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PHY 218: Electricity and Magnetism II
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2019

Problem Set 4

Due Friday, February 22, by 4pm in the homework locker

  • Problem 1 [10 points]

    Consider a plane polarized simple harmonic wave, traveling in the x direction with E polarized in the y direction. Compute all elements of the Maxwell stress tensor. Is your result reasonable? How is the electromagnetic momentum current density related to the electromagnetic energy density in this case?

  • Problem 2 [15 points]

    Suppose the electric field in spherical coordinates is given by

    E(r,θ,φ,t) = A sin θ
    r     		[ cos (kr-ωt) - (1/kr) sin (kr-ωt) ] eφ
    where eφ is the unit vector in the φ direction, and ω=ck. This is the form of the electric field for a radiated spherical wave.

    a) Show that E obeys all four of Maxwell's equations in a vacuum, and find the associated magnetic field B(r,θ,φ,t).

    b) Calculate the Poynting vector S. Average S over a full cycle of oscillation in time to get the intensity vector I = <S>. Comment on its direction and dependence on the radial distance r.

    c) Find the flux of I over the surface of a sphere of radius R to determine the total power flowing out from the origin.

  • Problem 3 [15 points]

    Four electromagnetic waves are represented by the following complex exponential expressions, in which the coefficients Ei are all real numbers:
    wave 1:     E1(r, t) = E1xei(kz-ωt)     B1(r, t) = (E1/c)yei(kz-ωt)
    wave 2:     E2(r, t) = E2yei(kz-ωt+δ)     B2(r, t) = −(E2/c)xei(kz-ωt+δ)
    wave 3:     E3(r, t) = E3xei(kz-ωt+δ)     B3(r, t) = (E3/c)yei(kz-ωt+δ)
    wave 4:     E4(r, t) = E1xei(-kz-ωt)     B4(r, t) = −(E1/c)yei(-kz-ωt)
    where x and y are the unit vectors in the x and y directions respectively.

    a) Calculate the time-average Poynting vector <S> for the superposition of wave 1 and wave 2. Show that this quantity is just the sum of <S> for the waves taken separately. Why is that so?

    b) Calculate <S> for the superposition of wave 1 and wave 3. Compare with the results of part (a) and explain the difference.

    c) Calculate <S> for the superposition of wave 1 and wave 4. Interpret the result. Calculate the time-dependent energy density uEM(t). Keep track to the electric and magnetic contributions separately and show that they oscillate in space and time, but are out of phase with each other.