advertisement

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 π€ππ‘π‘/ππ2. If an interplanetary “sailplane” had a sail of mass 10−4 π/ππ2 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 π0 ππ 2 π= π(πΎ0 − ππ) 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 [4ππ(π) − ππ]π(π, π) = 0. (b) Using the representation π(π) = π0 /(1 − πππ), where π0 = ππ2 π/4π and π is a damping time, show that in the approximation ππ π β« 1 any initial disturbance will oscillate with the plasma frequency and decay in amplitude with a decay constant π = 1/2π. Note that if you use π(π) ≈ π(0) = π0 in part (a), you will find no oscillations and extremely rapid damping with the (wrong) decay constant π = 4ππ0 . [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, π’(π₯, 0) = π(π₯)exp(ππ0 π₯), with π(π₯) the modulation envelope. For each of the forms π(π₯) below, calculate the wave-number spectrum |π΄(π)|2 of the packet, sketch |π’(π₯, 0)|2 and |π΄(π)|2, evaluate explicity the rms deviation from the means, βπ₯ and βπ (defined in terms of the intensities |π’(π₯, 0)|2 and |π΄(π)|2 ), and test inequality βπ₯βπ ≥ 1/2. (a) π(π₯) = πexp(−πΌ|π₯|/2) (b) π(π₯) = πexp(−πΌ 2 π₯ 2 /4) 4. Consider the nonlocal (in time) connection between π« and π¬, π«(π, π‘) = π¬(π, π‘) + ∫ πππΊ(π)π¬(π, π‘ − π) with the πΊ(π) appropriate for the single-resonance model, π(π) = 1 + ππ2 (π02 − π2 − ππΎπ)−1 . (a) Convert the nonlocal connection between π« and π¬ into an instantaneous relation involving derivatives of π¬ with respect to time by expanding the electric field in the integral in a Taylor series in π. Evaluate the integrals over πΊ(π) explicitly up to at least π 2 π¬/ππ‘ 2 . (b) Show that the series obtained in (a) can be obtained formally by converting the frequency-representation relation, π«(π, π) = π(π)π¬(π, π), into a space-time relation, π π«(π, π‘) = π (π ) π¬(π, π‘), ππ‘ where the variable π in π(π) is replaced by π → π(π/ππ‘).