Some school physics books say that energy is the ability to do work. But that’s plain wrong. First, does this mean that a body with zero energy can’t do any work? That’s false, for example, two opposite-signed charges have a negative energy and still can do any amount of work, which will reduce the energy even more. Second, heat energy can do work only under certain circumstances. Third, how is work defined if not as the amount of energy transferred?
How can one define energy, then? Feynman has an excellent parable explaining what energy is. If we digest it into a definition, here is what we get: Energy is a physical quantity constituting the sum of several separate quantities from different areas of physics (called “forms of energy”), such that this sum is always conserved.
This sounds so axiomatic that it just tempts one to start looking for corollaries. Can we prove the existence of such a quantity? No. Indeed, physics has no proof that the Law of the Conservation of Energy won’t break down in some new field of physics. Is this quantity unique? If we multiply all energy by a constant factor, it will still satisfy the definition. Apart from that, I suppose it can be proved that energy is unique.
It is interesting that in daily life people talk about “using energy”, “wasting energy” etc. People seem to think about energy as something that is spent, not conserved. Why? Because in daily life and in economics, “energy” actually means useful energy, i.e., energy that can do useful work. For example, I can use some of the energy stored at a power plant to heat my room. Of course, electric energy transforms into thermal energy, but I can’t use thermal energy to do more useful work, so for practical purposes it doesn’t exist. So I’ve spent that electric energy.
Energy can almost never be reused, and if it is, then only a small part. Regenerative braking reuses part of the energy, but most of it is lost, so people still talk about spending fuel, which is equivalent to energy, at least in a given engine.
This means that useful energy actually does equal the amount of net useful work it allows to do, so in a sense the “ability to do work” definition is correct after all. But it has no place in physics books. Instead, those books should mention the difference between energy and “useful energy” to keep students from getting confused.