1. Tardigrade
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  3. Physics
  4. A spherically symmetric charge distribution is characterised by a charge density having the following variation: ρ(r)=ρ0(1-(r/R)) for r < R ρ (r)=0 for r ge R Where r is the distance from the centre of the charge distribution and ρ0 is a constant. The electric field at an internal point (r < R) is

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Q. A spherically symmetric charge distribution is characterised by a charge density having the following variation :
$\rho\left(r\right)=\rho_{0}\left(1-\frac{r}{R}\right)$ for $r < R$
$\rho \left(r\right)=0 \,$ for $r\ge R$
Where $r$ is the distance from the centre of the charge distribution and $\rho_{0}$ is a constant. The electric field at an internal point $(r < R)$ is

JEE MainJEE Main 2014Electric Charges and Fields

Solution:

$\oint \varepsilon . d s=\frac{\Sigma a}{\varepsilon_{0}}$
$\varepsilon\left(4 \pi r^{2}\right)=\rho \cdot \int_{0}^{r}\left(1-\frac{r}{R}\right) 4 \pi r^{2} d r$
$4 \pi \rho_{0}\left[\frac{r^{3}}{3}-\frac{r^{2}}{4 R}\right]_{0}^{q}=\varepsilon\left(4 \pi r^{2}\right)$
$\varepsilon=\frac{\rho_{0}}{\varepsilon_{0}}\left(\frac{r}{3}-\frac{r^{2}}{4 R}\right)$

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