17th of September, 2008

1. 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.)

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   Figure 1: A conducting sphere embedded in a dielectric in various ways.

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2. A dielectric slab is partially inserted between between planar capacitor plates. Find the force acting on the slab of dielectric!
3. 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:
   1. The cavity is spherical.
   2. A needle-shaped (long and thin) cavity, parallel to the field.
   3. A wafer-shaped cavity, perpendicular to the field.

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   (Problem 4.16 in Griffiths’s book.)

4. An emulsion is a mechanical mixture of two unblendable fluids. Consider an emulsion where droplets of a fluid of permittivity ε₁ are dispersed in a fluid of permittivity ε₀. The droplets amount to a fraction β of the total volume of the emulsion. Find the net permittivity of the emulsion!
5. We have a magnetic field whose magnitude is linearly varying in a direction perpendicular to the field lines: B = (B₀ + kx).
   1. Qualitatively describe the motion of a charged particle in this field.
   2. Find an approximate formula for the drift velocity. Assume that in the region where the particle is moving |B - B₀|≪ B₀.

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