Coulomb Law|What is Coulomb Law ?|Electrostatic Force

Coulomb’s law : In 1785, the French physicist Charles Augustin Coulomb (1736-1806) experimentally measured the electric forces between small charged spheres by using a torsion balance. He formulated his observations in the form of Coulomb’s law which is electrical analogue of Newton’s law of Universal Gravitation in mechanics.

Coulomb Law

Coulomb’s law states that the force of attraction or repulsion between two stationary point charges is

(i) Directly proportional to the product of the magnitudes of the two charges and

(ii) Inversely proportional to the square of the distance between them. This force acts along the line joining the two charges.

If two point charges q_{1}

and q_{2} are separated by distance r, then the electric forces F of attraction or repulsion between them is such that

\displaystyle F\propto q_{1}q_{2}\ and\ F\propto \frac{1}{r^{2}}

\displaystyle \therefore F\propto \frac{q_{1}q_{2}}{r^{2}}\ or\ F=k \frac{q_{1}q_{2}}{r^{2}}

where k is a constant of proportionality, called electrostatic force constant. The value of k depends on the nature of the medium between the two charges and the system of units chosen to measure F, q_{1}

,q_{2} and r.

For the two charges located in free space and in SI units, we have

\displaystyle k=\frac{1}{4\pi \epsilon _{0}}=9\times 10^{9}\ Nm^{2}C^{-2}

where \displaystyle \epsilon _{0}

is called permittivity of free space.

So we can express Coulomb’s law in SI units as

Units of charge

(i) The SI unit of charge is coulomb. In the above equation, if q_{1}=q_{2}=1\ C

and r=1 m, then

\displaystyle F=\frac{1}{4\pi \epsilon _{0}}=9\times 10^{9}\ N

So one coulomb is that amount of charge that repels an equal and similar charge with a force of 9x 10 N when placed in vacuum at a distance of one metre from it.

(ii) In electrostatic cgs system, the unit of charge is known as electrostatic unit of charge (e.s.u. of charge) or statcoulomb (stat C).

One e.s.u. of charge or one statcoulomb is that charge which repels an identical charge in vacuum at a distance of one centimetre from it with a force of 1 dyne.

\displaystyle 1\ Coulomb=3\times 10^{9}\ Statcoulomb=3\times 10^{9}\ e.s.u\ of\ charge

(iii) In electromagnetic cgs system, the unit of charge is abcoulomb or electromagnetic unit of charge (e.m.u. of charge).

\displaystyle 1\ coulomb=\frac{1}{10}abcoulomb=\frac{1}{10}\ e.m.u. \ of\ charge


Coulomb Law in vector form

As shown in Fig. 1.8, consider two positive point charges q_{1}

and q_{2}, placed in vacuum at distance r from each other. They repel each other.
Repulsive coulombian force
Repulsive coulombian force

In vector form, Coulomb’s law may be expressed as

\displaystyle \vec{F_{12}}= Force\ on\ charge\ q_{2}\ due\ to\ q_{1}

\displaystyle \Rightarrow \vec{F_{12}}=\frac{1}{4\pi \epsilon _{0}}\cdot \frac{q_{1}q_{2}}{r^{2}}\widehat{r_{21}}

where \displaystyle \widehat{r_{21}}=\frac{\vec{r_{21}}}{r}

, is a unit vector in the direction from \displaystyle q_{1}  to \displaystyle q_{2}.


\displaystyle \vec{F_{21}}= Force\ on\ charge\ q_{1}\ due\ to\ q_{2}

\displaystyle \Rightarrow \vec{F_{21}}=\frac{1}{4\pi \epsilon _{0}}\cdot \frac{q_{1}q_{2}}{r^{2}}\widehat{r_{12}}

where \displaystyle \widehat{r_{12}}=\frac{\vec{r_{12}}}{r}

, is a unit vector in the direction from \displaystyle q_{2}  to \displaystyle q_{1}.

The coulombian forces between unlike charges \displaystyle \left ( q_{1}q_{2}<0 \right )

are attractive, as shown in Fig.

Importance of vector form

The vector form of coulomb’s law gives the following additional information:

1. As \displaystyle \widehat{r_{21}}=-\widehat{r_{12}}

, therefore \displaystyle \vec{F_{12}}=-\vec{F_{21}} . This means that the two charges exert equal and opposite forces on each other. So Coulombian forces obey Newton’s third law of motion.

2. As the Coulombian forces act along \displaystyle \vec{F_{12}}

or \displaystyle \vec{F_{21}} i.e., along the line joining the centres of two charges, so they are central forces.
  •  What is the range over which Coulombian forces can act ? State the limitations of Coulomb’s law in electrostatics.

Range of coulombian forces

Coulombian forces act over an enormous range of separations (r) nuclear dimensions (r=\displaystyle 10^{-15}

) to macroscopic distances as large as \displaystyle <10^{18}\ m . Inverse square is valid over this range of separation to a high degree of accuracy.

Limitations of Coulomb Law

Coulomb’s law is not applicable in all situations. It is valid only under the following conditions:

1. The electric charges must be at rest.

2. The electric charges must be point charges i.e., the extension of charges must be much smaller than the separation between the charges.

3. The separation between the charges must be greater than the nuclear size (\displaystyle 10^{-15}

), because for distances \displaystyle <10^{-15}\ m , the strong nuclear force dominates over the electrostatic force. -15,

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