Self Energy and Interaction Energy

[maverick280857]

Hello

I have a query (Classical Electrodynamics) regarding Self Energy and Interaction Energy. I understand that the integral definition

$$U_{tot} = \frac{1}{2}\int_{all space}k\epsilon_{0}E^2d\tau = \frac{1}{2}\int Vdq$$

represents the total electrostatic potential energy of a system and that this equals

$$q_{1}V_{12}$$ or $$q_{2}V_{21}$$ (for the discrete charge case).

What I want to know are the exact definitions of self energy and interaction energy of a system of charges (the discrete and continuous case) as I have not found convincing explanations in my textbook and on the internet yet. Is it true that

$$U_{self} = \int Vdq$$
$$U_{int} = q_{1}V_{12} = q_{2}V_{21}$$

I would be grateful if someone could help and/or offer links to specific references.

Thanks and cheers
Vivek

 

[dextercioby]

Hello

I have a query (Classical Electrodynamics) regarding Self Energy and Interaction Energy. I understand that the integral definition

$$U_{tot} = \frac{1}{2}\int_{all space}k\epsilon_{0}E^2d\tau = \frac{1}{2}\int Vdq$$

represents the total electrostatic potential energy of a system and that this equals

$$q_{1}V_{12}$$ or $$q_{2}V_{21}$$ (for the discrete charge case).

What I want to know are the exact definitions of self energy and interaction energy of a system of charges (the discrete and continuous case) as I have not found convincing explanations in my textbook and on the internet yet. Is it true that

$$U_{self} = \int Vdq$$
$$U_{int} = q_{1}V_{12} = q_{2}V_{21}$$

I would be grateful if someone could help and/or offer links to specific references.

Thanks and cheers
Vivek

What is U_{self}??What is U_{tot}??

IIRC,that U_{tot}(in your weird notation) is the energy of the electrostatic field created by a charge distribution and is called the SELF ENERGY OF THE ELECTROSTATIC FIELD...So you got them all mixed up... :yuck:

As for the INTERACTION ENERGY BETWEEN A DISCRETE CHARGE DISTRIBUTION AND AN EXTERNAL ARBITRARY ELECTROSTATIC FIELD,the formula reads:

$$U_{int}=:\sum_{a}q_{a}\phi_{a}(\vec{r}_{a})$$

Daniel.

 

[maverick280857]

First of all dextercioby, thanks for your reply. Yes the notations are mixed up because they come out of a book which is unique in that it provides these (wrong) interpretations of energy.

As I understand now, there is just one relationship for electrostatic potential energy but once I came across a university site which said that the total electrostatic potential energy of a system of charges could be written as the sum of two terms: self inergy + interaction energy. Now I didn't quite understand this concept (and still don't because I can't find this distinction in any electrodynamics text like Cheng or Griffths) then but it eventually featured in my textbook (an Indian book but it is not a proper electrodynamics text and is for exam prep primarily).

It is disappointing that I cannot quote that site here as I have forgotten its link but I get questions like "find the interaction energy of a system of 3 charges +q, +2q, +3q placed at the corners of an equilateral triangle of side l. Also find the total electrostatic potential energy of the system." What do I do? I suppose I use the relationships for U_[total] mentioned in my post for the second part and for the first part in this case as well.

But what about the situation where there are two concentric spherical shells and the regions filled with vacuum (or air). A charge Q is placed at the center of the shell and a charge q' is given to the outer shell. Find the self energy of the system and the total electric potential energy of the system.??

Thanks and cheers
Vivek

 

[dextercioby]

First of all dextercioby, thanks for your reply. Yes the notations are mixed up because they come out of a book which is unique in that it provides these (wrong) interpretations of energy.

Ya,i hate bad books,too... :yuck:

As I understand now, there is just one relationship for electrostatic potential energy

Nope,there are 2...

but once I came across a university site which said that the total electrostatic potential energy of a system of charges could be written as the sum of two terms: self inergy + interaction energy.

What do you mean self-energy of charges?

Now I didn't quite understand this concept (...)

Which concept?

It is disappointing that I cannot quote that site here as I have forgotten its link

Shame,indeed...

but I get questions like "find the interaction energy of a system of 3 charges +q, +2q, +3q placed at the corners of an equilateral triangle of side l. Also find the total electrostatic potential energy of the system."

Okay...What's unclear about this one...??

What do I do? I suppose I use the relationships for U_[total] mentioned in my post for the second part and for the first part in this case as well.

Yap,u use the first one in a "friendly" form,in which only charges and distances should appear...Those are the energies charge-charge...Three terms.And use the second to find the charge-field energy (again three terms) and then add the 2 results...

But what about the situation where there are two concentric spherical shells and the regions filled with vacuum (or air). A charge Q is placed at the center of the shell and a charge q' is given to the outer shell. Find the self energy of the system and the total electric potential energy of the system.??

Apply Gauss theorem to find the field,then the potentials and then the charge-field interaction energy.The charge-charge interaction energy should not be too complicated...

 

[maverick280857]

Apply Gauss theorem to find the field,then the potentials and then the charge-field interaction energy.The charge-charge interaction energy should not be too complicated...

I am sorry but I don't understand what you mean by interaction energy. Can you give me a mathematical form for it so that I understand you better.

Thanks and cheers
vivek

 

[dextercioby]

1.The electrostatic potential energy of a system of point charges in vacuum is:
$$U=\frac{1}{8\pi\epsilon_{0}}\sum_{a\neq b} \frac{q_{a}q_{b}}{|\vec{r}_{a}-\vec{r}_{b}|}$$

These charges generate the electrostatic field...
And this U is called SELF ENERGY OF THE ELECTROSTATIC FIELD CREATED BY A DISCRETE DISTRIBUTION OF CHARGE...

Daniel.

 

[dextercioby]

2.The energy of interaction between a discrete system of charges and an arbitrary electrostatic field (created NOT by the system involved) is the hamiltonian of interaction between the particles and the field,namely:

$$U_{int}=H_{int}=-L_{int}=\sum_{a=1}^{n} q_{a}\phi_{a}(\vec{r}_{a})$$

Can u see the difference between the 2 notions?

Daniel.

 

[maverick280857]

Well not really because I am not familiar with Lagrangian formulations. I am not even in college by the way :-). So I would be grateful if you could offer me a simpler explanation though I am willing to understand what your expressions mean (you will however have to condescend to my level for that).

Cheers
vivek

 

[dextercioby]

Well not really because I am not familiar with Lagrangian formulations. I am not even in college by the way :-). So I would be grateful if you could offer me a simpler explanation though I am willing to understand what your expressions mean (you will however have to condescend to my level for that).

Cheers
vivek

OOPS,sorry,that changes everything...So you're still in HS.Well,in HS,you're only presented with a nonrigurous proof of the first formula (post No.6,if I'm not mistaking) and therefore you don't need to know the second part with lagrangians and hamiltonians...

Don't worry...Keep learning.But i think this part of classical electrodynamics is,mathematically speaking,a bit way over your head...No mean to offend,but there's a lotta maths and physics you need to master in order to comprehend where these things come from...

I wish you had told me from the start...

Daniel.

PS.I wouldn't have gotten into complicated stuff...

 

[maverick280857]

Hello Daniel (if I may address you by name)

Thanks for your replies. Well I am not exactly in school but am on the verge of entering college, so I have to know a few advanced things which I can comprehend with the limited knowledge of physics and mathematics I have. I looked up some books on electrodynamics and came to the conclusion that the original textbook wherefrom I quoted the two "messy" formulae was not correct. As I now understand the ideas are not totally similar for discrete and continuous charge distributions and indiscriminate usage of the terms "self energy" and "interaction energy" is misleading (and obviously distressing).

In our part of the world, we are not given a good electrostatics tutorial in schools because of several reasons. Yet it is useful to know where the relationships break down, where they are applicable and where they are not. (The past 24 hours have been a great learning experience.) And the exams we prepare for, (college entrance) certainly do not tolerate (thankfully) formula crammers.

Thank you so much.

Cheers
Vivek