A conducting sphere with inner radius R and outer radius 3 R has some alloys mixed in it due to which its resistivity changes with radial distance r according to the function ρ r = Kr , where K is a

Question

A conducting sphere with inner radius R and outer radius 3 R has some alloys mixed in it due to which its resistivity changes with radial distance r according to the function $ρ_{r} = Kr$ , where K is a +ve constant. Inner surface of the sphere is grounded while outer surface of sphere in maintained at potential $V_{0}$ with the help of a battery of emf $V_{0}$ . Battery is ideal. The potential of the sphere at radial distance of $2 R$ is $V_{0} \frac{ℓ na}{ℓ n} b$ . Find $(a + b)$ .

TestHub · Accepted Answer

V 0 ​ = I Req R eq ​ = ∫ dR R eq ​ = ∫ 4 π r 2 ρ r ​ dr ​ R eq ​ = ∫ R 3 R ​ 4 π r Kdr ​ ⇒ R eq ​ = 4 π K ​ ln ( 3 ) R ( r ) = 4 π K ​ ln ( R r ​ ) V ( r ) = 1 R ( r ) = R eq ​ V 0 ​ ​ × R ( r ) V ( 2 R ) = V 0 ​ ℓ n 3 ℓ n 2 ​

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Doubts & Discussion

JEE Advanced 2027 (Class 12+)