Question

(a) Using Gauss law, drive an expression for the electric field intensity at any point outside a uniformly charged thin spherical shell of radius R and charge density $\sigma C/m^2$. Draw the field lines when the charge density of the sphere is (i) positive, (ii) negative

(b)A uniformly charged conducting sphere of $2\times 5m$ in diameter has a surfaces density of $100mC/m^2$ . Calculate the (i) charge on the sphere (ii) total electric flux passing through the sphere.

Answer 1

Let a thin spherical shell be considered of radius R with a centre O. Let +q charge be uniformly distributed over the surface of shell.

Acc. to Gauss Law;

$\oint \overrightarrow{E}.\overrightarrow{ds}=\frac{q}{{\epsilon }_0}$

$\oint \overrightarrow{E}.\hat{n}ds=\frac{q}{{\epsilon }_0}$

E$\oint ds=\frac{q}{{\epsilon }_0}$ [$\hat{n}.\hat{n}=1$]

$E\left(4\pi r^2\right)=\frac{q}{{\epsilon }_0}$

$E=\frac{1}{4\pi r^2}.\frac{q}{{\epsilon }_0}$

At any point at the surface of the shell; $r=R$

$E=\frac{1}{4\pi R^2}.\frac{q}{{\epsilon }_0}$

$q=4\pi R^2\sigma$

$E=\frac{1}{4\pi r^2}.\frac{4\pi R^2\sigma }{{\epsilon }_0}$

$E=\frac{\sigma }{{\epsilon }_0}$

a) i) Refer fig. (2)

Field lines when charge density of sphere is positive.

ii) Refer fig. (3)

Field lines when charge density of sphere is negative.

b) Diameter of sphere $=2.5m$

Radius $=1.25m$

Charge density, $\sigma =100\mu C{m}^{-2}={10}^{-4}C/m^3$

i) Total charge, $q=4\pi R^2\sigma =1.96\times {10}^{-3}C$

ii) Total flux, $\varphi =\frac{q}{{\epsilon }_0}=2.21\times {10}^{8}N{m}^2{C}^{-1}$