9.1 Electric field due to a point charge

  Electrostatics

Let a point charge $q$ is place at point $O$.

So, for calculating the electric field at the point $P$ which is at a distance $r$ from point $O$, we will draw a symmetrical gaussian spherical surface of radius $r$ from the charge.

So, at every point, the electric field has the same magnitude $E$ and it is perpendicular to the surface.

So, we can apply Gauss's law as, $$ES = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$
where,

$S = 4\pi {r^2}:$ Surface area of sphere
${q_{in}} = q$: Net charge enclosed within the Gaussian surface

So, $$E\left( {4\pi {r^2}} \right) = \frac{q}{{{\varepsilon _0}}}$$$$E = \frac{q}{{4\pi {\varepsilon _0}{r^2}}} = \frac{{kq}}{{{r^2}}}$$
We get same equation for electric field due to a point charge $q$ at a distance $r$ from Coulomb's law.