Polarazation by an external electric field

[shoestring]

(All media are supposed to isotropic.)

A sphere placed in a homogeneous electric field will get a homogeneous polarization parallel to the external field, and this will be reflected in an induced surface charge proportional to cos(a) with a being the angle measured from the axis pointing in the same direction as the polarizing field, because the surface charge can be calculated as the scalar product of the polarization vector P and the normal unit vector n at the surface.

Now let's assume it's a dielectric cube that is placed in the homogeneous electric field E, with E being perpendicular to two (opposite) faces of the cube.

Will the resulting E field inside the cube still be homogeneous and parallel to the homogeneous initial external field, as was the case for the sphere, or will it perhaps get a more complex shape? What happens for example near the corners and edges of the cube?

The reason I'm asking is that it seems to me that even if we're dealing with a normal isotropic and linear medium, the resulting polarization P and electric field E inside will not necessarily be parallel to each other, as is often assumed.

If the polarization inside the cube is uniform, then the resulting E inside the cube won't be. Conversely, if the resulting electric field is uniform inside the cube, then the polarization can't be.

What are your thought on this? Will the often assumed proportionality between E and the displacement D still hold?

 

[yungman]

(All media are supposed to isotropic.)

A sphere placed in a homogeneous electric field will get a homogeneous polarization parallel to the external field, and this will be reflected in an induced surface charge proportional to cos(a) with a being the angle measured from the axis pointing in the same direction as the polarizing field, because the surface charge can be calculated as the scalar product of the polarization vector P and the normal unit vector n at the surface.

Now let's assume it's a dielectric cube that is placed in the homogeneous electric field E, with E being perpendicular to two (opposite) faces of the cube.

Will the resulting E field inside the cube still be homogeneous and parallel to the homogeneous initial external field, as was the case for the sphere, or will it perhaps get a more complex shape? What happens for example near the corners and edges of the cube?

The reason I'm asking is that it seems to me that even if we're dealing with a normal isotropic and linear medium, the resulting polarization P and electric field E inside will not necessarily be parallel to each other, as is often assumed.

If the polarization inside the cube is uniform, then the resulting E inside the cube won't be. Conversely, if the resulting electric field is uniform inside the cube, then the polarization can't be.

What are your thought on this? Will the often assumed proportionality between E and the displacement D still hold?

I am no expert on electro static. This is my guess

If you place a dielectric cube in a uniform E field ( I assume that is what you mean homogeneous ) with two surface perpendicular to the E field, the induced E field is parallel but opposite direction as the external E field just by the boundary condition that the tangential E is zero and you only have normal E. If the shape is not even or the cube do not have surface perpendicular to the external E field, then the internal induced E field is not only not parallel to the external E field, the induced E field inside is not uniform. You have to apply E boundary to all the different surfaces. Then do the summation...get complicated.

I see it as boundary condition problem. Like normal D is continuous cross boundary if there is no surface charge on the boundary surface...yada yada.

 

[shoestring]

Yes, you're right, "uniform" is a better word for the E field.

I'm not sure if the tangential component of E has to be zero when dealing with a dielectric as it must be for a conductor. Had it been a conductor, the internal E would be zero, which means that the the induced E cancels the background E perfectly inside, no matter what shape the conductor is. In the case of a conductor, the induced field is no doubt uniform inside, because it has exactly the same strength as and opposite direction to the background field.

I wonder if what happens to a dielectric is similar to what happens to a conductor, with the only difference that the background field is only partially compensated for. If that's the case, the internal induced field has to be uniform, and the resulting internal field too. In the "limit" of a conductor, the internal field would simply be weakened down to zero. - Is it possible to think of a conductor as the limit of a dielectric like that, when considering polarization by a background electric field?

 

[yungman]

Yes, you're right, "uniform" is a better word for the E field.

I'm not sure if the tangential component of E has to be zero when dealing with a dielectric as it must be for a conductor. Had it been a conductor, the internal E would be zero, which means that the the induced E cancels the background E perfectly inside, no matter what shape the conductor is. In the case of a conductor, the induced field is no doubt uniform inside, because it has exactly the same strength as and opposite direction to the background field.
I only meant in the case you mentioned that two of the surface of the cube is perpendicular to the applied E field. If you turn the cube a little, there will be tangential component.
I wonder if what happens to a dielectric is similar to what happens to a conductor, with the only difference that the background field is only partially compensated for. If that's the case, the internal induced field has to be uniform, and the resulting internal field too. In the "limit" of a conductor, the internal field would simply be weakened down to zero. - Is it possible to think of a conductor as the limit of a dielectric like that, when considering polarization by a background electric field?

Actually you make me consider the electric field inside the dielectric is not uniform. It is only uniform if it is simple shape like the cube where even if you turn the cube and have tangential components, the refraction of the lines into the dielectric is still parallel. BUT if it is a more complex surface. the uniform external field would have different angle of refraction when enter into the dielectric and they start to sum and subtract each other and it would not be uniform inside.

 

[Meir Achuz]

The E field inside the cube is not uniform and is quite complicated.
E at either end of the cube must equal E_0 (the outside field) divided by epsilon, because of the boundary condition on D_normal.
E at the side of the cube must equal E_0 because of the boundary condition on E_tangential.

