What is the radius of the imaginary concentric sphere that divides the electrostatic field of a metal sphere of a radius 20 c m and charge of 8 μC in two regions of identical energy?

Step-by-step solution

E n e r g y b e t w e e n s h e l l 1 & 2 = e n e r g y outside s h e l l 2 . energy between shell 1 & 2 = ∫ r = a r = b 1 2 ε 0 K Q r 2 2 × 4 π r 2 d r = ε 0 K 2 Q 2 2 × 4 π - 1 r a b = ε 0 K 2 Q 2 2 1 a - 1 b E n e r g y b e y o n d s h e l l 2 = ∫ r = b r = ∞ 1 2 ε 0 K Q r 2 2 × 4 π r 2 d r = 1 2 ε 0 K 2 Q 2 × 4 π 1 b ∴ 1 2 ε 0 K 2 Q 2 × 4 π 1 a - 1 b = 1 2 ε 0 K 2 Q 2 × 4 π 1 b 1 a = 2 b ⇒ b = 2 a a = 20 cm b = 40 cm