A small marble of charge q and mass m can slide without friction along a long, thin vertical rod passing through the center of a horizontal conducting ring of radius R, mounted on an insulating support. What is the magnitude of the minimum charge Q placed on the ring that would allow the marble to oscillate along the rod?

An electron at a distance d = 1 mm is projected parallel to a grounded
perfectly conducting sheet with an energy of 100 electron volts. Let the
grounded conducting sheet lie in the xy-plane and the initial velocity of the
electron point in the x-direction. Let z(0) = d.

(a) Find the distance that the electron travels until it hits the plate.
Neglect the force of gravity.

(b) Find the magnitude and direction of a magnetic field parallel to the
surface of the plate and perpendicular to the electron velocity that keeps the
electron from hitting the plate.

Give numerical answers.

Hint: z = dsin^{2}θ may be a useful change of variable. ∫sin^{2}θdθ
= θ/2 - sin(2θ)/4.

Three small metal charged balls have equal charges q and masses m, 4m and m.

The balls are connected by light non-conducting strings of length d each and
placed on a horizontal non-conducting frictionless table. Initially, the balls
are at rest and form a straight line as shown. Then, a quick horizontal
push
gives the central ball a speed v directed perpendicular to the strings
connecting the balls.

(a) What is the total energy of the system?

(a) What is the kinetic energy associated with the motion of the CM?

(c) What is the minimum subsequent distance between the balls of mass m?

A capacitor composed of two parallel infinite conducting sheets separated by
a distance d is connected to a battery. The lower plate is maintained at some
potential V_{1} and the upper plate is maintained at some potential V.
A small hemispherical boss of radius a << d is introduced on the lower plate.
State the boundary conditions for this problem. (Hint: Consider the limit as
the distance between the plates becomes large.) Find the potential between the
plates and the surface charge density on the plates.

An aperture of radius a in a thin plate separates a region in which the electric
field is E_{1} from a region in which it is E_{2}. E_{1}
and E_{2} are perpendicular to the plate. Let E_{1} > E_{2}. A beam of particles of charge q
and energy qV_{0} comes to a focus at a distance z_{1} in
front of the plate, goes through the aperture, and comes to a second focus at a distance
z_{2}
behind the aperture.

Show that 1/z_{1} + 1/z_{2} = (E_{1}
- E_{2})/4V_{0}.

Assume
V_{0} >> E_{1}z_{1}, E_{2}z_{2}; a
<< z_{1}, z_{2}.

Hint: Use Gauss' law to estimate the radial impulse given to a particle by the field near
the aperture.