First slide

Gauss's Law

Question

Let there be a spherically symmetric charge distribution with charge density varying as $ρ (r) = ρ_{0} (\frac{5}{4} - \frac{r}{R})$ upto r = R, and $ρ$ (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 :

Moderate

A

$\frac{ρ_{0} r}{3 e_{0}} (\frac{5}{4} - \frac{r}{R})$
B

$\frac{4 {πρ}_{0} r}{3 e_{0}} (\frac{5}{3} - \frac{r}{R})$
C

$\frac{ρ_{0} r}{4 e_{0}} (\frac{5}{3} - \frac{r}{R})$
D

$\frac{4 ρ_{0} r}{3 e_{0}} (\frac{5}{4} - \frac{r}{R})$

Solution

Total charge in r < R region
$q = \int ρdv = \int_{0}^{r} ρ_{0} (\frac{5}{4} - \frac{r}{R}) 4 {πr}^{2} dr = 4 {πρ}_{0} [\frac{5}{4} (\frac{r^{3}}{3}) - \frac{1}{R} (\frac{r^{4}}{4})] = {πρ}_{0} r^{3} (\frac{5}{3} - \frac{r}{R})$
Electric field E = $\frac{kq}{r^{2}} = \frac{ρ_{0} r}{4 e_{0}} (\frac{5}{3} - \frac{r}{R})$

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