A parallel plate capacitor has square plates of width w and plate separation d.
A square dielectric also of width w and thickness d, with permittivity ε and
mass m, is inserted between the plates a distance x into the capacitor and held
there. The plates are connected to a battery with battery voltage V_{0}.

(a) Derive a formula for the force exerted on the dielectric as a function of x .
Neglect edge effects. Assume that the battery stays connected and the dielectric is released from rest
at x = w/2.

Describe its subsequent motion. What is the range of the dielectric's motion,
and within that range, what is the dielectric's speed as a function of position x?
What is the maximum speed v_{max} of the dielectric?
Express your answers in terms of ε_{0}, ε, V_{0}, w, d and m.

(Neglect friction, ohmic heating, radiation.)

(b) Now assume that the dielectric is released from rest at x = w/2 after the battery has been disconnected.
Derive a formula for the force exerted on the dielectric for w/2 < x < w. Neglect edge effects. What is the maximum speed of the dielectric?
Express your answer in terms of ε_{0}, ε, V_{0}, w, d and m.

(c) Find the ratio v_{max} (case a) to v_{max} (case b) in
terms of ε_{0} and ε.

A "dielectric" material consists of a number of brass spheres of diameter d, spaced 3d apart, in a regular lattice. Assuming that each sphere is influenced only by the imposed external electric field, i.e. neglecting the effect of the neighboring spheres or the redistribution of induced charges, find the dielectric constant k for this material.

In the circuit shown in the figure R_{1} = 6 Ω, R_{2} = 4 Ω,
and R_{3} = 2Ω .

(a) Find the currents I_{1}, through resistor R_{1}, I_{2},
through resistor R_{2}, and I_{3}, through resistor R_{3},
in the circuit
shown in the Figure.

(b) Indicate in what direction each current flows. Give the results in
terms of the junctions labeled in the circuit.

For example, you need to say whether I_{1} goes from b to c or from c to
b.

(c) Find the potential difference between junctions b and c. Clearly
indicate which junction has the higher potential.

The space between the plates of a parallel-plate capacitor (see figure) is
filled with two slabs of linear dielectric material. Each slab has
thickness s, so the total distance between the plates is 2s. Slab 1 has a
dielectric constant of 2, and slab 2 has a dielectric constant of 1.5. The
free charge density on the top plate is σ and on the bottom plate −σ.

(a) Find the electric displacement **D** in each slab.

(b) Find the electric field **E** in each slab.

(c) Find the polarization **P** in each slab.

(d) Find the potential difference between the plates.

(e) Find the location and amount of all the bound charge.

(f) Now that you know all the charge (free and bound), recalculate the
field in each slab, and compare with your answer to (b).

Five 1 Ω resistors are connected as shown in the figure. The resistance in the conducting wires (fully drawn lines) is negligible. Determine the resulting resistance R between A and B.