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Please answer the following question with explanation

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Asked by Deepak 10th September 2018, 2:40 PM
Answered by Expert
Answer:
As shown in figure consider a spherical continuous charge distribution with charge density varies radially as
  begin mathsize 12px style rho open parentheses r close parentheses space equals space rho subscript 0 open parentheses 1 space minus space r over R close parentheses end style  when r ≤ R , otherwise zero
In order to find electrical field at a point P which is at a distance d from center and inside the sphere,
let us consider a spherical shell of radius r and thickness dr as shown in figure.
 
electrical field dE at a point P is given by, dE  begin mathsize 12px style equals space fraction numerator q left parenthesis r right parenthesis over denominator 4 pi epsilon subscript o d squared end fraction space equals space fraction numerator 4 pi r squared d r over denominator 4 pi epsilon subscript o d squared end fraction space rho subscript 0 open parentheses 1 minus r over R close parentheses space equals space fraction numerator rho subscript 0 over denominator epsilon subscript o d squared end fraction subscript 0 open parentheses 1 minus r over R close parentheses space d r end style
 
total electrical fieldbegin mathsize 12px style equals space integral subscript 0 superscript d space fraction numerator rho subscript 0 over denominator epsilon subscript o d squared end fraction subscript 0 open parentheses 1 minus r over R close parentheses space d r space equals space rho subscript 0 over epsilon subscript o open parentheses d over 3 space minus space fraction numerator d squared over denominator 4 R end fraction close parentheses end style
( upper limit of integration is set as d, because if we consider the shell after the point p,
then point P is inside a charged spherical shell. In that case electrical field is zero)
Answered by Expert 10th September 2018, 5:33 PM
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