Assignment 9

Problem 1:

(a)  A positive charge Q is spread over an semicircular arc with radius R as shown.
(1)  What is the work required to bring in a charge -q from infinity to the center of the arc?
(2)  Calculate the magnitude and direction of the force on a charge –q at the center of the arc.

(b)  The potential of a uniformly charged spherical shell of radius R centered at the origin is
V(r) = q/(4πε0r) r ≥ R,   V(r) = q/(4πε0R) r < R, 
where q denotes the total charge of the sphere.
Calculate the energy that it requires to deposit a charge Q on an initially neutral conducting spherical shell with radius R.  Use two different approaches to come to the result.
(3)  Calculate the energy by incrementally adding a charge dq to the sphere.
(4)  Obtain the energy by considering the resulting electric field of the spherical shell.

Problem 2:

A model of the hydrogen atom was proposed before the advent of quantum mechanics, which consists of a single electron of mass m and an immobile uniform spherical distribution of positive charge with radius R.  Assume that the positive charge interacts with the electron via the usual Coulomb interaction but otherwise does not offer any resistance to the motion of the electron.
(a)  Explain why the electron’s equilibrium position is at the center of the positive charge.
(b)  Show that the electron will undergo simple harmonic motion if it is displaced a distance d < R away from the center of the positive charge.  Calculate its frequency of oscillation.
(c)  How big would the atom need to be in order to emit red light with a frequency of 4.57*1014 Hz?  Compare your answer with the radius of the hydrogen atom.
(d)  If the electron is displaced a distance d > R from the center, will it oscillate in position?  Will it undergo simple harmonic motion?  Explain!

Problem 3:

A charge array consists of two charges, each of magnitude +q, located on the z-axis at (0, 0, ±a).
(a)  Find the potential V(0,0,z) at an arbitrary point z > a; then expand V(0, 0,z) in a power series in z.
(b)  Using this series as a “boundary condition”, find the potential V(r,θ,φ) at an arbitrary location (r,θ,φ) with r > a.  An infinite series is acceptable.
(c)  Characterize the first three terms in 1/rn.

Problem 4:

Compute the force of attraction between a neutral metallic sphere of radius a and a point charge q positioned a distance r from the center of the sphere, where r > a.
How does the force of attraction behave as a function of r if r >> a?

Problem 5:

A spherical charge distribution is given by
ρ = ρ0(1 - r/a),    r < a,
ρ = 0,    r > a.
(a)  Calculate the total charge Q.
(b)  Find the electric field and potential for r > a.
(c)  Find the electric field and potential for r < a.
(d)  Find the electrostatic energy of this charge distribution.