# Q & A: deriving E=mc^2 via units?

Q:
I believe there is probably a simple answer to my question but I can't find it.I received a university engineering degree. One thing that was beat into our heads (must be the same for physics majors) is to always check your units. If you are looking for an answer that is supposed to be a distance and your units turn out to be seconds then you have a problem!OK, E=mc^2. Bravo Einstein. Units: Joules, kg x m^2 x s^2.If one plays with SI and derived units, one can easily come up with this:E=k x m x c^2. k is an unknown dimensions-less constant. c is speed of light squared. m = mass, E is energy. Now we don't know what "k" is (it turns out to be 1 of course). This could have been done well before Einstein. Correct? Doing experiments could (probably) determined that "k" is pretty close to 1. Yes?Could Maxwell's equations have helped us figure out what the constant "k" was. [Never ever got comfortable with his equations.]I guess my question is: by playing with SI Units [I know they were not all agreed on until after Einstein - I think] could we not have come up with the power concept of Einsteins famous equation?Further: by further playing with SI units could we not come up with other "insights" about the most basic things in physics? I assume we can't or it would already have been done. I just can't see where the "logic"of my thoughts breaks down. I assume there must be, I just can't see it.Thanks in advance to anyone who helps me find the answer.
- Merton Hale (age 69)
Belgium
A:

Those dimensional analysis arguments are wonderful but not magic. If there are two possible velocties that might appear in the energy formula, dimensional arguments can never tell you what combination of the two appears. So v2, c2, cv, etc. are all equivalent dimensionally.

Wit regard to Maxwell's equations, they can and did give a relation E=pc between the energy and momentum of an electromagnetic wave traveling in one direction. If you write the momentum p  as a product of the velocity and an inertial mass m, then that gives E=mc2. That equation in effect defines m, for light. Maxwell's equations themselves don't say that it should work for anything else. They also don't say whether that m has any significance other than just providing a way of re-writing E=pc. Other arguments are needed to show that this m is what shows up as a source of gravity.

Mike W.

(published on 05/10/2015)