Integral of product of infintesimal volume

[vijaypandey93]

I'm following the book ''introduction to electrodynamics by D.J. Griffiths''. As he has written that the formula ''integral of product of infintesimal volume with the square of electric field'' gives us the energy contained in a charge configuration that is always positive because we're encountering square.He has justified it by saying that it even considers the fabrication of charges and as we know the fabrication of point charge involves infinite energy.But why do i get a definite value when i calculate the energy of a charge configuration like spherical or cylindrical,rather if I'm following his argument correctly then in spherical,this formula should take into account of fabrication of all the charges and that's certainly infinite.what's that in his argument that I'm not getting?Please help!

 

[Simon Bridge]

The energy stored in a charge configuration is the amount of work needed to get it into that state.
http://www.masteringphysicssolution...ions-energy-stored-in-a-charge-configuration/

Point charges are just models - all spherical charge distributions look like point charges from the outside.

 

[vijaypandey93]

okay but what'd we get if we calculate the energy due to equal and opposite charges?should it be positive or negative?according to the formula statrd above,it'll always be positive.and using the formula [k(q1)(q2)]/r,it's negative.can you please justify how?

 

[Simon Bridge]

Energy stored in an electric dipole.
https://www.physicsforums.com/showthread.php?t=260877


Also see:
https://www.physicsforums.com/showthread.php?t=631225

The work to get the + and - charges into position does not depend on how they got there (conservative force).
If you started with two neutral spheres, and moves positive charge from one to the other, you would have to do some work right?

 

[pmacias]

Your confusion is understandable, as the concept of infinite energy can be difficult to understand. However, in this context, the term "infinite energy" does not refer to an actual infinite amount of energy, but rather to the fact that the energy of a point charge is theoretically infinite due to its infinitesimal size. This means that the energy calculated using the formula for the integral of the product of infinitesimal volume and the square of the electric field will always be positive, regardless of the charge configuration.

In the case of a spherical or cylindrical charge configuration, the formula still holds true, but the energy calculated will be a finite value. This is because the charges in these configurations are not point charges, they have a finite size and therefore do not have infinite energy. The formula takes into account the distribution of charges within the volume, rather than considering each charge as a point.

So, while the formula may seem to contradict the concept of infinite energy, it is simply a mathematical representation of the energy contained in a charge configuration, and the actual energy will always be a finite value. I hope this helps to clarify the concept for you.