17th of September, 2008

- A conducting sphere is kept at potential ${V}_{0}$.
Show that if we embed the sphere in a dielectric that occupies the region
$z<0$
(figure 1, top), the potential $V\left(r\right)$
will stay exactly the same everywhere in space as in the absence of the
dielectric! What about configurations (a) and (b) (figure 1, bottom)?
How will $V\left(r\right)$ change if not the potential of the sphere, but its total charge $Q$ is kept at a fixed value?

(Note: Based on 4.35 and 4.36 in Griffiths’s book.)

- A dielectric slab is partially inserted between between planar capacitor plates. Find the force acting on the slab of dielectric!
- A small cavity is carved inside a large piece of dielectric
material. Suppose that there is a uniform electric field,
${\overrightarrow{E}}_{0}$,
in the dielectric. Find the electric field inside the cavity in the following
cases:
- The cavity is spherical.
- A needle-shaped (long and thin) cavity, parallel to the field.
- A wafer-shaped cavity, perpendicular to the field.

(Problem 4.16 in Griffiths’s book.)

- An emulsion is a mechanical mixture of two unblendable fluids. Consider an emulsion where droplets of a fluid of permittivity ${\epsilon}_{1}$ are dispersed in a fluid of permittivity ${\epsilon}_{0}$. The droplets amount to a fraction $\beta $ of the total volume of the emulsion. Find the net permittivity of the emulsion!
- We have a magnetic field whose magnitude is linearly
varying in a direction perpendicular to the field lines:
$B=({B}_{0}+kx)\widehat{z}$.
- Qualitatively describe the motion of a charged particle in this field.
- Find an approximate formula for the drift velocity. Assume that in the region where the particle is moving $|B-{B}_{0}|\ll {B}_{0}$.