Understanding the Derivation of Energy in Dielectric Systems

In summary: If ##\epsilon## is not constant, then the 1/2 term will vary with position, and the result will not be a vector but rather a scalar.
  • #1
almarpa
94
3
Hello all.

I have a doubt about the derivation of energy in dielectrics formula (Griffiths pages 191 - 192).

In a certain step of the formula derivation, we encounter the following operation:

(view formula below).

I do not undertand that operation.

Can someone help me?
Dibujo.JPG
 
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  • #2
What means the [itex]\bigtriangleup[/itex]? It looks like it only works on the first term of D·E.
 
  • #3
USeptim said:
What means the [itex]\bigtriangleup[/itex]? It looks like it only works on the first term of D·E.
Laplace operator
 
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  • #4
Thanks zoki85, I used to see for the Laplacian [itex]\bigtriangledown^2[/itex].

Almarpa. The link you have set it's a bit out of context. It only shows that you can conmute the lapace operator and the dot product since D and E differ only by a constant [itex]\epsilon[/itex].
 
  • #5
2(E.E)=▽2E.E+E.▽2E

you can think the ▽2 as a scalar, but it also is a differential operator like d/dx
 
  • #6
It is not the laplacian operator. It represents an incremental variation of the quantity this symbol goes with.
 
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  • #7
It is not the laplace operator. It is just an increment.
 
  • #8
athosanian said:
2(E.E)=▽2E.E+E.▽2E
you can think the ▽2 as a scalar, but it also is a differential operator like d/dx
##\nabla^2({\bf E\cdot E})## is not that simple.
 
  • #9
##\Delta## is just an infinitesimal variation. That step is only valid if ##\epsilon##
does not vary with position anywhere in space. Then the step just says
##\Delta({\bf E\cdot D)=E\cdot(\Delta D)+(\Delta E)\cdot D}##.
 
  • #10
Meir Achuz said:
##\Delta## is just an infinitesimal variation. That step is only valid if ##\epsilon##
does not vary with position anywhere in space. Then the step just says
##\Delta({\bf E\cdot D)=E\cdot(\Delta D)+(\Delta E)\cdot D}##.
If so, he should wrote it d , not Δ
 
  • #11
##\Delta## is commonly used, with the limit ##d=lim\Delta\rightarrow 0##.
 
  • #12
Sorry, but I still do not get it.

What happens with the 1/2 term? It vanishes, but I can not see why.
 
  • #13
almarpa said:
Sorry, but I still do not get it.
What happens with the 1/2 term? It vanishes, but I can not see why.
##\Delta({\bf E\cdot D)=E\cdot(\Delta D)+(\Delta E)\cdot D}=2\epsilon{\bf E\cdot E}##
if ##\epsilon## is constant.
 

1. What is the role of dielectric materials in energy systems?

Dielectric materials play a crucial role in energy systems by acting as insulators, preventing the flow of electric current. This allows for the efficient storage and transfer of energy within the system.

2. How does the dielectric constant affect energy storage in a dielectric system?

The dielectric constant, also known as relative permittivity, measures the ability of a material to store electric charge. A higher dielectric constant means the material can store more charge, resulting in higher energy storage capacity in a dielectric system.

3. What is dielectric breakdown and how does it impact energy systems?

Dielectric breakdown occurs when the insulating properties of a material fail and it begins to conduct electricity. This can lead to short circuits and damage to the energy system, reducing its efficiency and potentially causing safety hazards.

4. What are the different types of dielectric materials used in energy systems?

There are several types of dielectric materials used in energy systems, including ceramics, polymers, and gases. Each type has different properties that make them suitable for specific applications, such as high voltage insulation or energy storage.

5. How do temperature and frequency affect the performance of dielectric materials in energy systems?

Temperature can significantly impact the performance of dielectric materials in energy systems. Higher temperatures can cause materials to lose their insulating properties and may lead to dielectric breakdown. Similarly, the frequency of the electric field can affect the dielectric constant and energy storage capacity of a material.

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