by Walter Lewin
These scientists—Ampère, Faraday, and Maxwell—knew they were on the brink of a total revolution. Researchers had been trying to understand electricity in a serious way for a century, but now these guys were constantly breaking new ground. I sometimes wonder how they managed to sleep at night.
Maxwell’s equations, because of what they brought together in 1861, were really the crowning achievement of nineteenth-century physics, most certainly for all physics between Newton and Einstein. And like all profound discoveries, they pointed the way for further efforts to try to unify fundamental scientific theories.
Ever since Maxwell, physicists have spent incalculable efforts trying to develop a single unified theory of nature’s four fundamental forces: the electromagnetic, strong nuclear, weak nuclear, and gravitational forces. Albert Einstein spent the last thirty years of his life in a failed effort to combine electromagnetism and gravity in what became known as a unified field theory.
The search for unification goes on. Abdus Salam, Sheldon Glashow, and Steven Weinberg won the Nobel Prize in 1979 for unifying electromagnetism and the weak nuclear force into what’s known as the electro-weak force. Many physicists are trying to unify the electroweak force and the strong nuclear force into what is called a grand unified theory, or GUT, for short. Achieving that level of unification would be a staggering accomplishment, on a par with Maxwell’s. And if, somehow, somewhere, a physicist ever manages to combine gravity with GUT to create what many call a theory of everything—well, that will be the holiest of Holy Grails in physics. Unification is a powerful dream.
That’s why, in my Electricity and Magnetism course, when we finally see all of Maxwell’s equations in their full glory and simplicity, I project them all around in the lecture hall and I celebrate this important milestone with the students by handing out flowers. If you can handle a little suspense, you will read more about this in chapter 15.
CHAPTER 9
Energy Conservation—Plus ça change…
One of the most popular demonstrations I’ve done through the years involves risking my life by putting my head directly in the path of a wrecking ball—a mini version of a wrecking ball, it must be said, but one that could easily kill me, I assure you. Whereas the wrecking balls used by demolition crews might be made from a bob, or spherical weight, of about a thousand kilos, I construct mine with a 15-kilo bob—about 33 pounds. Standing at one side of the lecture hall, with my head backed up against the wall, I hold the bob in my hands, snug up to my chin. When releasing it I must be extremely careful not to give it any kind of a push, not even a tiny little bit of a shove. Any push at all and it will surely injure me—or, as I say, possibly even kill me. I ask my students not to distract me, to make no noise, and even to stop breathing for a while—if not, I say, this could be my last lecture.
I have to confess that every time I perform this demonstration, I feel an adrenaline rush as the ball comes swinging back my way; as sure as I am that the physics will save me, it is always unnerving to stand there while it comes flying up to within a whisker of my chin. Instinctively I clench my teeth. And the truth is, I always close my eyes too! What, you may ask, what possesses me to perform this demonstration? My utter confidence in one of the most important concepts in all of physics—the law of the conservation of energy.
One of the most remarkable features of our world is that one form of energy can be converted into another form, and then into another and another, and even converted back to the original. Energy can be transformed but never lost, and never gained. In fact, this transformation happens all the time. All civilizations, not only ours but even the least technologically sophisticated, depend on this process, in many variations. This is, most obviously, what eating does for us; converting the chemical energy of food, mostly stored in carbon, into a compound called adenosine triphosphate (ATP), which stores the energy our cells can use to do different kinds of work. It’s what happens when we light a campfire, converting the chemical energy stored in wood or charcoal (the carbon in each combines with oxygen) into heat and carbon dioxide.
