# Where has the energy gone?

Setup: Two small bodies, let's say pingpong balls, are situated in deep space in an area free of any substantial fields of any kind. One pingpong ball is negativly charged, the other is positively charged. (equal but opposite charges) With a spatial separation between the balls, there exists a force which attracts them together and this force is an energy of sorts. When this force acts and reduces the separation to zero, IE touching, the net charge of the body composed of the pingpong balls becomes zero, or more precisely , a neutral charge, on a single composite body. What happened to the energy of the attractive force present under separation?

• "and this force is an energy of sorts." No. The relationship between force and energy is well defined and it is not any kind of equality ... these quantities are different kinds of things. There is an energy in the system because there is force in the system, but they are not the same thing and you will only confuse yourself if you don't keep that distinction clear. – dmckee Jan 6 '17 at 7:31

First, force is not the same as energy. Force is force and energy is energy. If, for example, you hold the charged balls in your hands, the net force on them is zero but they still have the electrostatic energy.

Therefore, it is correct to say that the system of the two charges has energy. If they start at rest at a distance $r$ with equal and opposite charges $\pm Q$, then their potential energy is

$$U = - \frac{Q^2}{r}$$

which is negative, and gets more negative the closer the charges are. Since the total energy (kinetic plus potential) must be a constant, as the potential energy decreases the kinetic energy must increase. This makes total sense: the electric force is attractive, so as the balls get closer they accelerate and go faster and faster.

The answer to "where did the energy go?" depends on what precise moment you're talking about. An instant before the balls collide, they have a lot of kinetic energy to compensate for the fact that their potential energy is very negative (in fact it diverges as $r\to 0$).

After the collision, the balls are stuck together with the same energy they had when they started, which if you remember is negative. The potential energy, however, is much lower than this (if two point charges were exactly in the same point, it would be negative infinity). The positive kinetic energy is still there but now it's in the form of internal movement of molecules, i.e., heat. The balls are hotter as a result of having been smacked together.

• If held separated in my hands there would still be a force between them, would there not? My hands would restrain the force, but it's still there! – RaSullivan Jan 6 '17 at 5:20
• Also, how can forces not have energy? – RaSullivan Jan 6 '17 at 5:21
• Additionally, the net charge of the composite body becomes zero, a neutral charge. So I still don't understand. – RaSullivan Jan 6 '17 at 5:25
• 1. Yes there would still be a force, but since you're holding the balls in your hands the net force on each of them is zero. 2. A force can be related to a potential energy like here, but forces themselves do not have energy. Energy is a property of a system. 3. I'm not sure what it is you don't understand. The net charge is zero but the final object, like everything else, is made up of lots of very small positive and negative charges. – Javier Jan 6 '17 at 14:44
• Javier. If I let go the force suddenly appears? – RaSullivan Jan 7 '17 at 0:50