Coulomb's Law and Point Charges: Real World vs Theory

[manofphysics]

1)What is the formal definition of a point charge? It is known, that coulomb's law , is valid for point charges? But the real world does not consist of point charges. And coulomb himself used two spheres in his experiments...

2)Another point, in many books I have found that they say, when spatial dimension of charge is much much less than the distance between the charges, they can be treated as point charges...Is this true? For eg. By this, can we say that an irregularly shaped boulder (with charge on it's surface) can be treated as point charge if the distance between it and a test charge is large (~ 500m.)? Won't we have to perform integration to find the exact field at the point of test charge?

 

[Vanadium 50]

A point charge is an approximation, like stretchless ropes, frictionless planes, etc. It's a useful concept, even if it is not exactly realized in nature.

 

[kanato]

When you get to studying the multipole expansion of the electrostatic potential, you will see the details of the charge distribution become less important at large distances. The multipole expansion is essentially an expansion in 1/r for a given charge distribution, and what you find is the monopole term (total charge of the object) is decays as 1/r, the dipole term (sum of charges times positions) decays as 1/r^2 and so on, with higher order contributions decaying faster. For instance, if your charge distribution occupies a region with a characteristic length a, at r = 10a the total charge of the object will contribute 10 times the magnitude of the simplest contribution coming from the details of the charge distribution. You can do the integration to find the exact field, but at large enough r you will always find that the details of the charge distribution contribute very little compared to the amount of total charge in your source. Now, if you had a charge distribution which was neutral, then the monopole term disappears and the first contribution is the dipole term. Then you can't treat the object as a point charge with charge = 0. But still, at large enough r, only the leading term is important.

 

[espen180]

1)What is the formal definition of a point charge? It is known, that coulomb's law , is valid for point charges? But the real world does not consist of point charges. And coulomb himself used two spheres in his experiments...

Using Gauss' Law it is easy to show that a charge on a spherical conducting surface (charge is uniformly distributed) creates an electric field that is zero inside the sphere, and outside it is equal to that of a point charge at the center of the sphere.

2)Another point, in many books I have found that they say, when spatial dimension of charge is much much less than the distance between the charges, they can be treated as point charges...Is this true? For eg. By this, can we say that an irregularly shaped boulder (with charge on it's surface) can be treated as point charge if the distance between it and a test charge is large (~ 500m.)? Won't we have to perform integration to find the exact field at the point of test charge?

True, but this is an approximation. I am sure that in your book(s) you have encountered examples/problems in which the author/you calculate the electric fields created by rods, loops, discs etc. with uniformly distributed charge. If you study the results, you will see that when you move far away (theoretically infinately far away) the equations approach that of a point charge.

 

[manofphysics]

Thanks a lot, kanato & espen180 ! I have understood the concept now after reading your answers.