Let there be a spherically symmetric charge distribution with charge density varying as ρ ( r ) = ρ 0 ( 4 5 − R r ) upto r = R , and ρ ( r ) = 0 for r R , where r is the distance from the orig

Question

Let there be a spherically symmetric charge distribution with charge density varying as ρ ( r ) = ρ 0 ​ ( 4 5 ​ − R r ​ ) 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

TestHub · Accepted Answer

Apply shell theorem the total charge upto distance r can be calculated as followed d q = 4 π r 2 ⋅ d r . ρ ​ = 4 π r 2 ⋅ d r ⋅ ρ 0 ​ [ 4 5 ​ − R r ​ ] = 4 π ρ 0 ​ [ 4 5 ​ r 2 d r − R r 3 ​ d r ] ∫ d q = q = 4 π ρ 0 ​ ∫ 0 r ​ ( 4 5 ​ r 2 d r − R r 3 ​ d r ) = 4 π ρ 0 ​ [ 4 5 ​ 3 r 3 ​ − R 1 ​ 4 r 4 ​ ] E = r 2 k q ​ = 4 π ε 0 ​ 1 ​ r 2 1 ​ ⋅ 4 π ρ 0 ​ [ 4 5 ​ ( 3 r 3 ​ ) − 4 R r 4 ​ ] E = 4 ε 0 ​ ρ 0 ​ r ​ [ 3 5 ​ − R r ​ ] ​

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