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A ball of radius R carries a positive charge whose volume density at a point is given by Where ρ 0 is a constant and r is distance from the centre. The value of maximum electric field due to this configuration is
A ball of radius R carries a positive charge whose volume density at a point is given by  Where ρ0 is a constant and r is distance from the centre. The value of maximum electric field due to this configuration is

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7 months ago

Arun
25768 Points
```							(a) We assume the ball to be divided into infinite number of concentric thin shell of thickness dr. Let us assume such a shell at a radial distance r.Volume of this shell =pdV=4πr 2 ρdr=4πr 2 ρ 0​  (1− Rr​  )drNet charge enclosed in the sphere of radius rq=∫pdv= 0∫​  r​  4πρ 0​  r 2 (1− Rr​  )dr=4πρ 0​  ( 3r 3 ​  − 4Rr 4 ​  ) Electric field at radial distance rE= 4πε 0​  r 2 q​  = 4πε 0​  r 2 1​  ×4πρ 0​  ( 3r 3 ​  − 4Rr 4 ​  )= 3ε 0​  ρ 0​  r​  (1− 4R3r​  ) Now for those points outside the sphere, we have to take into account the total charge contained in the sphere of radius R.Total charge Q= 0∫​  R​  ρ 0​  (1− Rr​  )4πr 2 dr= 3πρ 0​  R 3 ​  Electric field, using Gauss's laws, at a point which is at distance r(>R) from the centre; ∫E.ds= ε 0​  Q​  ⟹E= 4πε 0​  r 2 Q​  = 12ε 0​  r 2 ρ 0​  R 3 ​   (b) Again let us consider the expression for electric field within the sphere of radius R:E= 3ε 0​  ρ 0​  r​  (1− 4R3r​  )For maximum electric field:drdE​  =0⟹1− 4R6r​  =0⟹r= 32R​  Putting, r= 32R​  , value of maximum electric field, E max​  = 9∈ 0​  ρ 0​  R​
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7 months ago
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