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 (r) 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 (r) 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,
_{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 ε
_{1}are dispersed in a fluid of permittivity ε_{0}. The droplets amount to a fraction β 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).- 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}|≪ B_{0}.