A spherically symmetric charge distribution is considered with charge density varying as ρ r = ρ 0 3 4 − r R for r ≤ R Zero for r R Where, r r

Question

A spherically symmetric charge distribution is considered with charge density varying as

$ρ (r) = \{\begin{cases} ρ_{0} (\frac{3}{4} - \frac{r}{R}) & for r \leq R \\ Zero & for r > R \end{cases}$

Where, $r (r < R)$ is the distance from the centre $O$ (as shown in figure). The electric field at point $P$ will be :

TestHub · Accepted Answer

According to Gauss's law, the electric flux through a closed surface area is equal to the charge inside the surface divided by $ε_{0}$ .
$\Rightarrow \oint \vec{E \cdot} d \vec{s} = \frac{Q_{in}}{ε_{0}}$
$\Rightarrow E 4 π r^{2} = \frac{\int_{0}^{r} ρ_{0} (\frac{3}{4} - \frac{r}{R}) 4 π r^{2} d r}{ε_{0}}$
$\Rightarrow E 4 π r^{2} = \frac{ρ_{o} 4 π}{ε_{o}} (\frac{3}{4} \frac{r^{3}}{3} - \frac{r^{4}}{4 R})$
$\Rightarrow E = \frac{ρ_{o} r}{4 ε_{o}} (1 - \frac{r}{R})$

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