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  4. 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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Q. 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

AIIMSAIIMS 2018Electric Charges and Fields

Solution:

$\rho\,\propto\,r;\,\rho=Kr$
$dQ=\rho\,V$
$\quad=Kr\left(2\pi rdr\right)\,\times l$
Using Gauss’s Law,
$E.2\pi rl=\frac{\displaystyle \int_0^r\,Kr.2\pi rl.dr}{\varepsilon_{0}}$
$\Rightarrow \quad E=K \frac{r^{2}}{3\varepsilon_{0}}$
$\therefore \,\,E\,\propto\,r^{2}$

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