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Question:

A spherically symmetric charge distribution is characterised by a charge density having the following variation: ρ(r) = ρ₀(1 - r/R) for r < R; ρ(r) = 0 for r ≥ R. Where r is the distance from the centre of the charge distribution and ρ₀ is a constant. The electric field at an internal point (r < R) is:

ρ₀/(3ε₀)(r³ - r²/4R)

ρ₀/(12ε₀)(r³ - r²/4R)

ρ₀/(4ε₀)(r³ - r²/4R)

ρ₀/(ε₀)(r³ - r²/4R)

Solution:

Answer is B. Electric field at the internal point r is the sum of fields due to outer and inner spheres. However, the contribution due to the outer part is zero. For the inner part, consider a shell of thickness dr' at a distance of r' from the center of the sphere. Therefore, charge contained in it dq = 4πr'²dr'ρ₀(1 - r'/R). Or, Electric field dE is dE = 1/(4πε₀)r² * 4πr'²dr'ρ₀(1 - r'/R). Integrating above, we get E = ρ₀/(ε₀r²) ∫₀ʳ (r'² - r'³/R)dr' E = ρ₀/(ε₀)(r³/3 - r⁴/(4R))