Problem 1 [10 points]
Consider Maxwell's equations in the presence of charge and current sources. Show that E and B now satisfy the inhomogeneous wave equation
∇2E − μoεo∂2E/∂t2 = g(r, t) and
∇2B − μoεo∂2B/∂t2 = h(r, t)
and express the functions g(r, t) and h(r, t) in terms of the charge density ρ(r, t) and the current density j(r, t).
Problem 3 [10 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.
Problem 4 [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) eik ⋅ 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 eik ⋅ r
|
b) [5 points] Using the result of part (a), show that the Fourier transform of the complex exponential eiko⋅r is just a Dirac delta function in k-space, δ(k − ko).
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)/εo. 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/(4πεo|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, fL(r) and fT(r), as given by Helmholtz's theorem:
f(r) = fL(r) + f T(r)
where
| fL(r) = − ∇
|
[
|
1
4π
|
|
∫
|
d3r′
|
∇′ ⋅ f(r′)
|r-r′|
|
|
]
|
and
|
fT(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 fL(k) and fT(k) and the Fourier transform f(k) of f(r).