devarshi-dt-logo

Question:

Within a spherical charge distribution of charge density ρ(r), N equipotential surfaces of potential V0, V0+ΔV, V0+2ΔV,....., V0+NΔV (ΔV>0), are drawn and have increasing radii r0, r1, r2,..... rN, respectively. If the difference in the radii of the surfaces is constant for all values of V0 and ΔV then :

ρ(r)∝1/r

ρ(r)∝1

ρ(r)=constant

ρ(r)∝1/r2

Solution:

Considering a spherical gaussian surface and applying Gauss law, E(4πr²)=∫r0ρ(4πr²)dr/ε0
E=∫r0ρr²dr/ε0r²
It is given that dV/dr is constant.
But E=-dV/dr ⇒ E is constant ⇒ ∫ρr²dr∝r² ⇒ ρr²∝r ⇒ ρ∝1/r