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)

(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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