3.1 Coulomb's law in vector relations

  Electrostatics

Coulomb's law can be written as, $$\overrightarrow F = \frac{1}{{4\pi {\varepsilon _0}}}\frac{{{q_1}{q_2}}}{{{r^2}}}\widehat r$$ As we know, $$\widehat r = \frac{{\overrightarrow r }}{{\left| {\overrightarrow r } \right|}} = \frac{{\overrightarrow r }}{r}$$ So, $$\overrightarrow F = \left( {\frac{1}{{4\pi {\varepsilon _0}}}\frac{{{q_1}{q_2}}}{{{r^3}}}} \right)\overrightarrow r $$
Also, $${\overrightarrow F _{AB}} = - {\overrightarrow F _{BA}}$$$$\left| {{{\overrightarrow F }_{AB}}} \right| = \left| { - {{\overrightarrow F }_{BA}}} \right|$$
The above equation shows that the force between the charged particles are equal in magnitude but opposite in direction.

For the above configuration of charged particles, the Coulomb's law can be written as, $$\overrightarrow F = \frac{1}{{4\pi {\varepsilon _0}}}\frac{{{q_1}{q_2}}}{{{{\left| {{{\overrightarrow r }_2} - {{\overrightarrow r }_1}} \right|}^3}}}\left( {{{\overrightarrow r }_2} - {{\overrightarrow r }_1}} \right)$$

where,

$\overrightarrow r = \left( {{{\overrightarrow r }_2} - {{\overrightarrow r }_1}} \right)$

$r = \left| {{{\overrightarrow r }_2} - {{\overrightarrow r }_1}} \right|$