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CBSE Class 12-science Answered

Consider a Sphere of Radius R with a North Pole at point P. Draw a circle of radius r on the surface of the sphere centered at P. Let θ be the polar co-ordinate defined from the north pole .The galaxies are distributed evenly on this surface with number density n per unit area. Show that the total number N of galaxies inside a the circle of radius r is given by N = 2πnR^2 [1 − cos (r/R)]
Asked by vaishnavi0508 | 18 Jul, 2018, 03:12: PM
answered-by-expert Expert Answer
As shown in figure, It is required to find the number of galaxies in the circular area of radius r that was on top of the sphere of radius R.
In figure spherical coordinate system is shown. In this spherical coordinate system, a small area element dA is written as
 
dA = R2sinθdθ dφ ...........................(1)

if n is the number density per unit area, then number of galaxies in this area dA = n×R2sinθdθ dφ 

then number of galaxies in the circular area of radius r  begin mathsize 12px style equals space n R squared integral subscript 0 superscript theta sin theta space d theta space integral subscript 0 superscript 2 pi end superscript d phi space equals space 2 pi space n space R to the power of 2 space end exponent open parentheses 1 minus cos theta close parentheses end style
since θ = r/R, then number of galaxies in the circular area of radius r begin mathsize 12px style equals space space 2 pi space n space R to the power of 2 space end exponent open parentheses 1 minus cos open parentheses r over R close parentheses close parentheses end style
Answered by Thiyagarajan K | 15 Aug, 2018, 09:22: AM

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