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-t

FacebookTweetPin

Question Number 48460 by mr W last updated on 24/Nov/18

$$Apointsourcehasadistanced$$$$tothecenterofabigspherewithradius$$$$R.Anothersmallerspherewithradius$$$$risplacedbetweenthepointsource$$$$andthebigsphere.Ifthedistance$$$$betweenthetwospheresisconstant,$$$$say{it}^{\prime }sc.$$$$Findthemaximalshadowareaofthe$$$$smallsphereonthesurfaceofthe$$$$bigsphere.Findalsotheminimal$$$$completeshadowofthesmallsphere$$$$onthesurfaceofthebigsphere.$$$$$$$$Assumethesmallsphereismuch$$$$smallerthanthebigspheresuchthat$$$$thebigspherewillnevercompletelystay$$$$intheshadowofthesmallsphere.$$

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

$$\sin \theta =\frac{r}{d-c}$$$$\tan \theta =\frac{R\sin \varphi }{d-R\cos \varphi }=\lambda \left(say\right)$$$$\Rightarrow d\sin \theta =R(\sin \varphi \cos \theta +\cos \varphi \sin \theta )$$$$\Rightarrow R\sin (\varphi +\theta )=d\sin \theta$$$$\varphi ={\sin }^{-1}\frac{rd}{R(d-c)}-{\sin }^{-1}\frac{r}{d-c}$$$$\dots .\left(i\right)$$$$shadowareaonbiggersphere$$$$beS={\int }_0^{\varphi }\left(2\pi R\sin \phi \right)\left(Rd\phi \right)$$$$\Rightarrow S=2\pi R^2(1-\cos \varphi )\dots \left(ii\right)$$$$S=2\pi R^2[1-\cos ({\sin }^{-1}\frac{rd}{R(d-c)}-{\sin }^{-1}\frac{r}{d-c})].$$

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

$$thankyousir!thisistheminimalcomplete$$$$shadowofthesmallsphereonthe$$$$bigsphere.Pleasealsofindthe$$$$maximalshadowareawhichoccurs,$$$$Ithink,whenthesmallsphereisaside$$$$likethis:$$

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

Terms of Service
Privacy Policy
Contact: info@tinkutara.com

FacebookTweetPin