Question Number 48460 by mr W last updated on 24/Nov/18
$${A}\:{point}\:{source}\:{has}\:{a}\:{distance}\:{d} \\ $$$${to}\:{the}\:{center}\:{of}\:{a}\:{big}\:{sphere}\:{with}\:{radius} \\ $$$${R}.\:\:{An}\:{other}\:{smaller}\:{sphere}\:{with}\:{radius} \\ $$$${r}\:{is}\:{placed}\:{between}\:{the}\:{point}\:{source} \\ $$$${and}\:{the}\:{big}\:{sphere}.\:{If}\:{the}\:{distance} \\ $$$${between}\:{the}\:{two}\:{spheres}\:{is}\:{constant}, \\ $$$${say}\:{it}'{s}\:{c}.\: \\ $$$${Find}\:{the}\:{maximal}\:{shadow}\:{area}\:{of}\:{the} \\ $$$${small}\:{sphere}\:{on}\:{the}\:{surface}\:{of}\:{the} \\ $$$${big}\:{sphere}.\:{Find}\:{also}\:{the}\:{minimal} \\ $$$${complete}\:{shadow}\:{of}\:{the}\:{small}\:{sphere} \\ $$$${on}\:{the}\:{surface}\:{of}\:{the}\:{big}\:{sphere}. \\ $$$$ \\ $$$${Assume}\:{the}\:{small}\:{sphere}\:{is}\:{much} \\ $$$${smaller}\:{than}\:{the}\:{big}\:{sphere}\:{such}\:{that} \\ $$$${the}\:{big}\:{sphere}\:{will}\:{never}\:{completely}\:{stay} \\ $$$${in}\:{the}\:{shadow}\:{of}\:{the}\:{small}\:{sphere}. \\ $$
Commented by mr W last updated on 24/Nov/18
Commented by ajfour last updated on 24/Nov/18
Answered by ajfour last updated on 24/Nov/18
![sin θ = (r/(d−c)) tan θ = ((Rsin φ)/(d−Rcos φ)) = λ (say) ⇒ dsin θ = R(sin φcos θ+cos φsin θ) ⇒ Rsin (φ+θ) = dsin θ φ = sin^(−1) ((rd)/(R(d−c)))−sin^(−1) (r/(d−c)) ....(i) shadow area on bigger sphere be S = ∫_0 ^( φ) (2πRsin ϕ)(Rdϕ) ⇒ S = 2πR^( 2) (1−cos φ) ...(ii) S = 2πR^2 [1−cos (sin^(−1) ((rd)/(R(d−c)))−sin^(−1) (r/(d−c)))] .](https://www.tinkutara.com/question/Q48487.png)
$$\mathrm{sin}\:\theta\:=\:\frac{{r}}{{d}−{c}} \\ $$$$\mathrm{tan}\:\theta\:=\:\frac{{R}\mathrm{sin}\:\phi}{{d}−{R}\mathrm{cos}\:\phi}\:\:\:=\:\lambda\:\left({say}\right) \\ $$$$\Rightarrow\:\:{d}\mathrm{sin}\:\theta\:=\:{R}\left(\mathrm{sin}\:\phi\mathrm{cos}\:\theta+\mathrm{cos}\:\phi\mathrm{sin}\:\theta\right) \\ $$$$\Rightarrow\:{R}\mathrm{sin}\:\left(\phi+\theta\right)\:=\:{d}\mathrm{sin}\:\theta \\ $$$$\:\:\phi\:=\:\mathrm{sin}^{−\mathrm{1}} \frac{{rd}}{{R}\left({d}−{c}\right)}−\mathrm{sin}^{−\mathrm{1}} \frac{{r}}{{d}−{c}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\left({i}\right) \\ $$$${shadow}\:{area}\:{on}\:{bigger}\:{sphere} \\ $$$${be}\:{S}\:=\:\int_{\mathrm{0}} ^{\:\:\phi} \left(\mathrm{2}\pi{R}\mathrm{sin}\:\varphi\right)\left({Rd}\varphi\right)\:\: \\ $$$$\:\Rightarrow\:\:{S}\:=\:\mathrm{2}\pi{R}^{\:\mathrm{2}} \left(\mathrm{1}−\mathrm{cos}\:\phi\right)\:\:\:\:…\left({ii}\right) \\ $$$$\:{S}\:=\:\mathrm{2}\pi{R}^{\mathrm{2}} \left[\mathrm{1}−\mathrm{cos}\:\left(\mathrm{sin}^{−\mathrm{1}} \frac{{rd}}{{R}\left({d}−{c}\right)}−\mathrm{sin}^{−\mathrm{1}} \frac{{r}}{{d}−{c}}\right)\right]\:\:. \\ $$
Commented by mr W last updated on 24/Nov/18
$${thank}\:{you}\:{sir}!\:{this}\:{is}\:{the}\:{minimal}\:{complete} \\ $$$${shadow}\:{of}\:{the}\:{small}\:{sphere}\:{on}\:{the}\: \\ $$$${big}\:{sphere}.\:{Please}\:{also}\:{find}\:{the} \\ $$$${maximal}\:{shadow}\:{area}\:{which}\:{occurs}, \\ $$$${I}\:{think},\:{when}\:{the}\:{small}\:{sphere}\:{is}\:{aside} \\ $$$${like}\:{this}: \\ $$
Commented by mr W last updated on 24/Nov/18
