Within a spherical charge distribution of charge density ρ (r), N… | JEE Main 2016 (10 Apr, Shift 1)

Q102JEE Main 2016 · 10 APR · SHIFT 1Difficult

Electric Potential & Equipotentials

Within a spherical charge distribution of charge density $\rho$ (r), N equipotential surfaces of potential V0, V0 + $\Delta$ V, V0 + 2 $\Delta$ V, .......... V0 + N $\Delta$ V ( $\Delta$ V > 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 $\Delta$ V then:

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Answer: (c)

Equal $\Delta V$ at equal $\Delta r$ means $E = \dfrac{\Delta V}{\Delta r}$ is constant.
Gauss: $E \cdot 4\pi r^2 = \dfrac{q(r)}{\varepsilon_0} \Rightarrow q(r) \propto r^2$ .
$dq = \rho \cdot 4\pi r^2\,dr \propto 2r\,dr \Rightarrow \rho \propto \dfrac{1}{r}$
So the answer is (c) $\rho \propto \dfrac{1}{r}$ .

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