It’s what drives an arrow through the air once it’s been shot from a bow, converting the potential energy, built up when you pull the bowstring back into kinetic energy, propelling the arrow forward. In a gun, it’s the conversion of chemical energy from the gunpowder into the kinetic energy of rapidly expanding gas that propels bullets out of the barrel. When you ride a bicycle, the energy that pushes the pedals began as the chemical energy of your breakfast or lunch, which your body converted into a different form of chemical energy (ATP). Your muscles then use that chemical energy, converting some of it into mechanical energy, in order to contract and release your muscles, enabling you to push the pedals. The chemical energy stored in your car battery is converted to electric energy when you turn the ignition key. Some electric energy goes to the cylinders, where it ignites the gasoline mixture, releasing the chemical energy released by the gasoline as it burns. That energy is then converted into heat, which increases the pressure of the gas in the cylinder, which in turn pushes the pistons. These turn the crankshaft, and the transmission sends the energy to the wheels, making them turn. Through this remarkable process the chemical energy of the gasoline is harnessed to allow us to drive.
Hybrid cars rely in part on this process in reverse. They convert some of the kinetic energy of a car—when you step on the brakes—into electric energy that is stored in a battery and can run an electric motor. In an oil-fired furnace, the chemical energy of the oil is converted into heat, which raises the temperature of water in the heating system, which a pump then forces through radiators. In neon lights, the kinetic energy of electric charges moving through a neon gas tube is converted into visible light.
The applications are seemingly limitless. In nuclear reactors, the nuclear energy that is stored in uranium or plutonium nuclei is converted into heat, which turns water into steam, which turns turbines, which create electricity. Chemical energy stored in fossil fuels—not only oil and gasoline but also coal and natural gas—is converted into heat, and, in the case of a power plant, is ultimately converted to electrical energy.
You can witness the wonders of energy conversion easily by making a battery. There are lots of different kinds of batteries, from those in your conventional or hybrid car to those powering your wireless computer mouse and cell phone. Believe it or not, but you can make a battery from a potato, a penny, a galvanized nail, and two pieces of copper wire (each about 6 inches long, with a half-inch of insulation scraped off at each end). Put the nail most of the way into the potato at one end, cut a slit at the other end for the penny, and put the penny into the slit. Hold the end of one piece of wire on the nail (or wrap it around the nail head); hold the other piece of wire on the penny or slide it into the slit so it touches the penny. Then touch the free ends of the wires to the little leads of a Christmas tree light. It should flicker a little bit. Congratulations! You can see dozens of these contraptions on YouTube—why not give it a try?
Clearly, conversions of energy are going on around us all of the time, but some of them are more obvious than others. One of the most counterintuitive types is that of what we call gravitational potential energy. Though we don’t generally think of static objects as having energy, they do; in some cases quite a bit of it. Because gravity is always trying to pull objects down toward the center of the Earth, every object that you drop from a certain height will pick up speed. In doing so, it will lose gravitational potential energy but it will gain kinetic energy—no energy was lost and none was created; it’s a zero sum game! If an object of mass m falls down over a vertical distance h, its potential energy decreases by an amount mgh (g is the gravitational acceleration, which is about 9.8 meters per second per second), but its kinetic energy will increase by the same amount. If you move the object upward over a vertical distance h, its gravitational potential energy will increase by an amount mgh, and you will have to produce that energy (you will have to do work).
If a book with a mass of 1 kilogram (2.2 pounds) is on a shelf 2 meters (about 6.5 feet) above the floor, then, when it falls to the floor, its gravitational potential energy will decrease by 1 × 9.8 × 2 = 19.6 joules but its kinetic energy will be 19.6 joules when it hits the floor.
I think the name gravitational potential energy is an excellent name. Think of it this way. If I pick the book up from the floor and place it on the shelf, it takes 19.6 joules of my energy to do so. Is this energy lost? No! Now that the book is 2 meters above the floor, it has the “potential” of returning that energy back to me in the form of kinetic energy—whenever I drop it on the floor, be it the next day or the next year! The higher the book is above the floor, the more energy is “potentially” available, but, of course I have to provide that extra energy to place the book higher.
