Understanding the Derivation of Energy in Dielectric Systems

[almarpa]

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?

 

[USeptim]

What means the $\bigtriangleup$? It looks like it only works on the first term of D·E.

 

[zoki85]

What means the $\bigtriangleup$? It looks like it only works on the first term of D·E.

Laplace operator

 

[USeptim]

Thanks zoki85, I used to see for the Laplacian $\bigtriangledown^2$.

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 $\epsilon$.

 

[athosanian]

▽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

 

[almarpa]

It is not the laplacian operator. It represents an incremental variation of the quantity this symbol goes with.

 

[almarpa]

It is not the laplace operator. It is just an increment.

 

[Meir Achuz]

▽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.

 

[Meir Achuz]

$\Delta$ 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
$\Delta({\bf E\cdot D)=E\cdot(\Delta D)+(\Delta E)\cdot D}$.

 

[zoki85]

$\Delta$ 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

 

[Meir Achuz]

$\Delta$ is commonly used, with the limit $d=lim\Delta\rightarrow 0$.

 

[almarpa]

Sorry, but I still do not get it.

What happens with the 1/2 term? It vanishes, but I can not see why.

 

[Meir Achuz]

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.