Question

Let there be a spherically symmetric charge distribution with charge density varying as $\rho \left(r\right)={\rho }_0(\frac{5}{4}-\frac{r}{R})$ upto $r=R$, and $\rho \left(r\right)=0$ for $r>R$, where $r$ is the distance from the origin. Find the electric field at a distance $r(r<R)$ from the origin.

• A. $\frac{3{\rho }_{0}r}{4{ϵ}_0}(\frac{5}{3}-\frac{r}{R})$
• B. $\frac{4{\rho }_{0}r}{3{ϵ}_0}(\frac{5}{3}-\frac{r}{R})$
• C. $\frac{{\rho }_{0}r}{4{ϵ}_0}(\frac{5}{3}-\frac{r}{R})$
• D. $\frac{3{\rho }_{0}r}{4{ϵ}_0}(\frac{5}{4}-\frac{r}{R})$

Answer 1

The correct option is C $\frac{{\rho }_{0}r}{4{ϵ}_0}(\frac{5}{3}-\frac{r}{R})$

Imagine a gaussion surface that is a sphere r<R .

charge enclosed by surface

$dQ=\rho .dv\int dQ=\int \rho .dvQ=\int {\rho }_o[\frac{5}{4}-\frac{r}{R}]4\pi r^{2}dr={\rho }_{o}4\pi {[\frac{5r^3}{12}-\frac{r^4}{4R}]}_0^r{\rho }_{o}4\pi \left[R\frac{5r^3-3r^4}{12R}\right]\frac{Q}{{\epsilon }_0}=E.4\pi r^{2}E=\frac{Q}{{4\pi r^2\epsilon }_0}E=\frac{{r\rho }_o}{{4\epsilon }_0}[\frac{5}{3}-\frac{r}{R}]$