Question 21.9: Clipper Ships of Space GOAL Relate the intensity of light to...
Clipper Ships of Space
GOAL Relate the intensity of light to its mechanical effect on matter.
PROBLEM Aluminized Mylar film is a highly reflective, lightweight material that could be used to make sails for spacecraft driven by the light of the Sun. Suppose a sail with area 1.00 km² is orbiting the Sun at a distance of 1.50 × 10^{11} m. The sail has a mass of 5.00 × 10³ kg and is tethered to a payload of mass 2.00 × 10^{4} kg. (a) If the intensity of sunlight is 1.34 × 10³ W and the sail is oriented perpendicular to the incident light, what radial force is exerted on the sail? (b) About how long would it take to change the radial speed of the sail by 1.00 km/s? Assume the sail is perfectly reflecting. (c) Suppose the light were supplied by a large, powerful laser beam instead of the Sun. (Such systems have been proposed.) Calculate the peak electric and magnetic fields of the laser light.
STRATEGY Equation 21.30
p={\frac{2U}{c}} (complete reflection) [21.30]
gives the momentum imparted when light strikes an object and is totally reflected. The change in this momentum with time is a force. For part (b), use Newton’s second law to obtain the acceleration. The velocity kinematics equation then yields the necessary time to achieve the desired change in speed. Part (c) follows from Equation 21.27 and E = Bc.
I={\frac{E_{\operatorname*{max}}B_{\operatorname*{max}}}{2\mu_{0}}} [21.27]
(a) Find the force exerted on the sail.
Write Equation 21.30 and substitute U = PΔt = IA Δt for the energy delivered to the sail:
\Delta p={\frac{2U}{c}}={\frac{2P\Delta t}{c}}={\frac{2I A\Delta t}{c}}
Divide both sides by Δt, obtaining the force Δp/Δt exerted by the light on the sail:
F={\frac{\Delta p}{\Delta t}}={\frac{2I A}{c}}={\frac{{{2}}(1340{~ W}/{ m}^{2})(1.00~\times~{10}^{6}{~m}^{2})}{3.00~\times~{10^{8}{~ m}/s}}}
= 8.93 N
(b) Find the time it takes to change the radial speed by 1.00 km/s.
Substitute the force into Newton’s second law and solve for the acceleration of the sail:
a={\frac{F}{m}}={\frac{8.93\,\mathrm{N}}{2.50~\times~10^{4}\,\mathrm{kg}}}=3.57\times10^{-4}\,\mathrm{m}/s^{2}
Apply the kinematics velocity equation:
v=a t+v_{0}
Solve for t:
t={\frac{v~-~v_{0}}{a}}={\frac{1.00~\times~10^{3}\,{\mathrm{m/s}}}{3.57~\times~10^{-4}\,{\mathrm{m/s}}^{2}}}=\;{2.80\times10^{6}\mathrm~{s}}
(c) Calculate the peak electric and magnetic fields if the light is supplied by a laser.
Solve Equation 21.28 for E_{\mathrm{max}}:
I=\frac{E_{\mathrm{max}}^{2}}{2\mu_{0}c}\quad\rightarrow\quad E_{\mathrm{max}}=\sqrt{2\mu_{0}cI}
E_{\operatorname*{max}}={\sqrt{2(4\pi\times10^{-7}{\textrm{~N}}\cdot s^{2}/{\mathrm{C}^{2}})(3.00\times10^{8}{\textrm{~m}}/{\mathrm{s}})(1.34\times10^{3~}{\textrm{W/m²}})}}
= 1.01 × 10³ N/C
I =\frac{E^2_{max}}{2\mu_0c}=\frac{c}{2\mu_0}B^2_{max} [21.28]
Obtain B_{\mathrm{max}} using E_{\operatorname*{max}}=B_{\operatorname*{max}}c:
B_{\mathrm{max}}={\frac{{ E}_{\mathrm{max}}}{c}}={\frac{1.01~\times~10^{3}\,\mathrm{N}/\mathrm{C}}{3.00~\times~10^{8}\,\mathrm{m}/s}}=\mathrm{~{\mathrm{~3.37}}\times10^{-6}\,\mathrm{T}}
REMARKS The answer to part (b) is a little over a month. While the acceleration is very low, there are no fuel costs, and within a few months the velocity can change sufficiently to allow the spacecraft to reach any planet in the solar system. Such spacecraft may be useful for certain purposes and are highly economical, but require a considerable amount of patience.