CBSE Physics for Class 12 | Electrostatics | Coulomb’s Law

Coulomb’s Law.

If two stationary and point charges $\displaystyle Q{}_{1}$ and $\displaystyle Q{}_{2}$ are kept at a distance r, then it is found that force of attraction

or repulsion between them is $\displaystyle F\propto \frac{Q{}_{1}Q{}_{{{2}_{{}}}}}{{{r}^{2}}}$ i.e., $\displaystyle F=\frac{kQ{}_{1}Q{}_{2}}{{{r}^{2}}}$ ; (k = Proportionality constant)

(1) Dependence of k :

Constant k depends upon system of units and medium between the two charges.

(i) Effect of units

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

(b) In S.I. for air $\displaystyle k=\frac{1}{4\pi {{\varepsilon }_{0}}}=9\times {{10}^{9}}\frac{N-m{}^{2}}{C{}^{2}}$ , $\displaystyle F=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}$ Newton (1 Newton = 10⁵
Dyne)

Note :

$\displaystyle {{\varepsilon }_{0}}=$ Absolute permittivity of air or free space = $\displaystyle 8.85\times {{10}^{-12}}\frac{{{C}^{2}}}{N-{{m}^{2}}}$ $\displaystyle \left( =\frac{Farad}{m} \right)$ . It’s Dimension is $\displaystyle [M{{L}^{-3}}{{T}^{4}}{{A}^{2}}]$
$\displaystyle {{\varepsilon }_{0}}$ Relates with absolute magnetic permeability ( $\displaystyle {{\mu }_{0}}$ ) and velocity of light (c) according to the following relation $\displaystyle c=\frac{1}{\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}}$

(ii) Effect of medium

(a) 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 known as dielectric constant or specific inductive capacity (SIC) of the medium, K is also called relative permittivity e_r of the medium (relative means with respect to free space).

Hence in the presence of medium $\displaystyle {{F}_{m}}=\frac{{{F}_{air}}}{K}=\frac{1}{4\pi {{\varepsilon }_{0}}K}.\,\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}$

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

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

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

(2) Vector form of coulomb’s law :

Vector form of Coulomb’s law is

$\displaystyle {{\overrightarrow{F\,}}_{12}}=K.\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{3}}}{{\overrightarrow{\,r}}_{12}}=K.\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}{{\hat{r}}_{12}},$

where $\displaystyle {{\hat{r}}_{12}}$ is the unit vector from first charge to second charge along the line joining the two charges.

(3) A comparative study of fundamental forces of nature

S.No.	Force	Nature and formula	Range	Relative strength
(i)	Force of gravitation between two masses	Attractive F = Gm₁m₂/r², obey’s Newton’s third law of motion, it’s a conservative force	Long range (between planets and between electron and proton)	1
(ii)	Electromagnetic force (for stationary and moving charges)	Attractive as well as repulsive, obey’s Newton’s third law of motion, it’s a conservative force	Long (upto few kelometers)	$\displaystyle {{10}^{37}}$
(iii)	Nuclear force (between nucleons)	Exact expression is not known till date. However in some cases empirical formula $\displaystyle {{U}_{0}}{{e}^{r/{{r}_{0}}}}$ can be utilized for nuclear potential energy $\displaystyle {{U}_{0}}$ and $\displaystyle {{r}_{0}}$ are constant.	Short (of the order of nuclear size 10^–15 m)	10³⁹ (strongest)
(iv)	Weak force (for processes like b decay)	Formula not known	Short (upto 10^–15m)	10²⁴

Note :

Coulombs law is not valid for moving charges because moving charges produces magnetic field also.
Coulombs law is valid at a distance greater than $\displaystyle {{10}^{-15}}m.$
A charge $\displaystyle {{Q}_{1}}$ exert some force on a second charge $\displaystyle {{Q}_{2}}$ . If third charge $\displaystyle {{Q}_{3}}$ is brought near, the force of $\displaystyle {{Q}_{1}}$ exerted on $\displaystyle {{Q}_{2}}$ remains unchanged.
Ratio of gravitational force and electrostatic force between (i) Two electrons is 10^–43/1. (ii) Two protons is 10^–36/1 (iii) One proton and one electron 10^–39/1.
Decreasing order to fundamental forces $\displaystyle {{F}_{Nuclear}}>{{F}_{Electromagnetic}}>{{F}_{Weak}}>{{F}_{Gravitational}}$

(4) 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 $\displaystyle {{Q}_{1}}$ , $\displaystyle {{Q}_{2}}$ , $\displaystyle {{Q}_{3}}$ …are applying force on a charge Q

Net force on Q will be

$\displaystyle {{\vec{F}}_{net}}={{\vec{F}}_{1}}+{{\vec{F}}_{2}}+..........+{{\vec{F}}_{n-1}}+{{\vec{F}}_{n}}$