JEE Main & Advanced Physics Electrostatics & Capacitance Coulomb's Law

If two stationary and point charges \[Q{}_{1}\] and \[Q{}_{2}\] are kept at a distance r, then it is found that force of attraction or repulsion between them is

\[F\propto \frac{Q{}_{1}Q{}_{{{2}_{{}}}}}{{{r}^{2}}}\] i.e., \[F=\frac{kQ{}_{1}Q{}_{2}}{{{r}^{2}}}\](k = Proportionality constant)

In C.G.S. (for air ) \[k=1,\] \[F=\frac{{{Q}_{1}}\,{{Q}_{2}}}{{{r}^{2}}}\] Dyne

In S.I. (for air) \[k=\frac{1}{4\pi {{\varepsilon }_{0}}}=9\times {{10}^{9}}\frac{N\text{-}m{}^{2}}{C{}^{2}}\]

\[\Rightarrow \] \[F=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}\] Newton (1 Newton \[={{10}^{5}}\] Dyne)

\[{{\varepsilon }_{0}}=\]Absolute permittivity of air or free space

\[=8.85\times {{10}^{-12}}\frac{{{C}^{2}}}{N-{{m}^{2}}}\]\[\left( =\frac{Farad}{m} \right)\]. It's Dimensional formula is \[[{{M}^{-1}}{{L}^{-3}}{{T}^{4}}{{A}^{2}}]\]

(1) Vector form of coulomb's law : Vector form of Coulomb's law is \[{{\overrightarrow{F\,}}_{12}}=K.\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{3}}}{{\overrightarrow{\,r}}_{12}}=K.\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}{{\hat{r}}_{12}},\] where \[{{\hat{r}}_{12}}\] is the unit vector from first charge to second charge along the line joining the two charges.

(2) Effect of medium : When a dielectric medium is completely filled in between charges rearrangement of the charges inside the dielectric medium takes place and the force between the same two charges decreases by a factor of K (dielectric constant)

i.e.   \[{{F}_{medium}}=\frac{{{F}_{air}}}{K}\]\[=\frac{1}{4\pi {{\varepsilon }_{0}}K}.\,\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}\]

(Here \[{{\varepsilon }_{0}}K={{\varepsilon }_{0}}\,{{\varepsilon }_{r}}=\varepsilon \] = permittivity of medium)

If a dielectric medium (dielectric constant K, thickness t) is partially filled between the charges then effective air separation between the charges becomes \[(r-t\,+t\sqrt{K})\]

Hence force \[F=\frac{1}{4\pi {{\varepsilon }_{0}}}\,\frac{{{Q}_{1}}{{Q}_{2}}}{{{(r-t+t\sqrt{K})}^{2}}}\]

(3) Principle of superposition : According to the principle of super position, total force acting on a given charge due to number of charges is the vector sum of the individual forces acting on that charge due to all the charges.

Consider number of charge \[{{Q}_{1}}\],\[{{Q}_{2}}\],\[{{Q}_{3}}\] ... are applying force on a charge Q.

Net force on Q will be

\[{{\overrightarrow{F}}_{net}}={{\overrightarrow{F}}_{1}}+{{\overrightarrow{F}}_{2}}+....+{{\overrightarrow{F}}_{n-1}}+{{\overrightarrow{F}}_{n}}\]

The magnitude of the resultant of two electric forces is given by

\[{{F}_{net}}=\sqrt{F_{1}^{2}+F_{2}^{2}+2{{F}_{1}}{{F}_{2}}\cos \theta }\]

and        \[\tan \alpha =\frac{{{F}_{2}}\sin \theta }{{{F}_{1}}+{{F}_{2}}\cos \theta }\]

For problem solving remember following standard results.

Fundamental forces of nature