In a similar way, it takes energy to pull the string of a bow back when you want to shoot an arrow. That energy is stored in the bow and it is “potentially” available, at a time of your chosing, to convert that potential energy into kinetic energy, which gives the arrow its speed.
Now, there is a simple equation I can use to show you something quite wonderful. If you bear with me for just a bit of math, you’ll see why Galileo’s most famous (non)experiment works. Recall that he was said to have dropped balls of different mass (thus different weight) from the Leaning Tower of Pisa to show that their rate of falling was independent of their mass. It follows from Newton’s laws of motion that the kinetic energy (KE) of a moving object is proportional both to the mass of the object and to the square of its speed; the equation for that is KE = 1/2 mv2. And since we know that the change in gravitational potential energy of the object is converted to kinetic energy, then we can say that mgh equals 1/2 mv2, so you have the equation mgh = 1/2mv2. If you divide both sides by m, m disappears from the equation entirely, and you have gh = 1/2v2. Then to get rid of the fraction we multiply both sides of the equation by 2, to get 2gh = v2. This means that v, the speed, which is what Galileo was testing for, equals the square root of 2gh.* And note that mass has completely disappeared from the equation! It is literally not a factor—the speed does not depend on the mass. To take a specific example, if we drop a rock (of any mass) from a height of 100 meters, in the absence of air drag it will hit the ground with a speed of about 45 meters per second, or about 100 miles per hour.
Imagine a rock (of any mass) falling from a few hundred thousand miles away to the Earth. With what speed would it enter the Earth’s atmosphere? Unfortunately, we cannot use the above simple equation that the speed is the square root of 2gh because the gravitational acceleration depends strongly on the distance to Earth. At the distance of the Moon (about 240,000 miles), the gravitational acceleration due to Earth is about 3,600 times smaller than what it is close to the surface of the Earth. Without showing you the math, take my word for it, the speed would be about 25,000 miles per hour!
Perhaps you can now understand how important gravitational potential energy is in astronomy. As I will discuss in chapter 13, when matter falls from a large distance onto a neutron star, it crashes onto the neutron star with a speed of roughly 100,000 miles per second, yes, per second! If the rock had a mass of only 1 kilogram, its kinetic energy would then be about 13 thousand trillion (13 × 1015) joules, which is roughly the amount of energy that a large (1,000 MW) power plant produces in about half a year.
The fact that different types of energy can be converted into one another and then back again is remarkable enough, but what is even more spectacular is that there is never any net loss of energy. Never. Amazing. This is why the wrecking ball has never killed me.
When I pull the 15 kilogram ball up to my chin over a vertical distance h, I increase its gravitational potential energy by mgh. When I drop the ball, it begins to swing across the room due to the force of gravity, and mgh is converted into kinetic energy. Here, h is the vertical distance between my chin and the lowest position of the bob at the end of the string. As the ball reaches its lowest point in the swing, its kinetic energy will be mgh. As the ball completes its arc and reaches the upper limit of its swing, that kinetic energy is converted back into potential energy—which is why, at the very height of a pendulum swing, the ball stops for a moment. If there’s no kinetic energy, there’s no movement. But that is for just the slightest moment, because then the ball goes back down again, on its reverse swing, and potential energy is converted again into kinetic energy. The sum of kinetic energy and potential energy is called mechanical energy, and in the absence of friction (in this case air drag on the bob), the total mechanical energy does not change—it is conserved.
This means that the ball can go no higher than the exact spot from which it was released—as long as no extra energy is imparted to it anywhere along the way. Air drag is my safety cushion. A very small amount of the mechanical energy of the pendulum is sucked away by air drag and converted into heat. As a result, the bob stops just one-eighth of an inch from my chin, as you can see in the video of lecture 11 from course 8.01. Susan has seen me do the demonstration three times—she shivers each time. Someone once asked me if I practiced a lot, and I always answer with what is true: that I do not have to practice as I trust the conservation of energy, 100 percent.
