Question
The electric potential at a point in free space due to a charge $$Q$$ coulomb is $$Q \times {10^{11}}V.$$ The electric field at that point is
A.
$$4\pi {\varepsilon _0}\,Q \times {10^{22}}\,V/m$$
B.
$$12\pi {\varepsilon _0}\,Q \times {10^{20}}\,V/m$$
C.
$$4\pi {\varepsilon _0}\,Q \times {10^{20}}\,V/m$$
D.
$$12\pi {\varepsilon _0}\,Q \times {10^{22}}\,V/m$$
Answer :
$$4\pi {\varepsilon _0}\,Q \times {10^{22}}\,V/m$$
Solution :
As potential at any point due to a point charge is given by,
$$\eqalign{
& V = \frac{{kQ}}{r}\,\,\left[ {k = \frac{1}{{4\pi {\varepsilon _0}}}} \right] \cr
& V = \frac{1}{{4\pi {\varepsilon _0}}} \times \frac{Q}{r} \cr
& E = \frac{1}{{4\pi {\varepsilon _0}}} \times \frac{Q}{{{r^2}}} \cr
& E = \frac{{4\pi {\varepsilon _0}{V^2}}}{Q} = 4\pi {\varepsilon _0} \times \frac{{{Q^2} \times {{10}^{22}}}}{Q} \cr
& E = 4\pi {\varepsilon _0}Q \times {10^{22}}\,V/m \cr} $$