Tarea 3
1. A transverse plane wave is incident normally in vacuum on a perfectly
absorbing flat screen.
(a) From the law of conservation of linear momentum show that the pressure
(called radiation pressure) exerted on the screen is equal to the field energy per
unit volume in the wave.
(b) In the neighborhood of the earth the flux of electromagnetic energy from the
sun is approximately 0.14 𝑤𝑎𝑡𝑡/𝑐𝑚2. If an interplanetary “sailplane” had a sail of
mass 10−4 𝑔/𝑐𝑚2 of area and negligible other weight, what would be its
maximum acceleration in centimeters per second squared due to the solar
radiation pressure? How does this compare with the acceleration due to the
solar “wind” (corpuscular radiation)?
2. The time dependence of electrical disturbances in good conductors is governed
by the frequency-dependent conductivity
𝑓0 𝑁𝑒 2
𝜎=
𝑚(𝛾0 − 𝑖𝜔)
Consider longitudinal electric fields in a conductor, using Ohm’s law, the
continuity equation, and the differential form of Coulomb’s law.
(a) Show that the time-Fourier-transformed charge density satisfies the
equation
[4𝜋𝜎(𝜔) − 𝑖𝜔]𝜌(𝒙, 𝜔) = 0.
(b) Using the representation 𝜎(𝜔) = 𝜎0 /(1 − 𝑖𝜔𝜏), where 𝜎0 = 𝜔𝑝2 𝜏/4𝜋 and 𝜏 is
a damping time, show that in the approximation 𝜔𝑝 𝜏 ≫ 1 any initial disturbance
will oscillate with the plasma frequency and decay in amplitude with a decay
constant 𝜆 = 1/2𝜏. Note that if you use 𝜎(𝜔) ≈ 𝜎(0) = 𝜎0 in part (a), you will
find no oscillations and extremely rapid damping with the (wrong) decay
constant 𝜆 = 4𝜋𝜎0 . [W. M. Saslow and G. Wilkinson, Am. J. Phys. 39, 1244
(1971)].
3. An approximately monochromatic plane wave packet in one dimension has the
instantaneous form, 𝑢(𝑥, 0) = 𝑓(𝑥)exp(𝑖𝑘0 𝑥), with 𝑓(𝑥) the modulation
envelope. For each of the forms 𝑓(𝑥) below, calculate the wave-number
spectrum |𝐴(𝑘)|2 of the packet, sketch |𝑢(𝑥, 0)|2 and |𝐴(𝑘)|2, evaluate explicity
the rms deviation from the means, ∆𝑥 and ∆𝑘 (defined in terms of the intensities
|𝑢(𝑥, 0)|2 and |𝐴(𝑘)|2 ), and test inequality ∆𝑥∆𝑘 ≥ 1/2.
(a) 𝑓(𝑥) = 𝑁exp(−𝛼|𝑥|/2)
(b) 𝑓(𝑥) = 𝑁exp(−𝛼 2 𝑥 2 /4)
4. Consider the nonlocal (in time) connection between 𝑫 and 𝑬,
𝑫(𝒙, 𝑡) = 𝑬(𝒙, 𝑡) + ∫ 𝑑𝜏𝐺(𝜏)𝑬(𝒙, 𝑡 − 𝜏)
with the 𝐺(𝜏) appropriate for the single-resonance model,
𝜖(𝜔) = 1 + 𝜔𝑝2 (𝜔02 − 𝜔2 − 𝑖𝛾𝜔)−1 .
(a) Convert the nonlocal connection between 𝑫 and 𝑬 into an instantaneous
relation involving derivatives of 𝑬 with respect to time by expanding the electric
field in the integral in a Taylor series in 𝜏. Evaluate the integrals over 𝐺(𝜏)
explicitly up to at least 𝜕 2 𝑬/𝜕𝑡 2 .
(b) Show that the series obtained in (a) can be obtained formally by converting
the frequency-representation relation, 𝑫(𝒙, 𝜔) = 𝜖(𝜔)𝑬(𝒙, 𝜔), into a space-time
relation,
𝜕
𝑫(𝒙, 𝑡) = 𝜖 (𝑖 ) 𝑬(𝒙, 𝑡),
𝜕𝑡
where the variable 𝜔 in 𝜖(𝜔) is replaced by 𝜔 → 𝑖(𝜕/𝜕𝑡).