Spring 2018 Homework Assignments
The electric field of a wave traveling in the +z-direction at time t=0 is given as E = i Eo sin(z/L) exp(-z2/d2) where i is the unit vector in the x-direction, and Eo, L, and d are constants. The wave is propagating in a medium with dispersion relation given by ω = α k2 where α is a constant. Find the expression for the electric fieild for t>0. Take L=1, d=5 and α=1 to obtain plots of the electric field time t=0 and at several points in time: When the wave has traveled to z=10 and z=20.
Find the modes corresponding to lowest 10 cutoff frequencies for a cylindrical waveguide of radius 1cm. Give the frequencies in units of GigaHertz.
Find the multipole expansion for the fields of a loop antenna of radius R on the x-y plane, centered at the origin. The current distribution in the antenna is of the form I(φ) = Io cos(φ). Find the power radiated and the radiation pattern for the lowest mode in the expansion. Make the necessary approximations to carry out the computations.
A spaceship is providing shuttle service between Earth and a planet of the closest star to the Sun, Proxima Centauri approximately D = 4.2 light-years away. The position z of the spaceship with respect to Earth is given by z = D/2 [ 1 - cos(ωt) ] . Let TE and TS represent the round-trip travel times with respect to Earth and the spaceship respectively. (a) What are the smallest possible values of TE and TS (corresponding to largest possible ω)? (b) Write down an expression for TS as a function of ω. (c) Evaluate this expression (numerically if necessary) and obtain a plot of the ratio TE/TS as a function of TE.
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.