Problem 1:

A circuit below consists of four resistors connected by ideal wires. The circuit draws power P if an ideal battery is connected to either points A and D or points B and C.  If the same battery is connected to either points A and B or points C and D, the circuit draws power 2P.  What power would be drawn if the same battery is connected to points A and C?

Problem 2:

Two conducting spheres of radii a and b < a, respectively, are connected by a thin metal wire of negligible capacitance.  The centers of the two spheres are at a distance d >> a > b from each other.  A total net charge Q is located on the system.  Evaluate to zeroth order approximation, neglecting the induction effects on the surfaces of the two spheres,
(a)  how the charge Q is partitioned between the two spheres,
(b)  the value V of the electrostatic potential of the system (assuming zero potential at infinity) and the capacitance C = Q/V,
(c)  the electric field at the surface of each sphere, comparing the magnitudes and discussing the limit b --> 0.

Problem 3:

An electric dipole p is located at a distance r from a point charge q, as shown in the figure.  The angle between p and r is θ.  Evaluate the electrostatic force on the dipole.

Problem 4:

The volume between two concentric spherical surfaces of radii a and b (a < b) is filled with an inhomogeneous dielectric with permittivity
ε = ε₀/(1 + κr),
where ε₀ and κ are constants and r is the radial coordinate.
Thus D(r) = εE(r).  A charge Q is placed on the inner surface, while the outer surface is grounded.  Find:
(a)  The displacement D(r) and the field E(r) in the region a < r < b.
(b)  The capacitance of the device.
(c)  The volume polarization charge density in the region a < r <  b.
(d)  The surface polarization charge density at r = a and at r = b.

Problem 5:

A rectangular block of dielectric material with permittivity ε is partially inserted between two parallel plane conducting plates.  The plates are square, of side l, and are separated by a distance d, with d << l.  The dielectric is also square, of side l, and has a thickness of almost d.  A potential V₀ is applied across the plates by a battery.

(a)  When the dielectric has a length x inserted between the plates, calculate the force on the dielectric, including its direction.  If x is now changed, does this force depend on x?
(b)  When the dielectric has a length x inserted between the plates, the battery is disconnected.  Calculate the force on the dielectric, including its direction.  If x is now changed, does this force depend on x?