Assignment 3

Problem 1:

A spherical, shiny holiday decoration ball is acting as a convex mirror.  The sphere has a radius of 4 cm.  Your eye is 10 cm from the mirror.  How much bigger or smaller is the image of your eye than the actual size of your eye?  Is the image real or virtual, upright or inverted?

Problem 2:

For a symmetrical prism (one in which the apex angle lies at the top of an isosceles triangle), the total deviation angle ϕ of a light ray is minimized when the ray inside the prism travels parallel to the prismís base.  
Assume that a beam of light passes through a glass equilateral prism with refractive index 1.5.  The prism is in air and is mounted on a rotation stage, as shown in the figure.  When the prism is rotated, the angle by which the beam is deviated changes.  What is the minimum angle ϕ bywhich the beam is deflected?

Problem 3:

The region 0 ≤ z ≤ z0 is filled with a dielectric material of permittivity ε = 4ε0 and permeability μ = μ0.  A linearly polarized wave of amplitude E = E0i and angular frequency ω is incident normally on the interface at z = 0 from the region z < 0.  Show that the ratio of the reflected intensity to the incident intensity in the z < 0 region is
[1 + (16/9)csc2(2ωz0/c)]-1.

Problem 4:

An electromagnetic wave with circular frequency ω propagates in a medium of dielectric constant ε, magnetic permeability μ, and conductivity σ.
(a)  Show that there is a plane wave solution in which the amplitude of the E and B fields decreases exponentially along the direction of propagation, and find the characteristic decay length.
(b)  Simplify by assuming that σ is great enough so that σ/(εω) >> 1.

Problem 5:

In a double-slit experiment with 500 nm light, the slits each have a width of 0.1 mm.
(a)  If the interference fringes are 5 mm apart on a screen which is 4 m from the slits, determine the separation of the slits.
(b)  What is the distance from the center of the pattern to the first diffraction minimum on one side of the pattern?
(c)  How many interference fringes will be seen within the central maximum in the diffraction pattern?  Draw a sketch of the pattern.