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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 er 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}}    

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