Boundary condition of EM field

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Discussion Overview

The discussion revolves around the boundary conditions of electromagnetic fields, specifically focusing on the continuity of the tangential components of the electric displacement field \(\vec{D}\) and polarization \(\vec{P}\) at the interface of two regions. Participants explore the implications of differing dielectric constants on these conditions and the relationship between \(\vec{D}\) and the electric field \(\vec{E}\).

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the tangential components of \(\vec{D}\) and \(\vec{P}\) are continuous across the boundary, similar to the electric field \(\vec{E}\).
  • Another participant raises the issue of whether the two regions have the same dielectric constant, suggesting that this affects the continuity of \(\vec{D}\) and \(\vec{P}\).
  • It is noted that if the dielectric constants differ, the continuity of \(\vec{D}\) may not hold, even if \(\vec{E}\) is continuous.
  • A participant references the relationship between \(\vec{D}\) and \(\vec{E}\) and implies that changes in dielectric constant influence the behavior of \(\vec{D}\).
  • There is a mention of the polarization charge density \(\sigma_p\) and its relationship to bound charges, raising questions about terminology consistency across texts.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the continuity of \(\vec{D}\) and \(\vec{P}\) at the boundary, particularly in relation to differing dielectric constants. There is no consensus on whether these components are continuous under the stated conditions.

Contextual Notes

The discussion highlights the dependence on dielectric constants and the definitions of terms like bound charges and polarized charge density, which may vary across different texts. There are unresolved questions regarding the boundary conditions for \(\vec{H}\) as well.

KFC
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On the boundary (surface) of two regions, the tangential components of electric fields on above and below surface are continuous. I wonder if it is also true for displacement [tex]\vec{D}[/tex] and polarization [tex]\vec{P}[/tex]? That is, can I say:
the tangential component of [tex]\vec{D}[/tex] or [tex]\vec{P}[/tex] on above and below surface are continuous?

For magnetic field, the statement of the magnetic field about [tex]\vec{B}[/tex] is:

[tex](\vec{B}_{above} - \vec{B}_{below} )\cdot\hat{n} = 0[/tex]
and
[tex](\vec{B}_{above} - \vec{B}_{below} )\times \hat{n} = \mu_0\vec{K}[/tex]

I wonder if [tex]\vec{K}[/tex] means the free current surface density? What is the boundary conditions for [tex]\vec{H}[/tex]?
 
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Do the two regions have the same dielectric constant? Think about the formula that relates D and E...
 
Xezlec said:
Do the two regions have the same dielectric constant? Think about the formula that relates D and E...

That is the question. In the text, it said the tangential components of the electric fields on the boundary are continuous. But it doesn't tell if the tangential components of the displacement or polarization are also continuous or not. So if the dielectric constants in these two regions not the same, does it mean they will not be continuous even along the tangential direction?

By the way, in some text, it reads

[tex](\vec{P}_2-\vec{P}_1)\cdot\hat{n} = -\sigma_p[/tex]

and [tex]\sigma_p[/tex] is what we call the density of polarized charges. I wonder if this is the same name as bound charges which is used in other text?
 
Last edited:
KFC said:
That is the question. In the text, it said the tangential components of the electric fields on the boundary are continuous. But it doesn't tell if the tangential components of the displacement or polarization are also continuous or not. So if the dielectric constants in these two regions not the same, does it mean they will not be continuous even along the tangential direction?

I was just saying that by looking at the formula that relates D and E, you will see the answer to that question. The dielectric constant is the constant of proportionality between D and E, so if E is continuous, but the dielectric constant changes, what is going to happen to D? See what I mean?
 

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