Two parallel plane sheets $1$ and $2$ carry uniform charge densities ${\sigma }_1$ and ${\sigma }_2,$ as shown in figure. The magnitude of the resultant electric field in the region marked $\text{'}I\text{'}$ is $\left({\sigma }_1>{\sigma }_2\right)$

Using Gauss law, derive expression for electric field due to a spherical shell of uniform charge distribution $\sigma$ and radius $R$ at a point lying at a distance $x$ from the centre of shell, such that $0<x<R$.

Using Gauss law, derive expression for electric field due to a spherical shell of uniform charge distribution $\sigma$ and radius $R$ at a point lying at a distance $x$ from the centre of shell, such that $x>R$.

State Gauss's law in electrostatics and write in its mathematical form.Calculate the electric field $\overrightarrow{\mathrm{E}}$ at point $\mathrm{P}$ due to charges this spherical shell, shown in figure (1) What will be the field $\overrightarrow{E}$ shown in fig (2) and (3)? Given the surface charge density is $\sigma$.

Find an expression for the electric field intensity due to a uniformly charged thin spherical shell at a point outside the shell.