Boundary condition of EM field

AI Thread Summary
The discussion focuses on the boundary conditions of electromagnetic fields, specifically questioning the continuity of the tangential components of the displacement field \vec{D} and polarization \vec{P} across boundaries of different dielectric constants. It is established that while the tangential components of electric fields are continuous, the same may not hold for \vec{D} and \vec{P} if the dielectric constants differ. The relationship between \vec{D} and electric field \vec{E} is highlighted, indicating that a change in dielectric constant could disrupt the continuity of \vec{D}. Additionally, the role of surface current density \vec{K} in magnetic field conditions is examined, along with the concept of polarized charge density \sigma_p. The discussion ultimately emphasizes the need to consider dielectric properties when analyzing boundary conditions in electromagnetism.
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 \vec{D} and polarization \vec{P}? That is, can I say:
the tangential component of \vec{D} or \vec{P} on above and below surface are continuous?

For magnetic field, the statement of the magnetic field about \vec{B} is:

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

I wonder if \vec{K} means the free current surface density? What is the boundary conditions for \vec{H}?
 
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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

(\vec{P}_2-\vec{P}_1)\cdot\hat{n} = -\sigma_p

and \sigma_p 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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