Problem 1:

A neutral Na atom has eleven electrons.  Ten electrons are removed to form a Na10+ ion.  Calculate the frequency in Hz and the wavelength in nm of the n = 2 to n = 1 transition of the Na10+ ion.


Problem 2:

Assume a uniform electric field E = Ek exists in some region of space that contains no magnetic field.  Let the magnetic vector potential A(r,t)  be zero.  Write down an electrostatic potential Φ(r,t) for this field.
Define a gauge transformation, so that Φ'(r,t) = 0.  What is the corresponding vector potential A'?


Problem 3:

We connect a real inductor to a source of alternating voltage with an amplitude of 10 V and a frequency of 60 Hz.  A current with an amplitude of 16 mA flows through this circuit.  The amplitude of the current drops to 12 mA when we connect a resistor with a resistance of 500 Ω in series with the inductor.  Find the inductance L and the resistance RL of the inductor.


Problem 4:

Consider a two-state system.  The eigenbasis of the Hamiltonian H0 is {|1>, |2>}, and in that basis the matrix of the H0 is

H0 = ħ 

  ω0   0   
   0   ω0   




Let H = H0 + W be the Hamiltonian of the system when a small perturbation is applied, and let the matrix of W in the  {|1>, |2>} basis be

W = ħ 

   0   ε1   
   ε2   0   




Find the eigenvalues of H to first order using stationary perturbation theory, and find the corresponding zeroth order normalized eigenstates.


Problem 5:

A rectangular trough extends infinitely along the z direction, and has a cross section as shown in the figure. 


All the faces are grounded, except for the top one, which is held at a potential V(x) = V0.  Find the potential inside the trough.


Problem: 5

A very large sheet of electric charge densities +σ lies parallel to and a small distance d1 in front of a very large grounded conducting plane.  A point charge +q is placed a distance d2 < d1 in front of the plane, very far from the edges. The plane and the sheet are held in place.  Find the external force required to also keep the point charge at rest.