Problem 1:

A beam of monochromatic light with a wavelength of 500 nm is directed through an absorber having 5 equally narrow slits separated by 20 mm between adjacent slits.  The resulting diffraction pattern is observed on a screen that is perpendicular to the direction of light and 5 m from the slits.  The intensity of the central maximum is 1.3 W/m².
(a)  What are the distances from the central maximum to the first and second principal maxima on the screen?
(b)  What will be the intensity of the central maximum if there are only 4 equally narrow slits (of the same width as in part a), separated by 20 mm between adjacent slits? 

Problem 2:

(a)  Determine the speed of light in water, which has a dielectric constant of 1.78.
(b)  An electromagnetic wave in vacuum has an electric field amplitude of 220 V/m.  Calculate the amplitude of the corresponding magnetic field.
(c)  What is the energy of a photon of blue light (l = 450nm) and of a photon of red light (l = 700nm) in units of eV.

Problem 3:

It has been proposed to drive a spacecraft remotely by directing an intense electromagnetic beam to the craft.
(a)  Is it more efficient to absorb the beam or reflect it from the craft?
(b)  If the beam carries 10⁶ watts uniformly in an area of 5 m² and the beam is reflected, how long would it take for a 1000 kg spaceship to reach a final velocity of 10⁶ m/s, and how far would the craft travel in this time?

Problem 4:

Fresnel's reflectance formulas are given by

R = |sin(q_i-q_r)/sin(q_i+q_r)|²   or   R = |tan(q_i-q_r)/tan(q_i+q_r)|²

depending on the polarization of the incident wave with respect to the plane of incidence, and with q_i and q_r the angles of incidence and refraction, respectively.
(a)  Specify the boundary conditions across an interface.
(b)  For simplicity assume m₁ = m₂ = m₀, e₁ = e₀, e₂ = e, and derive the reflectance formulas.
(c)  For polarization parallel to the plane of incidence, find the Brewster angle for an index of refraction of n₂ = 1.50, and comment on the polarization of the reflected radiation for a wave of mixed polarization incident on a plane interface at the Brewster angle.

Problem 5:

A “tenuous” plasma consists of free electric charges of mass m and charge –e (where e is positive).  There are n charges per unit volume.  Assume that the density is uniform and that the interactions between the charges may be neglected.  Also assume that the charges can be treated classically.
A linearly-polarized electromagnetic wave of frequency w is incident on the plasma.
Let the electric field component of the plane wave be E = E₀exp(i(kx - wt)).
(a)  Solve the equation of motion for a single charge and find the current density j and the conductivity s of the plasma as a function of w.
(b)  Assume a plane wave of the form E = E₀exp(i(kx - wt)) propagate in the plasma with conductivity s.  Find the dispersion relation —the relation between k and w— for the electromagnetic wave in the plasma and the index of refraction as a function of w.

Problem 6:

Consider an electromagnetic traveling wave with electric and magnetic fields given by

E_x = E₀cos(kz – wt + f),

and

B_y = B₀cos(kz – wt + f).

Using Maxwell’s equations show that B₀ can be written in terms of E₀.

Problem 7:

Light of wavelength 640nm is shone on a double-slit apparatus and the interference pattern is observed on a screen.  When one of the slits is covered with a very thin sheet of plastic of index of refraction n = 1.6, the center point on the screen becomes dark.  What is the minimum thickness of the plastic?