Boundary value problems with dielectrics

Problem:

A conducting sphere of radius a carrying a charge q is submerged halfway into a non-conducting dielectric liquid of dielectric constant ε.  The other half is in air. 
Will the electric field be purely radial?  Explain.

image

Solution:


Azimuthal symmetry

Problem:

A dielectric sphere of radius a has a uniform isotropic permittivity, kε0, and is located in an electric field that is uniform at infinity.
(a)  Solve for the electric potential everywhere by boundary value methods.
(b)  Show that the electric field inside the sphere is uniform and find its value relative to the field E0 at infinity.

Solution:

Problem:

Consider a homogeneous dielectric ε, of infinite extent, in which there is a uniform field E0.  A spherical cavity of radius a is cut out of this dielectric.  Find:
(a)  Φ in the cavity and on its surface.
(b)  The polarization charge density σp on the walls.
(c)  The field outside the cavity.

Solution:

Problem:

(a)  A spherical dielectric of radius a has a uniform polarization P in the z-direction.
Show that the electric field inside the dielectric due to the polarization is given by E = -P/(3ε0).
(b)  A large capacitor in vacuum has parallel circular plates of radius R separated by a distance d (d << R).
The capacitor is charged to a potential difference V and disconnected from the source.  Find the energy stored in the capacitor.
(c)  Subsequently, a small sphere of radius a (a << d) and dielectric constant K is placed in the center of the capacitor between the plates.
Find the electric field inside the dielectric.
(d)  Will the capacitor have less, more, or the same energy than before the dielectric was inserted?  Explain!

Solution:

Problem:

Two very large metal plates are held a distance d apart, one at potential zero and the other at potential V0.  A metal sphere of radius a is sliced in two, and one hemisphere is placed on the grounded plate, so that its potential is likewise zero.  The radius of the sphere a is very small compare to the distance d between the plates, (a << d), so that you may assume that the electric field near the upper plate is constant.  If the region between the plates is filled with a weakly conducting material of conductivity σ, what current flows to the hemisphere?

image

Solution:


Cylindical coordinates:  Φ = Φ(ρ,φ)  (0 ≤ φ  ≤ 2π) independent of z

Problem:

A long cylindrical rod of radius a and dielectric constant k is placed in a uniform electric field E0 with its axis perpendicular to the field direction.
(a)  Find the potential Φ inside and outside the rod.
(b)  Find the electric field E inside and outside the rod.
(c)  Find the volume and surface polarization charge density.

Solution: