Negative Coulombic Energy

[Michelle Li 2B]

Hi! I know Dr. Lavelle mentioned this in lecture, but I would really appreciate some elaboration on this subject. Is the Coulombic energy always negative and if so, why? What factors make it more negative? Would it be when the electrons are closer to each other since r will be smaller and then the Coulombic energy will be a greater negative value?

[Darren Apostol 2L]

For like charges that attract, try to think of gravity. The higher up an object is, the more gravitational potential energy is has. When it drops to a lower level, it loses that potential energy (converted into kinetic energy during the fall). Every time it goes lower, it has less and less potential energy. For the Coulomb force between two like charges, like a proton and an electron, the same principle applies. The closer the electron gets, the more potential energy it loses. This is what is meant by the potential energy being negative (although, since there is no way of determining the actual potential energy of a system, we use the change in potential energy instead). We need to have a baseline for that negative i.e. where it is zero, and it's convention that set potential energy to be zero at very large distances (radius r = infinity).

For two like charges, like two electrons, you can think of compressing a spring (actually, instead of gravity, you could stretched string going back to normal for like charges). You must put energy in i.e. do work on the electrons to get them closer. The closer they get, the more potential energy they have because of the Coulomb force desperately wanting to push them apart. The moment the electrons are allowed to move freely, this potential energy is released in the form of kinetic energy. You can think of this as why the potential energy is positive for like charges, as they have energy that can potentially be released.

In terms of math, you can just use their signs in the electric potential energy formula involving two charges. If the signs are the same (repel), then the sign is positive. If the signs are different (attract), then the sign is negative. A smaller r value will lead to either a more negative or positive value depending on the signs.