Assignment 9, solutions

Problem 1:

imageA 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?

Problem 2:

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 = dsin2θ may be a useful change of variable.  ∫sin2θdθ = θ/2 - sin(2θ)/4.

Problem 3:

imageThree 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?

Problem 4:

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 V1 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.


Problem 5:

An aperture of radius a in a thin plate separates a region in which the electric field is E1 from a region in which it is E2.  E1 and E2 are perpendicular to the plate.  Let E1 > E2.  A beam of particles of charge q and energy qV0 comes to a focus at a distance z1 in front of the plate, goes through the aperture, and comes to a second focus at a distance z2 behind the aperture.
Show that 1/z1 + 1/z2 = (E1 - E2)/4V0.
Assume V0 >> E1z1, E2z2;  a << z1, z2.
Hint: Use Gauss' law to estimate the radial impulse given to a particle by the field near the aperture.