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Let there be a spherically symmetric charge distribution with charge density varying as $\rho (r)=\rho _{0}\left ( \frac{5}{4}-\frac{r}{R} \right )\; upto\; r=R,and\; \rho (r)=0\; for\; r> R,$ where $r$ is the distance from the origin. The electric field at a distance $r(r< R)$ from the origin is given by

Option 1)
$\frac{\rho _{0}r}{3\varepsilon _{0}}\left ( \frac{5}{4}-\frac{r}{R} \right )$

Option 2)
$\frac{4\pi \rho _{0}r}{3\varepsilon _{0}}\left ( \frac{5}{3}-\frac{r}{R} \right )$

Option 3)
$\frac{\rho _{0}r}{4\varepsilon _{0}}\left ( \frac{5}{3}-\frac{r}{R} \right )$

Option 4)
$\frac{4\rho _{0}r}{3\varepsilon _{0}}\left ( \frac{5}{4}-\frac{r}{R} \right )$

Answers (2)

best_answer

As we learnt in

Gauss's Law -

Total flux linked with a closed surface called Gaussian surface.

Formula:

$\phi = \oint \vec{E}\cdot d\vec{s}=\frac{Q_{enc}}{\epsilon _{0}}$

- wherein

No need to be a real physical surface.

Q_enc - charge enclosed by closed surface.

Volume of shell $dV=4\pi\:x^{2}}dx$

$Q_{in}=\int_{0}^{r}\rho\:dv=\int_{0}^{r}\rho _{o}(\frac{5}{4}-\frac{x}{R})4\pi\:x^{2}dx$

$=4\pi\:\rho _{o}\int_{0}^{r}\rho _{o}(\frac{5}{4} x^{2}-\frac{x^{3}}{R})dx$

$=4\pi\:\rho _{o}[\frac{5}{12} x^{3}-\frac{x^{4}}{4R}]_{o}^{r}$

$=4\pi\:\rho _{o}[\frac{5}{12} r^{3}-\frac{r^{4}}{4R}]$

$=\frac{4\pi\:\rho _{o}}{4}[\frac{5}{3} r^{3}-\frac{r^{4}}{R}]$

$=\pi\:\rho _{o}[\frac{5}{3} r^{3}-\frac{r^{4}}{R}]$

$E.\ 4\pi\:r^{2}=\frac{q_{in}}{\varepsilon _{o}}$

$E.4\pi\:r^{2}=\frac{\pi\rho _{o}}{\varepsilon _{o}}[\frac{5}{3}r^{3}-\frac{r^{4}}{R}]$

$E=\frac{\pi \rho_{o}r^{3}}{4\pi r^{2}\varepsilon_{o}}[\frac{5}{3}-\frac{r}{R}]$

$\Rightarrow E=\frac{\rho_{o}{r}}{4\varepsilon_{o}}[\frac{5}{3}-\frac{r}{R}]$

Option 1)

$\frac{\rho _{0}r}{3\varepsilon _{0}}\left ( \frac{5}{4}-\frac{r}{R} \right )$

This is incorrect option

Option 2)

$\frac{4\pi \rho _{0}r}{3\varepsilon _{0}}\left ( \frac{5}{3}-\frac{r}{R} \right )$

This is incorrect option

Option 3)

$\frac{\rho _{0}r}{4\varepsilon _{0}}\left ( \frac{5}{3}-\frac{r}{R} \right )$

This is correct option

Option 4)

$\frac{4\rho _{0}r}{3\varepsilon _{0}}\left ( \frac{5}{4}-\frac{r}{R} \right )$

This is incorrect option

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prateek

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