Electric Potential energy in dielectric medium

[rafaelpol]

Homework Statement

It is actually not a problem, but a formulation of the electric potential energy in a dielectric medium present in a book, which seems to me as inconsistent with the way it is usually defined in electromagnetism books. I can't see it as a typo or a mistake because the author uses the definition a few times to derive some properties of the system (charge density located in a cavity on a dielectric medium).

Homework Equations

The author defines the electric potential energy in the dielectric as:
$$E_R(\rho,\rho' )= \int_{R^3}\rho'(\overrightarrow{r}) V_R(\overrightarrow{r})d\overrightarrow{r}$$

V_R is the potential produced by pho' in the dielectric medium (determined by the Poisson equation)

The Attempt at a Solution

The equation seems inconsistent to me because according to books such as Greiner, for e.g., there should be a 1/2 before the integral so as not to double count interactions.

The way it is written above, it seems as the work to assemble the charge distribution (such that before that the infinitesimal elements of charge are all at infinite distance from each other, i.e., all of those are in places where V = 0) in space where the potential V_R is present before constructing the desired charge distribution. However, as stated, V_R is determined from the only source of electric field in this system, which is pho'. Hence, this intepretation does not follow, and also it is clear from the paper that the author is not referring to this, but to the energy of the system.

I really can't see anything other than this. Can anyone please help me to figure out why the author uses this definition?

 

[mark726]

Thank you for bringing up this interesting topic. it is important to question and analyze different formulations and definitions in order to fully understand a concept. After looking at the equation and your analysis, I do agree that the definition of electric potential energy in the dielectric medium presented in the book seems to be inconsistent with other sources. As you mentioned, the lack of a 1/2 factor in the integral could potentially lead to overcounting interactions.

It is possible that the author may have a different perspective or approach in defining the electric potential energy in this particular scenario. However, it is important to note that there is no one correct way to define a concept in science. Different authors and researchers may have different notations and formulations, but as long as they are mathematically sound and logically consistent, they can all be valid.

In this case, I would suggest reaching out to the author or consulting other experts in the field to get a better understanding of their reasoning behind this particular definition. It could be that there is a specific context or application where this formulation is more appropriate. As scientists, it is important to have open discussions and debates in order to further our understanding and knowledge in a particular subject.

I hope this helps in your understanding of the concept. Keep questioning and analyzing, as that is the essence of science.