# When does gravitational force = electromagnetic force?

1. Jun 25, 2009

### potmobius

Yes, gravity is WAYYYYYYYYYY weaker than the electromagnetic force in the quantum world, BUT when we talk about the big picture, the universe is held together by gravity. So... in that transition from the micro to the macro, can anybody tell me at what scale, roughly, the gravitational force will equal the electromagnetic force? If that's possible..... I pondered over it, and I thuoght why would it NOT be possible? You do see 2 extreme scenarios in 2 extreme frames of reference, so there must exist an equilibrium somewhere....

2. Jun 25, 2009

### diazona

It depends on the charge and mass of objects you're talking about.

The reason the universe is held together by gravity is that the positive and negative charges of elementary particles cancel out in macroscopic objects. More or less, for every proton that exerts an EM force, there's an electron orbiting it to exert the opposite EM force, and the two add up to no net force. But there are no positive and negative masses, so gravitational forces always add up, without any cancellations. So when you put a lot of particles together, you get a lot of gravity but essentially no EM attraction.

A little mathematical analogy:
1+1+1+1+1+1+1+1+1+... = a huge number (like gravity)
1-1+1-1+1-1+1-1+1-1+1-... = never greater than 1 (like electromagnetism)

3. Jun 25, 2009

### potmobius

perfect answer! Thanks so much!! So, realistically speaking, that wont happen. But, if I'm looking for a mathematical equilibrium of attractive forces between a body the mass of the earth, how many protons would you need to make a planet beside earth to equal the attractive force of gravity by earth? (assume that the earth-sized body is electrically neutral, has no magnetic field, and no atmosphere. assume that the protons are massless)

4. Jun 25, 2009

### potmobius

5. Jun 25, 2009

### diazona

Thanks

As for your question about protons... if I understand correctly, you're asking how many protons it would take to produce an electric force equivalent to Earth's gravity? That depends on what's being attracted to the protons.

Here's how you'd figure it out: the formula for gravitational force is
$$F_G = G \frac{Mm}{r^2}$$
and the one for electrostatic force is
$$F_E = k \frac{Qq}{r^2}$$
In this situation, G is the universal gravitational constant, M is the mass of the Earth, k is the electrostatic constant, and Q is the charge of the "electric Earth equivalent" (which is what you're asking for).
\begin{align*}G &= 6.67\times 10^{-11} \frac{\mathrm{N}\ \mathrm{m}^2}{\mathrm{kg}^2} \\ M &= 5.97\times 10^{24}\ \mathrm{kg} \\ k &= 8.99\times 10^{9} \frac{\mathrm{N}\ \mathrm{m}^2}{\mathrm{C}^2}\end{align*}
Now, if we're saying that the electrostatic force is equivalent to gravity,
$$F_E = F_G$$
or
$$k \frac{Qq}{r^2} = G \frac{Mm}{r^2}$$
or
$$Q = \frac{GM}{k}\frac{m}{q}$$
Now, you can easily calculate $\frac{GM}{k}$ by just multiplying/dividing those numbers I mentioned. But $\frac{m}{q}$ depends on the particle that's being attracted to the Earth or ball of protons. For example, if it were an electron, you'd use the mass of the electron $m = 9.1\times 10^{-31}\ \mathrm{kg}$ and its charge $q = -1.6\times 10^{-19}\ \mathrm{C}$, do the calculations, and find that it would take $2.5\times 10^{-7}\ \mathrm{C}$ of charge to attract an electron as hard as the Earth does. That's just over a trillion protons, which is practically nothing - a millionth of a speck of dust, maybe.

6. Jun 25, 2009

### Naty1

This is, in general, a false premise...but is accurate for atomic scales. Forces are scale dependent.

Take any particle, say an electron or photon, or a group of them in a box, and compress it...reduce it's volume. When it gets small enough, gravitational attraction collapses the object and voila a black hole forms from the extreme density.

As another example, a collapsing star greater than about 1.3 solar masses will also pull itself together and naturally form a black hole without any help from us. Whether this is a "quantum world" phenomena is another issue.

7. Jun 26, 2009

### potmobius

diazona: Awesome once again! That's exactly what I wanted to know! Thanks

naty1: that was random, dude! lol! :tongue2: