Assignment 2

Problem 1:

The ground state of the neutral lithium atom is doubly degenerate.  The first excited state is 6-fold degenerate, and it is at an energy 1.2 eV above the ground level.
(a)  In the outer atmosphere of the sun, which is at a temperature of 6000 K, what fraction of the neutral lithium is in the first excited level? 
Since all the other levels of Li are at much higher energy, it is safe to assume that they are not significantly occupied.
(b)  Find the average energy of a lithium atom at temperature T.  (Again, consider only the ground and first excited level.)
(c)  Find the contribution of these levels to the specific heat per mole, CV, and sketch CV as a function of 1/T.  Discuss the curve.

Problem 2:

A glowing incandescent lamp filament may be regarded as a black body.  If the filament is heated to 2400 K with a power of 100 W, find the surface area of the filament in cm2.

Problem 3:

Assume you have a sufficiently large concave spherical mirror with focal length F and outer diameter F.   Assume the reflectance of the coated aluminum mirror is 88%.  Is it possible to melt a small iron sphere of radius R by focusing the sun's rays with this mirror?   Let R be the radius of the image of the sun, formed by the mirror.  The melting point of iron is 1812 K, and the surface temperature of the sun is assumed to be TS = 5830 K. 

Problem 4:

(a)  Write down the set of Maxwell's equations in the form that applies to static fields.
(b)  Use the continuity equation to adapt Maxwell's ×B equation to dynamic fields.  (This can be done by using a term that Maxwell referred to as "displacement current".)
(c)  Write down the dynamical form of Maxwell's equations.
(d)  Introduce the vector and scalar potentials and show that this leads to two wave equations in these potentials.  (The Lorentz gauge may be used to answer this part.)
(e)  Define the Coulomb gauge and elaborate on some Coulomb gauge details.

Problem 5:

Assume magnetic charges exist and Maxwell's equations are of the form
E = ρ/ε0,   B = μ0ρm,   -×E = μ0jm + ∂B/∂t,   ∇×B = μ0j + μ0ε0E/∂t.
Assume a magnetic monopole of magnetic charge qm is located at the origin, and an electric charge qe is placed on the z-axis at a distance R from it.
(a)  Write down expressions for the electric field E(r) and the magnetic field B(r).  Make a sketch.
(b)  Write down expressions for the momentum density g(r) and angular momentum density (r) of the electromagnetic field.
(c)  Show that there is electromagnetic angular momentum Lz about the z-axis and derive an expression for it.  Show that Lz is independent of R.
Useful vector identity:  (a)n = (1/r)[a - n(an)].   Here n is r/r is the unit radial vector.