Question

the volume charge density within a volume v is p(r).what is the force on a small test charge Q place outside the volume having opposite vactor r with respect to the same orgin considered to specify the position vactor of the charge distribution within the volume ​

Answer 1

Answer:

Explanation:

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