Problem Set 4

Problem Set 4

Due Friday, March 4, 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 [10 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) ] e_φ

where e_φ 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 [30 points total]

In lecture we defined the three dimensional Fourier transform of a function f(k) = FT[f(r)] by:

f(k) = 1
(2π)³ +∞
∫
−∞ d³r f(r) e^{−ik ⋅ r} and f(r) = +∞
∫
−∞ d³k f(k) e^{ik ⋅ r}

a) [5 points] If δ(r) is the three dimensionsl Dirac delta function, show that its Fourier transform is a constant independent of k

FT[δ(r)] = 1
(2π)³ so that δ(r) = 1
(2π)³ +∞
∫
−∞ d³k e^{ik ⋅ r}

b) [5 points] Using the result of part (a), show that the Fourier transform of the complex exponential e^ik_o⋅r is just a Dirac delta function in k-space, δ(k − k_o).

c) [5 points] Consider the Poisson equation for the electrostatic potential of a point charge q located at the origin, −∇²V(r) = q δ(r)/ε_o. By taking the Fourier transform of this equation, and using the results of the previous parts, find the Fourier transform V(k) of the solution to this problem V(r).

You already know that the solution to this problem is V(r) = q/(4πε_o|r|). Use this fact and your result above to show that the Fourier transform of 1/|r| is 4π/((2π)³ k²). This is an easier way to find the Fourier transform of 1/|r| than by doing the Fourier transform integration of 1/|r| directly!

d) [15 points] In lecture we discussed the decomposition of any vector function f(r) into its longitudinal (curlfree) and transverse (divergenceless) parts, f_L(r) and f_T(r), as given by Helmholtz's theorem:

f(r) = f_L(r) + f _T(r)

where

f_L(r) = − ∇ [ 1
4π
∫ d³r′ ∇′ ⋅ f(r′)
|r-r′|
] and f_T(r) = ∇x [ 1
4π
∫ d³r′ ∇′ x f(r′)
|r-r′|
]

Using the results above, find expressions for the Fourier transforms of the longitudinal and transverse parts of f(r). Your results should tell you why these parts have the names that they do!

Hint: Just substitute into the above formulas the expressions for the Fourier transforms of f(r) and 1/|r−r'| and rearrange the order of the integrations, do the integrals that you can, and then identify the Fourier coefficients of the longitudinal and transvese parts of f(r) from the result. You should then have found a relatively simple relation between f_L(k) and f_T(k) and the Fourier transform f(k) of f(r).

Problem 4 [10 points]

Four electromagnetic waves are represented by the following complex exponential expressions, in which the coefficients E_i are all real numbers:

wave 1:     E₁(r, t) = E₁xe^i(kz-ωt)     B₁(r, t) = (E₁/c)ye^i(kz-ωt)

wave 2:     E₂(r, t) = E₂ye^i(kz-ωt+δ)     B₂(r, t) = −(E₂/c)xe^i(kz-ωt+δ)

wave 3:     E₃(r, t) = E₃xe^i(kz-ωt+δ)     B₃(r, t) = (E₃/c)ye^i(kz-ωt+δ)

wave 4:     E₄(r, t) = E₁xe^i(-kz-ωt)     B₄(r, t) = −(E₁/c)ye^i(-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 u_EM(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.