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I understand that both the electric and gravitational forces are inversely proportional to the squared distance from the point source and that the gravitational constant is around $10^{-20}$ times Coulomb's constant.

I don't understand why this suffices to conclude that the gravitational force is stronger than the electric force. The two constants that I see getting compared here have different units. One involves electric charges, while the other involves gravitational masses. To me, this makes as much sense as saying that a second is larger than a meter.

Is it related to the energy densities of these fields? If yes, how? I think that it is not the duplicate of the mentioned question because the point being asked in that question is why do we consider gravitational fields weaker even if they are more dominant in the universe at macroscopic levels. But, I want to ask that what are the criteria based on which we define the relative strength or weakness of forces/fields. What do we exactly mean by declaring a force/field stronger than the other? Is it related to their energy densities?


marked as duplicate by BMS, Jerry Schirmer, Bernhard, Qmechanic Jul 26 '14 at 19:54

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    $\begingroup$ Possible duplicate of physics.stackexchange.com/q/4243 $\endgroup$ – garyp Jul 26 '14 at 18:43
  • $\begingroup$ Finally someone is asking the right question! Comparing a dimension-full interaction (gravity) with a dimensionless interaction (standard model interactions) is meaningless if no circumstance is provided. Just to give you some food for thought: •If the electron mass is increased to the mass of a flea egg ($10^{-10}$ kg, the plank mass), the gravitational attraction between electrons will be in balance with the repulsive electronic force. In technical jargon, the Schwarzschild radius and the Compton wavelength are of the same order as the Planck length for this case. $\endgroup$ – MadMax Mar 14 at 14:02
  • $\begingroup$ We should rather ask: why are the masses of the elementary particles so small compared with the Planck scale? This leads you to the nagging issue of natureness/hierarchy problem. $\endgroup$ – MadMax Mar 14 at 14:07
  • $\begingroup$ @MadMax Yes, the question is actually pretty old and is marked as a duplicate to a question that I don't think resembles my question and so I cannot leave an answer there but I have come to an answer to this question along the same lines you mention. In particular, the question just translates to asking as to why the elementary particles are so lighter than the Planck mass, i.e., the hierarchy problem as you mention. Thanks for your comment! :) $\endgroup$ – Dvij Mankad Mar 14 at 14:15
  • $\begingroup$ @MadMax I guess you must be aware, but just in case, I would like to mention that there has been some great work by Nima, Motl, et al. in showing that gravity must be the weakest force (in the sense we discussed) in any consistent theory of quantum gravity. Reference: arxiv.org/abs/hep-th/0601001 $\endgroup$ – Dvij Mankad Mar 14 at 14:20

You're just comparing forces to forces, Newtons to Newtons. I guess if you want to be really nitpicky, you're talking about force per charge vs force per mass. But when one says that the electric force is much stronger than the gravitational force, one means that in dealing with the usual objects, like electron, the effects of gravity on the electron's motion are negligible when compared to the effects of the electromagnetic forces - negligible by many orders of magnitude.


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