Poynting’s theorem -- Check the relation of the energy balance

  • #1
Lambda96
233
77
Homework Statement
Check the relation of the energy balance
Relevant Equations
none
Hi

I have a problem with task c)


Bildschirmfoto 2024-11-18 um 21.52.38.png

For the Poynting vector and the energy density I got the following:




Then I calculated the following



and the i used the hint



Unfortunately I don't know what I can do with it, the results are quite a mess or have I miscalculated and also in the task part b) ?
 
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  • #2
The relation

might easen your calculation. In your calculation of



I find is strange in dimension.
 
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  • #3
Thank you anuttarasammyak for your help and the tip 👍


With the term now looks like this:



Unfortunately, I still can't get any further with this term and I have recalculated and again get the term for
 
  • #4
Lambda96 said:
I have recalculated and again get the term for
You clearly have an algebra error. The term is dimensionally inconsistent because .
 
  • #5
The problem statement expresses the fields in Gaussian units. In these units the Poynting vector is and the energy density is .

Using the expressions for the fields given in the problem I find This is the same as your result except for differences in the constants. I could have made mistakes, so you can see whether or not you can get the same expression. The reason for writing the last factor as is to prepare for using the divergence identity given in the problem statement.

The problem statement says . Actually, this should be , where the right side is the two-dimensional delta function and is the two-dimensional position vector for points in a plane perpendicular to the z-axis. The dimension of is inverse length; whereas, the dimension of is inverse area.

is useful for expressing the current density for this problem:
 
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  • #6
Very gross understanding towards poynting, what is j and E vector?
 
  • #7
graphking said:
Very gross understanding towards poynting, what is j and E vector?
E is the electric field.

j is the current density vector. For current in a wire, j is a vector with magnitude equal to the current per unit cross-sectional area of the wire and j points in the direction of the current. For the limiting case of an infinitely thin wire, j can be expressed in terms of the current and a Dirac delta function. See the last equation in post #5.
 
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  • #8
Thanks for your help and explanation TSny 👍👍

I have repeated the calculation again with Gaussian units and get the same result for the Poynting vector
:smile:

You were right, instead of it should be . But my lecturer wrote it as
 
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  • #9
Lambda96 said:
my lecturer wrote it as
Good.
 

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