Answer to Question #175776 in Electricity and Magnetism for sadeep

Question

Consider a hollow charged shell of inner radius a and outer radius b . The volume

charge density is ρ(r) = k

r

2

( k is constant) in the region a < r < b. The magnitude

of the electric field produced at distance r > a is ?

Answer 1

Given,

Radius of the inner spherical shell = a

Radius of the outer spherical shell = b

Volume charge density $(\rho)=kr^2$

$a<r<b$

Electric field at a distance $a<r$ ?

$dQ= \rho 4\pi r^2 dr$

Electric field at a distance r, from the center of the spherical shell,

$dE(r)=\frac{1}{4\pi \epsilon_o}\frac{dQ}{r^2}$

Now, taking the integration of both side,

$\int_0^E dE(r)=\int_a^r\frac{1}{4\pi \epsilon_o}\frac{dQ}{r^2}$

Now, substituting the values,

$E(r) =\frac{1}{4\pi \epsilon_o}\int_a^r\frac{\rho \times 4\pi r^2}{r^2}dr$

$\Rightarrow E(r) =\frac{1}{4\pi \epsilon_o}\int_a^r\frac{kr^2 \times 4\pi r^2}{r^2}dr$

$\Rightarrow E(r)=\frac{4\pi\times k}{4\pi \epsilon_o}\int_a^r(r^2)dr$

$\Rightarrow E(r) = \frac{k}{\epsilon_o}(\frac{r^3}{3})_a^r$

$\Rightarrow E(r)=\frac{k}{3\epsilon_o}(r^3-a^3)$