# What is the domain of validity of inverse square laws of newton and coulomb?

1. Nov 18, 2009

### jonjacson

¿what is the domain of validity of inverse square laws of newton and coulomb?

I think (correct me if this is false) that at the small scale these laws have been verified accurately , but ¿what happens with large distances? :uhh: .

¿Have been done an experiment which measured the electrical force between charges at kilometric distances? (for example) .

¿Are there any experiments designed to verify these laws now? .

I would like to know which is the experimental basis of these laws and most recent investigations about the topic, thanks for replying in advance.

2. Nov 18, 2009

### Count Iblis

Re: ¿what is the domain of validity of inverse square laws of newton and coulomb?

At small distances Coulomb's law breaks down due vacuum polarizaton effects. If you write the potential of an electron as:

eQ(r)/(4 pi epsilon_0 r)

Then Q(r) = 1 if Coulomb's law is exactly valid. However, one can show that for r much smaller than the electron Compton wavelength, we have:

Q(r) = 1- 2 alpha/(3 pi) [Log(m r) + gamma + 5/6 + ...]

where m is the inverse electron Compton wavelength:

m = me c/hbar

where me is the electron mass. alpha is the fine structure constant and gamma is Euler's constant.

For r much larger than the Compton wavelength, we have:

Q(r) = 1 + alpha/[4 sqrt(pi) (m r)^(3/2)] exp(-2 m r) +...

In case of long distances, Coulomb's law will become exactly valid, unless the photon is massive. In that case you'll have a
Q(r) = exp(-mr) with m the inverse photon-Compton wavelength.

So, you'll have to look at limits on the photon mass. There are results from direct tests of Coulombs law vbased on the fact that a deviaton of Couloms law laeds to a violation of Gauss' law, which in turn leads to electric fields inside hollow conductors. You can then perform very sensitive tests where to expose a hollow conductor to strong electric fields and attempt to detect a tiny electric field inside the hollow conductor. Null results frm such measurements have limited the photon mass to less than 10^(-14) eV

Another type of tests considers the fact that the vector potential has directly meaurable effects (via the mass term m A^2 in the Lagrangian). It is known that there is some ambient galactic magnetic field. That field is very weak. but it extends over huge distances (tens of thousands of lightyears). If you have a weak constant magnetic field extending over tens of thousands of lightyears, then because the magnetic field is the nabla times A, the vector potential will be of order B r, with r of the order of 10^4 lightyears.

The "photon mass term" m A^2 in the Lagrangian implies that a magnetized ring would experience a torque due to the huge galactic vector potential. Null results from experiments that attempt to detect such effects have limted the photon mass to much lower values.

However, it has been shown that the tests that depend on the galactic vector potential are only valid when assuming a specific model for the photon mass, see here:

http://arxiv.org/abs/hep-ph/0306245

So, we can be sure that Coulombs law is valid for distances smaller than hbar c/[10^(-14) eV] = 2*10^4 km

3. Nov 19, 2009

### jonjacson

Re: ¿what is the domain of validity of inverse square laws of newton and coulomb?

Very interesting, I didn't know that lines of thought.

And talking about the "mass of the photon", imagine that you have an spherical mirror, with a reflectivity of the 100 %, so there aren't losses of energy, and imagine that you have a photon inside the mirror of radius R , ¿what happens with the photon if when the R --> 0?

¿what would be the energy of the photon?