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A long solid non-conducting cylinder of radius $R$ is charged such that the volume charge density is proportional to $r$ where $r$ is the distance from axis. The electric field $E$ at a distance $r(r < R)$ will depend on $r$ as
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The correct answer is:
$E \propto r^2$
$\begin{aligned} & (\mathbf{b}): \rho \propto r ; \rho=K r \\ & d Q=\rho V \\ & \quad=K r(2 \pi r d r) \times l\end{aligned}$
Using Gauss's Law,
$\begin{aligned} & E .2 \pi r l=\frac{\int_0^r K r .2 \pi r l . d r}{\varepsilon_0} \\ & \Rightarrow E=K \frac{r^2}{3 \varepsilon_0} \\ & \therefore E \propto r^2\end{aligned}$
Using Gauss's Law,
$\begin{aligned} & E .2 \pi r l=\frac{\int_0^r K r .2 \pi r l . d r}{\varepsilon_0} \\ & \Rightarrow E=K \frac{r^2}{3 \varepsilon_0} \\ & \therefore E \propto r^2\end{aligned}$
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