# Can Newton’s law of gravitation be derived from Coulomb’s law? [duplicate]

I’m casually learning physics and have noticed that Newton’s law of gravitation and the electrostatic force formulas look similar. I’ve asked this question before but would really appreciate another response. Is it possible that the two laws are related? Can the law of gravitation be seen as the macroscopic averaging of Coulomb’s law? So atoms on average have negative charge (positive mass) and thus on a macroscopic scale we observe that two large bodies (eg planets) attract rather than repel. Would it help if we assume that masses can be positive as well as negative? Apologies as I’m not a physicist (rather a data analyst) and these are probably dumb questions.

• – John Rennie Aug 8 '18 at 5:57
• Keep in mind - we know that Newton's law of gravitation is wrong. It was an approximation, and we now have a better approximation that looks nothing like Coulomb's law - general relativity. Funnily enough, Coulomb's law is also replaced by a better approximation (quantum field theory), that by necessity works with general relativity. But in that case, the reason for the similarity is obvious - general relativity basically defines what space and time are, so any GR-compatible physical theory must include spacetime as understood by GR (e.g. be strictly local, curved etc.). – Luaan Aug 8 '18 at 10:34
• Can we think of Unification of forces in nature? – Sriharsha Aug 8 '18 at 12:02
• Related: physics.stackexchange.com/q/944/2451 , physics.stackexchange.com/q/54942/2451 and links therein. – Qmechanic Aug 8 '18 at 12:27
• The obvious logical point to make about Coulomb's law for particles is that if you scaled it up, so that particles in bulk, such as planets, had negative charges as suggested, gravity would not result: two planets, each negatively charged, would repel each other, since like charges repel. The similarity between Coulomb's law and Newtonian gravity is that each force declines in accordance with the inverse square law; but that is not unique to those two theories, it occurs much more widely, e.g. in electromagnetism, hence it does not imply an association between Coulomb's and Newton's theories. – Ed999 Feb 14 at 1:24

those two laws look similar because they both describe the propagation of a long-range field through three-dimensional space which produces a force that acts (in the simplest example) between pairs of objects, and in which the strength of the resulting force depends on some extensive property of each (charge in one case, mass in the other). the long-range-force part in three dimensions furnishes the 1/r^2 part, the influence of the extensive property furnishes the product m1 x m2 or q1 x q2, and the constant term (G or E) contains the fundamental strength of the coupling.

However, if we are talking about coulomb's law and newtonian gravity, those different fields do not couple or mix- which means you can't "make" gravity out of electrostatics or electrostatics out of gravity. Furthermore, gravity is always attractive because there are no negative masses, whereas electrostatic forces can be either attractive or repulsive because charges come in either + or - form.

This is a simplistic explanation. Note that there are far deeper reasons rooted in the underlying mathematics for why these things are the way they are and I invite the professionals here to weigh in on this.

To the best of our knowledge they are not deeply related although there is a theory called Kaluza–Klein theory that tried to interpret electro-magnetism as curvature of space-time much like gravity. There are, however, no real indications that this is correct.

To get back to the original question the relation is that the force equation has identical functional form with just different constants. This can be interpreted as coincident but is useful in mechanics since you can reuse many results for gravity in the case of charges that interact.

What if mass had a sign?

There are (let's keep it simple) Sun, Earth and Moon.

Earth goes around the Sun, so they have different signs.

What about the Moon? If it's attracted to the Earth, it would be repelled by Sun, and vice versa. This is not what happens.

• the "sign" of mass is imaginary. So Sun has ai mass, earth has bi mass and they repell each other with a force of abii=-ab/(earth_sun distance)^2. Moon has ci mass and earth and moon repell each other with -bc /(earth_moon distance)^2. Simillarily for the sun; all bodies attrackt each other. – Taemyr Aug 8 '18 at 9:55
• Ha-ha! Take a small mass and consider its equations of motion when it is affected not only by the gravity, but also by an electrostatic force (experience of Milliken, for example). The mass term is the same for all forces, so the mass is a positive-valued quantity. – Vladimir Kalitvianski Aug 8 '18 at 12:23
• @Taemyr Another proposition would be that Gravity has two signs, but the law is reversed - the SAME signs ATTRACT and opposite signs REPEL. – Maciej Aug 8 '18 at 13:09
• @Taemyr Are there ... any models that can make use of some thing like that or is it just a cool mathematical observation? – kaine Aug 8 '18 at 13:36
• @Empischon it doesn't have to be reversed, it's enough that the const factor is negative... – Ister Aug 9 '18 at 13:09

They look similar because both describe "self-generated" forces which a) act at a distance, and b) are conservative. That is, they conserve energy and produce stable orbits.

You'll need to learn calculus to do it, but you can show that, for a force which varies with distance, a non-circular closed loop in the system ONLY produces a zero net energy if the exponent of the force with distance is -2.

In other words, if the two forces did not obey the inverse square law, the universe would either explode or implode, as energy is either created or disappears. Since neither of these things happens, we're stuck with the existing form of the laws.

• It sounds like you're claiming the only possible conservative force is the $1/r^2$ force. That's very far from true. – knzhou Aug 8 '18 at 13:10