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Question

# The electric field at 2r from centre of a uniformly charged non conducting sphere of radius r andE .What is the electric field at distance r/2 from centre?

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Solution

## Let the total charge on the sphere be = Q. It is nonconducting dielectric sphere. It is uniformly charged. So charge density per volume = ρ = Q/[4/3 π R³] = 3Q/(4πR³) ---(1) Electric field at distance 2 R due to the sphere = E = 1/[4πε] * Q/(2R)² --- (2) For finding the electric field at a distance R/2 from the center of sphere, we need to take into account the charge enclosed within the sphere of radius R/2 only. This is obtained from Gauss's law for flux coming out of a closed spherical surface of radius R/2. Q1 = charge enclosed = ρ * volume = ρ * 4/3 π (R/2)³ = π/6 ρ R³ = π/6 * 3Q/(4πR³) * R³ = Q/8 E1 = 1/[4πε] * Q1/(R/2)² = 1/[4π] * (Q/8) * 4/R² = 1/[4π] * Q/(2R²) = 2 E

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