Consider a uniformly charged solid cylinder of large length and radius R. Now consider a cylinderical surface of radius 2R and length 2R coaxial with cylinder and electric flux through cylindrical sur

Question

Consider a uniformly charged solid cylinder of large length and radius R. Now consider a cylinderical surface of radius 2R and length 2R coaxial with cylinder and electric flux through cylindrical surface is ϕ 0 ​ . Now consider a spherical surface of radius 2R and one of the diameter along axis of cylinder and electric flux through spherical surface is ϕ . Find ϕ 0 ​ ϕ ​ .

TestHub · Accepted Answer

ϕ 0 ​ = ε 0 ​ ρ π R 2 .2 R ​ ⇒ ϕ 0 ​ = ε 0 ​ 2 ρ π R 3 ​ \begin{center} \end{center} ​ V 1 ​ = ∫ 0 π /6 ​ π 4 R 2 sin 2 θ 2 R sin θ d θ ​ = 4 8 π R 3 ​ ∫ 0 π /6 ​ ( 3 sin θ − sin 3 θ ) d θ ​ = 4 8 π R 3 ​ ( 3 8 ​ − 2 3 3 ​ ​ ) ​ Total volume V = ( V 1 ​ + V 2 ​ ) 2 ​ = 2 π R 3 ​ ( 3 8 ​ − 2 3 3 ​ ​ ) + π R 2 2 3 ​ R ​ = π R 3 [ 2 3 ​ − 4 3 3 ​ ​ + 3 4 ​ ] ⇒ ( 12 15 3 ​ + 16 ​ ) π R 3 ​ ϕ 0 ​ ϕ ​ = 24 15 3 ​ + 16 ​ ≈ 1.87 ​

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