Amber and lodestone

Newton’s way of thinking soon spread to other parts of physics, above all electricity and magnetism. People had known both for thousands of years. The Greeks knew that rubbed amber attracts light objects. In China, a lodestone spoon that turns to point south was described in early texts, and by 1088 the scholar Shen Kuo was describing magnetized needles used as compasses. China’s famous south-pointing chariot is sometimes mentioned too, but that was a geared machine that held its direction mechanically, not a magnet.

Yet for two thousand years, understanding stayed qualitative. This course points to two reasons. Production didn’t need electricity yet, since machines were enough. And nobody had found the underlying laws. After the Renaissance, both changed. Long-distance trade and rising production created new demands, and new instruments made new experiments possible. The Leyden jar, invented in the 1740s, stored electric charge far longer than rubbed amber could. For the first time, people had electricity to experiment with. Better instruments, once again, lead to better ideas.

The same law, twice

Opposite charges attract and like charges repel; people had long known that much. In 1785 the French engineer Charles-Augustin de Coulomb measured the force precisely, and it had a familiar shape:

Two lessons follow. First, look at the constants. Gravity’s is tiny, so you only notice gravity between enormous masses, like a planet and you. The electric constant is huge, so even tiny charges pull hard. Nature’s constants fence in the range of things we can experience.

Second, look at the shared form. This course calls it physical similarity: the laws of two different sciences can resemble each other even though their details differ, so methods from one can be carried into the other. You use the same trick at work whenever experience in one area helps you in another. Electrostatics followed Newton’s methods and advanced fast. Then, in 1800, Alessandro Volta invented the battery, which produced a steady electric current, and a flood of new discoveries followed.

Fields

Gravity pulls across empty space, and so does the electric force. How? Newton himself found action at a distance absurd, and some thinkers revived the aether as an invisible medium to carry it. The answer that lasted came from Michael Faraday.

Faraday had little mathematics, but a world-class power of imagination. He pictured the space around a magnet or a charge as filled with lines of force, the way iron filings sprinkled near a magnet arrange themselves into curves. Any other charge that enters this field feels its pull. The idea led to electric motors and generators.

Then came James Clerk Maxwell (1831–1879), a Scottish physicist with first-class mathematics. He gathered everything known about electricity and magnetism and looked for the simplest, most universal laws beneath it. In 1865 he published the equations that bear his name. What Newton’s laws are to mechanics, Maxwell’s equations are to electromagnetism: a handful of lines that cover every electrical and magnetic phenomenon.

The equations also predicted something nobody had seen: waves of electric and magnetic fields, traveling through space at the speed of light. Light itself, Maxwell concluded, is such a wave. In 1887–88 Heinrich Hertz produced them in his laboratory. A crowd shows what a wave is. Picture a stadium wave: each person only stands up and sits down in place, but the wave races around the stands. Nothing travels except the pattern, and the energy it carries. Electromagnetic waves are the same, except that what rises and falls, in step, are electric and magnetic fields. Their range runs from radio waves as long as a building, through microwaves and the infrared that night-vision goggles detect, to visible light, ultraviolet, and X-rays.

Why the inverse square?

Why do gravity and electricity both weaken with the square of the distance? The field picture gives a simple answer. Think of a candle or a small light bulb. Its light spreads out in every direction, over the surface of a growing sphere. A sphere’s area is 4πr², so at twice the distance the same light covers four times the area and is a quarter as bright. A field spreading from a mass or a charge thins out the same way. One idea from experiments and another from pure imagination arrive at the same law. In this reading, that agreement between independent routes is what gives science its meaning.

Einstein’s general theory of relativity (1915) took the idea furthest. Gravity is the bending of space and time itself around a mass. Even light, which has no mass, follows the bend. During a solar eclipse in 1919, astronomers measured starlight curving around the sun by the amount Einstein predicted. Force had become field, and field turned out to be the deeper idea.