Curl of Polarization in a bar electret

[Telis]

In Griffith's "Introduction to Electrodynamics" says that in a bar electret the curl of the polarization does not equal zero everywhere. Why is that ? Thanks in advance

 

[Charles Link]

What comes to mind very quickly is the corresponding magnetostatics problem with a cylinder of uniform magnetization $\vec{M}$. In that case, $\nabla \times \vec{M} =\vec{J}_m$ results in surface currents per unit length of $\vec{K}_m=\vec{M} \times \hat{n}$ on the surface of the cylinder. $\\$ In the simplest case of uniform $\vec{ P}$, if you take $\nabla \times \vec{P}$, you will get places where $\nabla \times \vec{P }$ diverges at parts of the surface/air interface. In general, if $\vec{P}$ is uniform, the derivative $\nabla \times \vec{P}$ vanishes, but this derivative can diverge when $\vec{P}$ undergoes a discontinuity such as at the surface/air interface.

 

[Telis]

Thank you, it is pretty clear for me now that at the surface/air interface will have $\nabla \times \vec{P} \neq 0$. Then Griffiths continues and says that "if the problem exhibits spherical, cylindrical, or plane symmetry then evidently in such cases $\nabla \times \vec{P} = 0$." So what does he mean about cylindrical symmetry and why a bar electret is not such a case ?

 

[Charles Link]

Griffith's seems to be talking about the polarization $\vec{P}$ in the material, and not considering edge effects. It is well known from E&M, that in the case of a dielectric sphere in a uniform electric field, the polarization inside the sphere is uniform. This also is the case for a cylinder that is transverse to the electric field. I'll see if I can find a couple of "links" to these very special cases. In general, for odd geometries, the resulting polarization $\vec{P}$ is not uniform when the dielectric object is placed in uniform electric field, and consequently, the derivative $\nabla \times \vec{P}$ does not vanish. Let me try to find a couple of "links" on the two special dielectric geometries: See: https://www.physicsforums.com/threads/electric-field-of-a-charged-dielectric-sphere.890319/ and https://www.physicsforums.com/threa...ormly-polarized-cylinder.941830/#post-5956930