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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.

$${A}\:{point}\:{source}\:{has}\:{a}\:{distance}\:{d} \\ $$$${to}\:{the}\:{center}\:{of}\:{a}\:{big}\:{sphere}\:{with}\:{radius} \\ $$$${R}.\:\:{An}\:{other}\:{smaller}\:{sphere}\:{with}\:{radius} \\ $$$${r}\:{is}\:{placed}\:{between}\:{the}\:{point}\:{source} \\ $$$${and}\:{the}\:{big}\:{sphere}.\:{If}\:{the}\:{distance} \\ $$$${between}\:{the}\:{two}\:{spheres}\:{is}\:{constant}, \\ $$$${say}\:{it}'{s}\:{c}.\: \\ $$$${Find}\:{the}\:{maximal}\:{shadow}\:{area}\:{of}\:{the} \\ $$$${small}\:{sphere}\:{on}\:{the}\:{surface}\:{of}\:{the} \\ $$$${big}\:{sphere}.\:{Find}\:{also}\:{the}\:{minimal} \\ $$$${complete}\:{shadow}\:{of}\:{the}\:{small}\:{sphere} \\ $$$${on}\:{the}\:{surface}\:{of}\:{the}\:{big}\:{sphere}. \\ $$$$ \\ $$$${Assume}\:{the}\:{small}\:{sphere}\:{is}\:{much} \\ $$$${smaller}\:{than}\:{the}\:{big}\:{sphere}\:{such}\:{that} \\ $$$${the}\:{big}\:{sphere}\:{will}\:{never}\:{completely}\:{stay} \\ $$$${in}\:{the}\:{shadow}\:{of}\:{the}\:{small}\:{sphere}. \\ $$

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)))]  .

$$\mathrm{sin}\:\theta\:=\:\frac{{r}}{{d}−{c}} \\ $$$$\mathrm{tan}\:\theta\:=\:\frac{{R}\mathrm{sin}\:\phi}{{d}−{R}\mathrm{cos}\:\phi}\:\:\:=\:\lambda\:\left({say}\right) \\ $$$$\Rightarrow\:\:{d}\mathrm{sin}\:\theta\:=\:{R}\left(\mathrm{sin}\:\phi\mathrm{cos}\:\theta+\mathrm{cos}\:\phi\mathrm{sin}\:\theta\right) \\ $$$$\Rightarrow\:{R}\mathrm{sin}\:\left(\phi+\theta\right)\:=\:{d}\mathrm{sin}\:\theta \\ $$$$\:\:\phi\:=\:\mathrm{sin}^{−\mathrm{1}} \frac{{rd}}{{R}\left({d}−{c}\right)}−\mathrm{sin}^{−\mathrm{1}} \frac{{r}}{{d}−{c}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\left({i}\right) \\ $$$${shadow}\:{area}\:{on}\:{bigger}\:{sphere} \\ $$$${be}\:{S}\:=\:\int_{\mathrm{0}} ^{\:\:\phi} \left(\mathrm{2}\pi{R}\mathrm{sin}\:\varphi\right)\left({Rd}\varphi\right)\:\: \\ $$$$\:\Rightarrow\:\:{S}\:=\:\mathrm{2}\pi{R}^{\:\mathrm{2}} \left(\mathrm{1}−\mathrm{cos}\:\phi\right)\:\:\:\:...\left({ii}\right) \\ $$$$\:{S}\:=\:\mathrm{2}\pi{R}^{\mathrm{2}} \left[\mathrm{1}−\mathrm{cos}\:\left(\mathrm{sin}^{−\mathrm{1}} \frac{{rd}}{{R}\left({d}−{c}\right)}−\mathrm{sin}^{−\mathrm{1}} \frac{{r}}{{d}−{c}}\right)\right]\:\:. \\ $$

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:

$${thank}\:{you}\:{sir}!\:{this}\:{is}\:{the}\:{minimal}\:{complete} \\ $$$${shadow}\:{of}\:{the}\:{small}\:{sphere}\:{on}\:{the}\: \\ $$$${big}\:{sphere}.\:{Please}\:{also}\:{find}\:{the} \\ $$$${maximal}\:{shadow}\:{area}\:{which}\:{occurs}, \\ $$$${I}\:{think},\:{when}\:{the}\:{small}\:{sphere}\:{is}\:{aside} \\ $$$${like}\:{this}: \\ $$

Commented by mr W last updated on 24/Nov/18

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