A non-conducting sphere of radius R has volume charge density ρ(r)=ρ₀(r/R). At what point strictly inside the sphere is the electric field equal in magnitude to the field just outside its surface?
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A non-conducting sphere of radius R has volume charge density ρ(r)=ρ₀(r/R). At what point strictly inside the sphere is the electric field equal in magnitude to the field just outside its surface?

G1 Physics · Accepted Answer

For r<R, Q(r)=4π∫₀ʳ ρ₀(r′/R)r′²dr′ = πρ₀r⁴/R. Hence E(r)=ρ₀r²/(4ε₀R). At r=R, E(R)=ρ₀R/(4ε₀). Equality requires r²=R², so no point strictly inside satisfies it.