# Gravitational force and Electromagnetic force?

I found this interesting note in one of my textbooks,

The enormous strength of the electromagnetic force compared to gravity is evident in our daily life. When we hold a book in our hand, we are balancing the gravitational force on the book due to the huge mass of the earth by the 'normal force' provided by our hand. The latter is nothing but the net electromagnetic force between the charged constituents of our hand and the book, at the surface in contact. If electromagnetic force were not intrinsically so much stronger than gravity, the hand of the strongest man would crumble under the weight of a feather! Indeed, to be consistent, in that circumstance, we ourselves would crumble under our own weight!

What does this mean?

• Nov 8, 2014 at 13:39

The notion of "weak gravity" is "not even wrong"! It's comparing oranges to apples: it's meaningless to compare a dimension-full interaction (gravity) with a dimensionless interaction (standard model interactions such as the electromagnetic force), if no circumstance (e.g the specific charges and masses in comparison) is provided.

The Nobel laureate Frank Wilczek actually wrote a whole book ( The Lightness of Being) about what should be the right question: Given that the "charge" of the gravitational force is mass (energy tensor), the correct question is 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 (the unnatural gap between the Planck mass and the weak scale/Higgs mass). As of yet, mortal physicists are still scratching their heads and fretting about this nasty "naturalness/hierarchy/fine-tuning problem". The world’s best minds are loosing sleeping on it (ask Lisa or Nima) and yet there is no answer.

To give you some thought experiment of "strong" gravity:

• If the electron mass is increased to the mass of a flea egg ($$10^{-8}$$ 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 for this case.

• If the electron mass is increased to the mass of a chicken egg, the
gravitational force between electrons will trump the electronic force and electrons will be crushed into an electron black hole by gravity. Of course, true quantum gravity effects will be the dominant one in this case, the conventional semi-quantum black hole reasoning (a la Stephen Hawking) shall be taken with a grain of salt.

• Lisa Randall & Nima Arkani-Hamed? The long-range, 1 Astronomical Unit +, effects of electromagnetism seem to be 'hidden' by the fact very high charge densities would generate particle-antiparticle pairs + a lot of energy, isn't it? Jul 31, 2020 at 15:55
• Yes, Lisa and Nima have offered some tentative explanation. And I am sure Eddie gave it a lot thought too. Jul 31, 2020 at 16:04

I think the part about electromagnetic force being exponentially stronger than gravitation got through to you!(An easy way to picture this is how static electricity can lift up things against the pull of the earth!) About the feather,the way you are able to support the feather against the gravitational pull of the earth is because of the electromagnetic forces exerted by your hand on the feather and vice versa. If these were much much weaker something as little as a feather would be able to crumple your hand, that is if it hadn't already been crumpled by gravity before that!

I've read in The Briefer History of Time (Stephen Hawking), that even very slight changes in the strength of the fundamental forces can cause great imbalance. If the electromagnetic force was stronger, atoms wouldn't be as stable as they are, and if it was weaker, atoms might not form at all, so the chances of life existing would be zero!

If there was a sudden change, hypothetically, this instant, the whole world could possibly crumble. But if that actually was the case we wouldn't even have been here.

• Suppose that I place a mobile phone vertically on a table and get my hand as close as possible to one side of the mobile, then by the repulsion of the charged particles in my hand and in the phone, the phone supposed to fall down horizontally on the table, but I observed that it is not happening? Nov 8, 2014 at 14:06
• Hi Rogue: I believe what you're not getting is this: you know when you "touch something"? Ie when one piece of wood pushes another piece of wood. in fact, fundamentally, that is achieved by electromagnetic force. That's what "ordinary, hard solids" are - what you're feeling is fundamentally the electromagnetic force. they are not talking in the quoted passage about anything at all to do with "electricity", static cling, etc. Just "normal everyday solids". Nov 8, 2014 at 14:56

The weak, electromagnetic, and strong fundamental forces can all be quite important, depending on the energy and distance scales of the phenomenon in question. How about gravitation? It is negligible, at least for what concerns the energies we can reach in particle accelerators (in fact, gravity is not even included in the the Standard Model). Gravity manifests its power only at the macroscopic scale or at the unreachable Planck scale.

This "puzzle" of "weak gravity" is an unsolved problem in theoretical physics, known as the Hierarchy Problem.

You can easily see that gravity is really "weak" at the fundamental level: in the static limit, the electric and gravitational force laws are both inverse square laws, so if one computes the ratio of the forces between two bodies, the distances cancel. Take two point particles with masses $$m_1$$, $$m_2$$ and charges $$q_1$$, $$q_2$$. You have the gravitational force and the electric Coulomb force (CGS units are used) $$F_G = G \frac{m_1 m_2}{d^2} \qquad \quad F_C = \frac{q_1 q_2}{d^2}$$ Clearly (depending on the actual values of $$m_1$$, $$m_2$$, $$q_1$$ ,$$q_2$$) you may have that $$F_G > F_C$$, or the other way round. However, for (charged) elementary particles, you always have that $$F_G \ll F_C$$. In particular, if you use two electrons ($$m_1=m_2=m_e$$, $$q_1=q_2=q_e$$), then you have

$$F_G / F_C = \frac{G m_e m_e}{q_e q_e} = G \left(\frac{m_e}{q_e} \right)^2 \sim 10^{-42}$$

where the quantity in the parentheses is the mass to charge ratio first measured by Thompson. As you see $$F_G \sim 10^{-42}F_C$$, independently on the distance $$d$$: the gravity attraction is really extremely weak.

In an Hydrogen atom you should compare the attraction between the electron and the proton (i.e. the Hydrogen nucleus). The proton is $$\sim 2000$$ times more massive than the electron so $$F_G$$ is $$2000$$ times bigger than the electron-electron gravitational attraction considered before. In this case you should find $$F_G/F_C \sim 10^{-39}$$. Still a super small number, telling us that there is no need to include gravity when studying the atomic structure.

Clearly if you don't use fundamental particles but, say, planets (that are almost neutral but have a huge mass), then $$F_G$$ wins. However, using planets and stars is, clearly, not in the spirit of the Hierarchy problem.

• As I like to say, the real question is not "why is gravity so weak" but rather "why are things so light" in my opinion. The charges, masses, and factors are just a matter of units. If you set units so that both Coulumb's law and Newton's gravity law have exactly the same form everything but charge and mass cancels. Clearly those force ratios will be the same, so it will be caused by the charge to mass ratio only. The question then is, why are electrons, and even protons, so light, or so charged, or just why is that ratio so high. Feb 28, 2023 at 3:39
• @PoissonAerohead I agree! Isn't this exactly what I am saying by writing $G (m_e/q_e )^2 \sim 10^{-42}$? Feb 28, 2023 at 10:15
• By using CGS, you have removed one of the factors, but you still have G. You would need to rescale so the G is also 1. Also, the Gmm/d^2 is in mks, so your ratio still is apples to oranges. To do this properly, you need to take both equations in the same units (say mks) and then scale the units until the form of the equation is the same (either 4pi in both denominators or neither and no arbitrary factors). You then have a meaningful "charge to mass" ratio. See the variety of natural units en.wikipedia.org/wiki/Natural_units Feb 28, 2023 at 19:09

This sort of thing just compares the strength of gravity to the strength of EM. It is a bit of a "wow" factor in popular science to say that all the earth's gravity is resisted by a few electrons as you hold something above the ground or whatever. As the top commenter pointed out, this is technically comparing apples to oranges, and the right question is not "why is gravity so weak" but rather "why are things so light" instead. That may be above your needs though.