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CBSE Class 12-science Answered

The volume charge density within a volume V is rho (r). What is the force on a small test charge q placed outside the volume having position vector r with respect to the same origin considered to specify the position vector of the charge distribution within the volume.
Asked by sd2021667 | 08 Dec, 2018, 10:41: AM
answered-by-expert Expert Answer
Let R be the radius of sphere. Let us consider a spherical shell of radius s and thickness ds.
Charge of this sphere is (4πs2 ds)×ρ(s). 

Electric field  dE at P due to this charge of spherical shell  is given by
 
 begin mathsize 16px style d E space equals space fraction numerator 4 πs squared space straight rho left parenthesis straight s right parenthesis space ds over denominator 4 pi epsilon subscript 0 space open vertical bar r with rightwards arrow on top close vertical bar squared end fraction space equals fraction numerator begin display style s squared space straight rho left parenthesis straight s right parenthesis space ds end style over denominator begin display style epsilon subscript 0 space open vertical bar r with rightwards arrow on top close vertical bar squared end style end fraction end style 
where begin mathsize 16px style r with rightwards arrow on top end style is the position vector of point P and begin mathsize 16px style open vertical bar r with rightwards arrow on top close vertical bar end style is the distance of point P from origin O (Centre of Sphere)
Electric field E at the point P due to the sphere is given by
 
begin mathsize 16px style E space equals space integral d E space equals space fraction numerator 1 over denominator epsilon subscript 0 space open vertical bar r with rightwards arrow on top close vertical bar squared end fraction space integral subscript 0 superscript R s squared space straight rho left parenthesis straight s right parenthesis space ds end style
Force on a test charge q placed at P is   begin mathsize 16px style q E space equals space space fraction numerator q over denominator epsilon subscript 0 space open vertical bar r with rightwards arrow on top close vertical bar squared end fraction space integral subscript 0 superscript R s squared space straight rho left parenthesis straight s right parenthesis space ds end style
Answered by Thiyagarajan K | 09 Dec, 2018, 10:40: AM

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