 

[shoestring]

Here's something I found online, in Becker, Electromagnetic Fields and Interactions, at google books. At the bottom of page 102 and top of page 103 it says:

§30. The homogenously polarized ellipsoid

If we place a body of permittivity $$\epsilon$$= 1 + 4$$\pi$$$$\chi$$ in a uniform electric field E0, we do not (as we have seen in the example of the sphere) obtain in general a polarization P = $$\chi$$E0, but rather the polarization P = $$\chi$$Ei - for the polatrization has just the consequence that the field Ei obtaining within the material is essentially different from E0. However, the equation P = $$\chi$$Ei has something to say about the effect of the field existing only at the particular place considered. For an arbitrary shape of the body to be polarized, the additional field E' = Ei - E0 produced within the material by P itself becomes a complicated function of position, and thus a homogenous E0 will not in general yield a homogenous P. A homogenous P results only when the body has the shape of an ellipsoid.

http://books.google.com/books?id=x5...ok_result&ct=result&resnum=1&ved=0CB0Q6AEwAA"

An ellipsoid has the equation (x/a)^2 + (y/b)^2 + (z/c)^2 = 1. The sphere is a special case with a=b=c, a plane can be seen as an extreme case of an ellipsoid with b=c=infinity, and in a similar way a rod with either a circular or elliptical cross section can be seen as a an ellipsoid with c=infinity.

Those are the cases that get a uniform polarization in a uniform background field. In all other cases, if the text is to be trusted, the resulting polarization of the dielectric is not uniform.

Will the complexity of the polarization of a dielectric cube mean that there can be a nonzero volume charge density inside the cube, or is there some condition that prevents that from happening?

 

[shoestring]

Is it fair to assume that the comformal mapping of a circle onto a square shown http://www.flickr.com/photos/sbprzd/362529354/" gives an idea of how the electric field might look inside an infinite rod with a square cross section? Assuming that the uniform polarizing field is perpendicular to one (well, two) of the faces. Perhaps that gives a hint of how the field inside the polarized cube will look like. It won't be the same, but it might share some of its characteristics.

I guess the questions posed in the first post have been answered. Neither the polarization nor the electric field inside the dielectric cube will be uniform, but the internal polarization and the internal field will be parallel to each other, and there will still be a proportionality between E and D.

What confused me was that I've seen illustrations in textbooks showing an unevenly shaped "arbitrary" object being uniformly polarized by a background field. Detailed calculations are typically made only for a plane and/or sphere, but there's no mentioning of how complex the situation gets for more general shapes, even for something as symmetric as a cube. I wish there were more mentioning of the fact that only ellipsoids, including the special or extreme cases of ellipsoids like for example spheres and planes, are uniformly polarized by a uniform background field. It's something I've never heard of before, but perhaps I simply haven't read the right books.

 

[yungman]

The E field inside the cube is not uniform and is quite complicated.
E at either end of the cube must equal E_0 (the outside field) divided by epsilon, because of the boundary condition on D_normal.
E at the side of the cube must equal E_0 because of the boundary condition on E_tangential.

As I explained, if you position a perfect cube in the field where the field is absolute normal to the surface, then the field inside the cube is uniform and equal to E in the external divided by epsilon. This is the only limited can. Any other position will cause uneven distribution because of e_tangential exist as you said.

 

[shoestring]

As I explained, if you position a perfect cube in the field where the field is absolute normal to the surface, then the field inside the cube is uniform and equal to E in the external divided by epsilon. This is the only limited can. Any other position will cause uneven distribution because of e_tangential exist as you said.

But if the field inside the cube is uniform, then the polarization inside the cube can't be, because a uniform polarization will lead to uniform surface charges on the two faces perpendicular to the field, and these will make the interior field non-uniform.

This leads to a paradox, because it's expected that the interior field and polarization are aligned and proportional to each other, at least for a linear, isotropic and homogeneous substance. It turns out that only a non-uniform interior field, polarization and surface charge distribution can make the interior field and polarization everywhere aligned with each other. At least that's the way I now understand it.

It seems that ellipsoids (including spheres and infinite planes) are the only cases where the surface charge from a uniform polarization will not make the resulting internal electric field non-uniform.

 

[shoestring]

As I explained, if you position a perfect cube in the field where the field is absolute normal to the surface, then the field inside the cube is uniform and equal to E in the external divided by epsilon. This is the only limited can. Any other position will cause uneven distribution because of e_tangential exist as you said.

Not sure if I follow. Any object placed in the initially uniform external field will distort it, most noticeably near the object. I didn't mean that the resulting external field had to be uniform. Once the external field is distorted by placing the dielectric cube in it, there'll be very few points where the resulting external field is normal to it (probably just two points, one at the middle of each face perpendicular to the initial field). I guess the resulting external field will make all kinds of angles, the whole range between 0 and pi, to the normal of the surface of the cube.

Jamming the cube between two capacitor plates would probably result in a fairly uniform internal field, because that would make the potential on each face touching a capacitor plate constant. But placing the dielectric cube in a large uniform field, even if the cube is aligned with the (initial) field, can't guarantee any of the two faces perpendicular to the field to have a constant potential, so in this case the internal field will probably deviate more from being uniform.

 

[yungman]

I went back and read your post, I think I was wrong in assuming the dielectric is linear homogeneiou and isotropic. I see too many "homogeneous" and got it wrong. What I said only apply to homogeneous, linear isotropic material that there is no volume charge density inside, all polarized charge present on the surface.

I am not expert, I jumped in only because I was thinking about the boundary condition of homogeneous isotropic dielectric that there would be no surface charge at the interface of the dielectric. D has only normal component and is continuous from space into the dielectric if the cube is absolutely normal. The external field and the surface that is perfectly parallel should theoretically have no interaction. So the field just enter from one side of the cube and exist from the other side.

Any more than that, I better no talk with you about it. Sorry.

 

[shoestring]

No need to apologize! I'm glad you're taking an interest. :)

It's not an easy problem, because dielectric cubes placed in a uniform field aren't much discussed in the books, for what I know. I tried to google for it, but haven't find any picture based on numerical calculations yet, and it's probably too difficult to do analytically.