Within a spherical charge distribution of charge density ρ(r), N equipotential surfaces of potential V₀, V₀+Δ V, V₀+2Δ V, … V₀+NΔ V (Δ V0) are drawn and have increasing radii r₀,r₁,r₂,… r_N respectively. If the difference in the radii of the surfaces is constant for all values of V₀ and Δ V, then

Question

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

Accepted Answer

(3) ρ(r)∝1/r. Equal potential steps Δ V for equal radial steps Δ r means |(dV)/(dr)| is the same at every radius, i.e. E(r) is constant. By Gauss's law E(r)=(q_enc(r))/(4πε₀ r²), so a constant E requires q_enc(r)∝ r². Differentiating, (dq_enc)/(dr)=ρ(r) 4π r²∝ r. Hence ρ(r)∝r/(r²)=1/r.

Step-by-step solution

Why the other options are wrong