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?

Answer 1

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