This is one of the most famous equations in history. When Einstein wrote it, he was announcing to the world that energy (E) and mass (M) were equivalent. The equation is merely a conversion table. With this idea, we can explain why the sun is able to emit the light and heat we observe - nuclear fusion of Hydrogen into Helium. More precisely, net reaction converts 4 hydrogen atoms into 1 helium atom, ... which produces a small loss in mass, ... which is converted into radiation (gamma rays). Ok, .... you can find all this in any astronomy book but let me throw a monkey wrench into the topic.
A hydrogen atom consists of an electron and a proton. The electrons play no role in the reaction, ... so we only have to deal with 4 protons. Now the mass of one proton is:
proton mass = 1.6726 x 10-27 kilograms
The helium nucleus that results from this fusion consists of 2 protons and 2 neutrons (an alpha particle). Again, the electrons play no role here. Now the twist! If you look up the mass of a neutron, you will see that it has a higher mass than a proton!
neutron mass = 1.6749 x 10−27 kg
On the surface, it looks like the net reaction produces a gain in mass, ... not a loss in mass. Your mission is to resolve this apparent paradox and make the sun shine (not suck up energy).
This is a tough one. Most students who attempt this question discover that the overall reaction (known as the p-p chain reaction) involve many individual steps and then discover that two positrons are given off in the overall reaction. See this graph. This is definitely a mass loss and I'll give some credit if you discover this ... but it is NOT the answer. Here is why: The exact value of the mass of positron is 9.11 x 10-31 kg. Now two positrons have twice this value. But the math still doesn't work.
4 x 1.6726 x 10-27 (four protons) - 2 x 9.11 x 10-31 (two positrons) = 6.688578 x 10-27 kg and this is still LESS than the mass of two individual protons and two individual neutrons ( 2 x 1.6726 x 10-27 + 2 x 1.6749 x 10-27 = 6.695 x 10-27 kg) .... so something is still wrong!
It turns out that the actual mass of the helium nucleus is slightly less than the mass of the individual particles that make up the nucleus. If you look up the mass of a helium nucleus (also known as an alpha particle) you will get 6.644657230 ×10−27 kg and this is LESS than calculated mass sum of the individual particles we just calculated above. What is going on?
This is known as the "mass defect" and this slight loss in mass is the key to resolving this paradox. How can this be?
There is energy (and therefore a mass equivalent) in the form of "nuclear binding energy" in the nucleus of a helium atom. This is energy similar to gravitational potential energy but in this case, it works inside the nucleus of the helium atom.
To help you see this let's use an example with gravity. Imagine two situations, a big rock sitting on the ground and the same rock on the top of a cliff. Which situation has more energy? If the rock falls off the cliff, it will fall to the ground and cause all kinds of damage (which takes energy). So the rock on the cliff has more "potential energy" than the rock on the ground. If we jump to the helium atom, the force holding the nucleus together is known as the strong nuclear force and acts like gravity to hold the particles together. However, to get them all together they must "fall" (like the rock) to a lower energy state. Lower energy equates to lower mass. This equivalent mass loss more than makes up for the small gain in mass between protons and neutrons. Paradox resolved.