Why Does E=mc2_ - Brian Cox [10]
Equations are the most powerful of tools available to physicists in their quest to understand nature. They also are often among the scariest things most people meet during their school years, and we feel it necessary to say a few words to the apprehensive reader before we continue. Of course, we know that not everyone will feel that way about mathematics, and we ask for a degree of patience from more confident readers and hope they won’t feel too patronized. At the simplest level, an equation allows you to predict the results of an experiment without actually having to conduct it. A very simple example, which we will use later in the book to prove all sorts of incredible results about the nature of time and space, is Pythagoras’ famous theorem relating the lengths of the sides of a right-angled triangle. Pythagoras states that “the square of the hypotenuse is equal to the sum of the squares of the other two sides.” In mathematical symbols, we can write Pythagoras’ theorem as x2 + y2 = z2, where z is the length of the hypotenuse, which is the longest side of the right-angled triangle, and x and y are the lengths of the other two sides. Figure 1 illustrates what is going on. The symbols x, y, and z are understood to be placeholders for the actual lengths of the sides and x2 is mathematical notation for x multiplied by x. For example, 32 = 9, 72 = 49 and so on. There is nothing special about using x, y, and z; we could use any symbol we like as a placeholder. Perhaps Pythagoras’ theorem looks more friendly if we write it as= ☺2. This time the smiley-face symbol represents the length of the hypotenuse. Here is an example using the theorem: If the two shorter sides of the triangle are 3 centimeters (cm) and 4 centimeters long, then the theorem tells us that the length of the hypotenuse is equal to 5 centimeters, since 32 + 42 = 52. Of course, the numbers don’t have to be whole numbers. Measuring the lengths of the sides of a triangle with a ruler is an experiment, albeit a rather dull one. Pythagoras saved us the trouble by writing down his equation, which allows us to simply calculate the length of the third side of a triangle given the other two. The key thing to appreciate is that for a physicist, equations express relationships between “things” and they are a way to make precise statements about the real world.
FIGURE 1
Maxwell’s equations are mathematically rather more complicated, but in essence they do exactly the same kind of job. They can, for example, tell you in which direction a compass needle will be deflected if you send a pulse of electric current through a wire without having to look at the compass. The wonderful thing about equations, however, is that they can also reveal deep connections between quantities that are not immediately apparent from the results of experiments, and in doing so can lead to a much deeper and more profound understanding of nature. This turns out to be emphatically true of Maxwell’s equations. Central to Maxwell’s mathematical description of electrical and magnetic phenomena are the abstract electric and magnetic fields Faraday first pictured. Maxwell wrote down his equations in the language of fields because he had no choice. It was the only way of bringing together the vast range of electric and magnetic phenomena observed by Faraday and his colleagues into a single unified set of equations. Just as Pythagoras’ equation expresses a relationship between the lengths of the sides of a triangle, Maxwell’s equations express relationships between electric charges and currents and the electric and magnetic fields they create. Maxwell’s genius was to invite the fields to emerge from the shadows and take center stage. If, for example, you asked Maxwell why a battery causes