Answer to Question #211534 in Electricity and Magnetism for Rocky Valmores

Question

 A hollow spherical shell carries charge density ρ = k/r2 in the region a < r < b (see Figure). Find the electric field in the three regions : (I) r < a (II) a < r < b, (III) r > b. Plot the magnitude of E~ as a function of r.

Answer 1

Gives $\rho=\frac{k}{r^2}$

r<a

$Q_r=0\rightarrow(1)$

r>b

$Q_r=Q$

$Q_r=\int _{s_r}\rho d\tau$

$Q_r=4 \pi\int_{a}^{r}\rho (s)s^2 ds$

$Q_r=4 \pi\int_{a}^{r}\frac{k}{s^2}s^2 ds$

$Q_r=4\pi k(r-a)\rightarrow(2)$

$a<r<b$

Similarly

$Q_r=4 \pi\int_{a}^{b}\frac{k}{s^2}s^2 ds$

$Q_r=4\pi k(b-a)\rightarrow(3)$

Gauss law

$$\oint E.da=\oint E.da=4\pi r^2( E_r)=\frac{Q_r}{\epsilon_0}$$

$E_r=\frac{Q_r}{4\pi\epsilon_0 r^2}\rightarrow(4)$

Put $Q_r$ Value

equation (1) and equation (4)

$r<a$

$E_r=\frac{0}{4\pi\epsilon_0 r^2}=0$

equation (2) and (4)

$a<r<b$

$E_r=\frac{4\pi k(r-a)}{4\pi\epsilon_0 r^2}$

$E_r=\frac{k}{\epsilon_0}(\frac{r-a}{r^2}),$

Equation (3) and (4)

$r>b$

$E_r=\frac{Q_r}{4\pi\epsilon_0 r^2}$

$E_r=\frac{4\pi k(b-a)}{4\pi\epsilon_0 r^2}$

$E_r=\frac{k}{\epsilon_0}(\frac{b-a}{r^2})$