In the figure, the inner (shaded) region A represents a sphere of radius $r_{A} = 1,$ within which the electrostatic charge density varies with the radial distance r from the center as $ρ_{A} = k r,$ where k is positive. In the spherical shell B of outer radius $r_{B},$ the electrostatic charge density varies as $ρ_{B} = \frac{2 k}{r} .$ Assume that dimensions are taken care of. All physical quantities are in their SI units. If the electric potential at $r_{B} = \frac{3}{2}$ is $\frac{n k}{ε_{0}}$ then the value of n = ______________ .

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$V = \frac{1}{4 π \in_{0}} \frac{q_{n e t}}{r} q_{n e t} = q_{A} + q_{B} q_{A} = \int_{0}^{1} k r (4 π r^{2} d r) = 4 π k {[\frac{r^{4}}{4}]}_{0}^{1} = π k q_{B} = \int_{1}^{r} \frac{2 k}{r} \times 4 π r^{2} d r = 8 π k (\frac{r^{2} - 1}{2}) = 4 π k r^{2} - 4 π k S o, q_{n e t} = q_{A} + q_{B} = 4 π k r^{2} - 3 π k V = \frac{1}{4 π \in_{0}} \frac{π k (4 r^{2} - 3)}{r} \Rightarrow V = \frac{k}{4 \in_{0}} (4 r - \frac{3}{r}) \Rightarrow V = \frac{k}{4 \in_{0}} (4 \times \frac{3}{2} - \frac{3}{(\frac{3}{2})}) V = \frac{k}{4 \in_{o}} (4) = \frac{k}{\in_{o}} n = 1$