Assignment 10

Problem 1:

imageA 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 V0.
(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 vmax of the dielectric?  Express your answers in terms of ε0, ε, V0, 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 ε0, ε, V0, w, d and m.

(c)  Find the ratio vmax (case a) to vmax (case b) in terms of ε0 and ε.

Problem 2:

imageA "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.

imageProblem 3:

In the circuit shown in the figure R1 = 6 Ω, R2 = 4 Ω, and R3 = 2Ω .
(a)  Find the currents I1, through resistor R1, I2, through resistor R2, and I3, through resistor R3, 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 I1 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.

imageProblem 4:

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).

imageProblem 5:

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.