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Electric Charges And Fields

Question
CBSEENPH12039566

Let there be a spherically symmetric charge distribution with charge density varying as  space straight rho left parenthesis straight r right parenthesis space equals space space straight rho subscript 0 space open parentheses 5 over 4 minus straight r over straight R close parentheses upto r = R, and ρ(r) = 0 for r > R, where r is the distance from the origin. The electric field at a distance r ( r < R) from the origin is given by

  • fraction numerator 4 space straight pi space straight rho subscript 0 straight r over denominator 3 straight epsilon subscript 0 end fraction space open parentheses 5 over 3 minus straight r over straight R close parentheses
  • fraction numerator begin display style space straight rho subscript 0 straight r end style over denominator begin display style 4 straight epsilon subscript 0 end style end fraction space open parentheses fraction numerator begin display style 5 end style over denominator begin display style 3 end style end fraction minus fraction numerator begin display style straight r end style over denominator begin display style straight R end style end fraction close parentheses
  • fraction numerator 4 space straight rho subscript 0 straight r over denominator 3 straight epsilon subscript 0 end fraction space open parentheses 5 over 3 minus straight r over straight R close parentheses
  • fraction numerator space straight rho subscript 0 straight r over denominator 3 straight epsilon subscript 0 end fraction space open parentheses 5 over 4 minus straight r over straight R close parentheses

Solution

B.

fraction numerator begin display style space straight rho subscript 0 straight r end style over denominator begin display style 4 straight epsilon subscript 0 end style end fraction space open parentheses fraction numerator begin display style 5 end style over denominator begin display style 3 end style end fraction minus fraction numerator begin display style straight r end style over denominator begin display style straight R end style end fraction close parentheses

Apply shell theorem, the total charge upto distance r can be calculated as follows
dq space equals space 4 πr squared dr. straight rho
space equals space 4 πr squared. dr. straight rho subscript 0 space open square brackets 5 over 4 minus straight r over straight R close square brackets
left parenthesis because space dq space equals space ρdv right parenthesis
space equals space 4 πρ subscript 0 space open square brackets 5 over 4 straight r squared dr space minus straight r cubed over straight R dr close square brackets
space equals space integral space space dq space equals space straight q space equals space 4 πρ subscript 0 integral subscript 0 superscript straight r open parentheses 5 over 4 straight r squared dr minus straight r cubed over straight R dr close parentheses
space equals space 4 πρ subscript 0 space open square brackets 5 over 4 straight r cubed over 3 minus 1 over straight R straight r to the power of 4 over 4 close square brackets
As space the space electric space field comma space straight E space space equals kq over straight r squared
equals space fraction numerator 1 over denominator 4 πε subscript 0 end fraction 1 over straight r squared.4 πρ subscript 0 space open square brackets 5 over 4 open parentheses straight r cubed over 3 close parentheses minus fraction numerator straight r to the power of 4 over denominator 4 straight R end fraction close square brackets
straight E space equals space fraction numerator straight rho subscript 0 straight r over denominator 4 straight epsilon subscript 0 end fraction open square brackets 5 over 3 minus straight r over straight R close square brackets