COULUMB’S LAW

The first quantitative measurements of electric force between point charge bodies was made by Charles Augustin de Coloumb in 1785 by means of a very sensitive balance (torsion balance).

It states that the magnitude of force between two point charges is directly proportional to the product of two charges and inversely proportional to the square of the distance between them.”

Let q1 and q2 be the two charges separated by a distance r in vacuum.

\begin{array}{l}<br /><br /> F \propto \,\frac{{{q_1}{q_2}}}{{{r^2}}}\\<br /><br /> F = \frac{{K{q_1}{q_2}}}{{{r^2}}}\,\,\, = \frac{1}{{4\pi { \in _0}}}\frac{{{q_1}{q_2}}}{{{r^2}}}<br /><br /> \end{array}

Where K is the proportionality constant.

K = \frac{1}{{4\pi { \in _0}}}    where    { \in _0}= absolute permittivity of free space = 8.85\, \times \,{10^{ - 12}}\,{C^2}{N^{ - 1}}{m^{ - 2}}

∴    K = \frac{1}{{4\pi \, \times \,8.85 \times {{10}^{ - 12}}}}\, = \,9 \times {10^9}N{m^2}{C^{ - 2}}

In any medium, force is:

{F_m}\, = \,\frac{1}{{4\pi { \in _0}{ \in _r}}}\,\frac{{{q_1}{q_2}}}{{{r^2}}}

Where { \in _r} is called relative permittivity of the dielectric medium or dielectric constant.

Note:

(a) Coulomb’s law hold good only for stationary charges.

(b) It hold good for point charges only i.e their linear dimension are much smaller than the distance.

(c) It is a central force i.e the force will act along the line joining the centres of two charged bodies.

(d) It is a conservative force.

(e) If q1q2 > 0, i.e like charges repel each other.

If q1q2 < 0, i.e unlike charges attract each other.

(f) If charges are placed in any other medium then {F_{med}}\, = \,\frac{{{F_{vaccum}}}}{{{ \in _r}}}.

(g) The variation of electrostatic force with respect to  \frac{1}{{{r^2}}}  (for q1q2 > 0 and q1q2 < 0) is as shown in the graph.

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(h) The variation of force with respect to distance r is as shown in the graph.

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(i) The electrostatic force between two charges is spherically symmetric.

(j) Electrostatic force between two charges is not affected by the presence of other charges. Hence electrostatic force is a two body interaction.

(k) Dimension of { \in _0}

\begin{array}{l}<br /><br /> \,\,F = \frac{1}{{4\pi { \in _0}}}\frac{{{q_1}{q_2}}}{{{r^2}}}\\<br /><br /> { \in _0}\,\, = \frac{{{q_1}{q_2}}}{{4\pi F{r^2}}}<br /><br /> \end{array}

\displaystyle \,\,\,=\frac{\left[ AT \right]\left[ AT \right]}{\left[ ML{{T}^{-2}} \right]\left[ {{L}^{2}} \right]}=\frac{\left[ {{A}^{2}}{{T}^{2}} \right]}{\left[ M{{L}^{3}}{{T}^{-2}} \right]}=\left[ {{M}^{-1}}{{L}^{-3}}{{T}^{4}}{{A}^{2}} \right]

 

COULOMB’S LAW IN VECTOR FORM:

Consider two point charges q1 and q2 are kept apart by a distance r in vacuum or air.

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{{\overrightarrow{F}}_{12}} = force on (1) due to (2)

\displaystyle {{\overrightarrow{F}}_{12}}\,=\,K\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}.{{\widehat{r}}_{21}}  [Here {\widehat r_{21}} is a unit vector pointing from q2 to q1].

Similarly,  \displaystyle {{\overrightarrow{F}}_{21}}\,=\,K\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}.{{\widehat{r}}_{12}}   [Here {\widehat r_{12}} is a unit vector pointing from q1 to q2].

Since {\widehat r_{21}}\, = \, - {\widehat r_{12}}

 {{\overrightarrow{F}}_{12}}\,=\,-{{\overrightarrow{F}}_{21}}

Thus the force extended by two charges on each other is equal and opposite i.e., they obey Newton’s third law of motion.

DEFINITION OF 1 COULOMB:

We have,

{F_{vacuum}} = \,\frac{{K{q_1}{q_2}}}{{{r^2}}}

If q1 = q2 = 1 C, r = 1m and { \in _r}\, = \,\,1,

Then F = K = 9 x 109 N

“Thus one coulomb is that quantity of charge which exerts a force of 9 x 109 N on each other when kept apart 1m in vacuum.

Post Author: E-Physics

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