But if I were to give the ball the slightest little push when I let it go—say I had coughed just then and that caused me to give the ball some thrust—it would swing back to a spot a little higher than where I released it from, smashing into my chin.
The conservation of energy was discovered largely due to the work of a mid-nineteenth-century English brewer’s son, James Joule. So important was his work to understanding the nature of energy that the international unit by which energy is measured, the joule, was named after him. His father had sent him and his brother to study with the famous experimental scientist John Dalton. Clearly Dalton taught Joule well. After Joule inherited his father’s brewery, he performed a host of innovative experiments in the brewery’s basement, probing in ingenious ways into the characteristics of electricity, heat, and mechanical energy. One of his discoveries was that electric current produces heat in a conductor, which he found by putting coils of different kinds of metal with current running through them into jars of water and measuring their changes in temperature.
Joule had the fundamental insight that heat is a form of energy, which refuted what had been the widely accepted understanding of heat for many years. Heat, it was thought, was a kind of fluid, which was called caloric—from which our contemporary word calorie derives—and the belief at the time was that this fluid heat flowed from areas of high concentration to low, and that caloric could never be either created or destroyed. Joule made note, though, that heat was produced in many ways that suggested it was of a different nature. For example, he studied waterfalls and determined that the water at the bottom was warmer than that at the top, and he concluded that the gravitational potential energy difference between the top and bottom of the waterfall was converted into heat. He also observed that when a paddle wheel was stirring water—a very famous experiment that Joule performed—it raised the temperature of the water, and in 1881 he came up with remarkably accurate results for the conversion of the kinetic energy of the paddle wheel into heat.
In this experiment Joule connected a set of paddles in a container of water to a pulley and a string from which he suspended a weight. As the weight lowered, the string turned the shaft of the paddles, rotating them in the water container. More technically, he lowered a mass, m, on a string over a distance, h. The change in potential energy was mgh, which the contraption converted into the rotational (kinetic) energy of the paddle, which then heated the water. Here is an illustration of the device:
What made the experiment so brilliant is that Joule was able to calculate the exact amount of energy he was transferring to the water—which equaled mgh. The weight came down slowly, because the water prevented the paddle from rotating fast. Therefore the weight hit the ground with a negligible amount of k
inetic energy. Thus all the available gravitational potential energy was transferred to the water.
How much is a joule? Well, if you drop a 1-kilogram object 0.1 meters (10 centimeters), the kinetic energy of that object has increased by mgh, which is about 1 joule. That may not sound like much, but joules can add up quite quickly. In order to throw a baseball just under 100 miles per hour, a Major League Baseball pitcher requires about 140 joules of energy, which is about the same amount of energy required to lift a bushel of 140 hundred-gram apples 1 full meter.*
One hundred forty joules of kinetic energy hitting you could be enough to kill you, as long as that energy is released quickly, and in a concentrated fashion. If it were spread out over an hour or two, you wouldn’t even notice it. And if all those joules were released in a pillow hitting you hard, it wouldn’t kill you. But concentrated in a bullet, say, or a rock or a baseball, in a tiny fraction of a second? A very different story.
Which brings us back to wrecking balls. Suppose you had 1,000-kilogram (1-ton) wrecking ball, which you drop over a vertical distance of 5 meters. It will convert about 50,000 joules of potential energy (mgh = 1,000 × 10 × 5) into kinetic energy. That’s quite a wallop, especially if it’s released in a very short time. Using the equation for kinetic energy, we can solve for speed too. At the bottom of its swing the ball would be moving at a speed of 10 meters per second (about 22 miles per hour), which is a pretty high speed for a 1-ton ball. To see this kind of energy in action, you can check out an amazing video online of a wrecking ball hitting a minivan that had strayed into a Manhattan construction zone, knocking the van over as though it were a toy car: www.lionsdenu.com/wrecking-ball-vs-dodge-mini-van/.