Question

The electric field at 2R from the centre of a uniformly charged non conducting sphere of radius R is E. The electric field at a distance R/2 from the centre will be 1) zero2) 2E3) 4E4) 16E

Answer 1

Answer is Option 2) 2 E.

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

Answer 2

Answer:

Explanation: surface charge density sigma=q/4/3Pi r^3 q=sigma×4/3pi r^3

E=Kq/r^2

Where r is distance surface charge and test charge according to question, sigma=q'/4/3pi r^3 q'=sigma×4/3pi (r/2)^3 = 8×sigma×4/3pi r^3/8=q/8 with the help of Gouce E'=Kq'/(r/2)^2 =Kq/8/(r^2/4) = 2E