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Question:

Let there be a spherically symmetric charge distribution with charge density varying as p(r) = p₀(54 - r/R) up to r = R, and p(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 :

4p₀r³ε₀(54 - r/R)

4πp₀r³ε₀(53 - r/R)

p₀r⁴ε₀(53 - r/R)

p₀r⁴ε₀(54 - r/R)

Solution:

Apply shell theorem the total charge upto distancer can be calculated as followed
dq = 4πr².dr.p = 4πr².dr.p₀[54 - r/R] = 4πp₀[54r²dr - r³Rdr]
∫dq = q = 4πp₀∫₀ʳ(54r²dr - r³Rdr) = 4πp₀[54r³/3 - r⁴/4R]
E = kq/r² = 1/4πε₀r².4πp₀[54(r³/3) - r⁴/4R]
E = p₀r⁴ε₀[53 - r/R]