Coulomb's Law And Electric Field

Electrostatics

Charge

Charge is that property of an object by virtue of which it apply electrostatic force of interaction on other objects.

Charges are of two types

(i) Positive charge

(ii) Negative charge

Like charge repel and unlike charges attract each other.

Conductors

Conductors are those substances which can be used to carry or conduct electric charge from one point to other, e.g. silver, copper, aluminium etc.

Insulators

Insulators are those substances which cannot conduct electric charge, e.g. glass, rubber, plastic etc.

Charge by induction

The process of changing a neutral body nearby it without making contact between two bodies is known as charging by induction.

Additivity of Charge

If a system consists of n charges q₁,q₂,q₃………q_n, then the total charge of the system will be q₁+q₂+q₃+q₄+……..q_n.

Quantization of Charge

Charge on any object can be an integer multiple of a smallest charge (e).

$Q=\pm ne$b

Where, n = 1,2,3,……and e = 1.6 × 10^–19 C.

Conservation of Charge

Charge can neither be created nor be destroyed, but can be transferred from one object to another object.

Recently a new particle has been discovered called 'Quark'. It contains charge $\pm \frac{e}{3},\pm \frac{2e}{3}.$

The protons and neutrons are combination of other entities called quarks, which have charges $\frac{1}{3}e.$ However, isolated quarks have not been observed, so, quantum of charge is still e.

Coulomb's Law of Electrostatics

Electrostatic force of interaction acting between two stationary point charges is given by

$F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$

Where, q₁,q₂ are magnitude of point charges, r is the distance between here,

$\frac{1}{4\pi {\epsilon }_0}=\left({10}^{-7}\frac{N-s^2}{C^2}\right)C^2$

Substituting value of c = 2.99792458 × 10⁸ m/s.

We get $\frac{1}{4\pi {\epsilon }_0}=8.99\times 10^{9}N-m^2/C^2$

In example and problem we will often use the approximate value,

$\frac{1}{4\pi {\epsilon }_0}=9\times 10^{9}N-m^2/C^2$

The value of ${\epsilon }_0$ is $8.85\times 10^{12}{C}^2/N-m^2$

If there is another medium between the point charge except air or vacuum, then ${\epsilon }_0$ is replaced by ${\epsilon }_{0}K$ and ${\epsilon }_0{\epsilon }_r$ or $\epsilon .$

Where K of ${\epsilon }_r$ is called dielectric constant or relative permittivity of the medium

$K={\epsilon }_r=\frac{\epsilon }{{\epsilon }_0}$

Where, $\epsilon =$ permittivity of the medium.

For air or vacuum, K = 1

For water, K = 81

For metals, $K=\infty$

Coulomb's Law in Vector Form

Let q₁ and q₂ are positive.

Force on q₂ due to q₁.

{F_{21}}\, = \,\dfrac{1}{{4\pi {\varepsilon _0}}}\,.\,\dfrac{{{q_1}{q_2}}}{{{r^2}}}\,{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{r} _{AB}}

$F_{21}=\frac{1}{4\pi {\epsilon }_0}.\frac{q_1{q}_2}{r^3}{r}_{AB}$

$F_{21}=\frac{q_1{q}_2}{4\pi {\epsilon }_0}.\frac{r_2-r_1}{{\left|r_2-r_1\right|}^3}$

The above equations give the Coulob's law in vector form.

$\because$ Force on q₁ due to q₂ = – Force on q₂ due to q₁

$\because$ F₁₂ = – F₂₁

$F_{12}=\frac{q_1{q}_2}{4\pi {\epsilon }_0}.\frac{r_1-r_2}{{\left|r_1-r_2\right|}^3}$

The forces due to two point charges are parallel to the line joining the point charges; such forces are called central forces and so electrostatic force are conservative forces.

Forces between Multiple Charges

Consider a system of n point charges q₁,q₂,q₃…………..q_n are distributed in space. Let the charges be q₂,q₃……q_n, exert forces F₁₂, F₁₃,…..F_1n on charge q₁. The total force on charge q₁ is given by

{F_1}\, = \,\dfrac{1}{{4\pi {\varepsilon _0}}}\,\left( {\dfrac{{{q_1}{q_2}}}{{r_{12}^2}}{{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{r} }_{21}} + \,\dfrac{{{q_1}{q_2}}}{{r_{13}^2}}{{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{r} }_{31}}\, + \,......... + \,\,\dfrac{{{q_1}{q_2}}}{{r_n^2}}{{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{r} }_{n1}}} \right)

Continuous Charge Distribution

The region in which charges are closely spaced is said to have continuous distribution of charge.

Linear Charge Density $\left(\lambda \right)$

It is defined as the charge per unit length of linear charge distribution. Its unit is coulomb/metre.

Surface Charge Density $\left(\sigma \right)$

It is defined as the charge per unit surface area of surface charge distribution. Its unit is coulomb/metre².

Volume Charge Density $\left(\rho \right)$

It is defined as the charge per unit volume of volume charge distribution. Its unit is coulomb/metre³.

Electric Field

The space in the surrounding of any charge in which its influence can be experienced by other charges is called electric field.

Electric Field Lines

"An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric field vector at that point.

The relative closeness of the lines at some place give an idea about the intensity of electric field at that point."

Two lines can never in intersect.

Electric field lines always begin on a positive charge and end on a negative charge and do not start or stop in mid space.

Electric Field Intensity (E)

The electrostatic force acting per unit positive charge on a point in electric field is called electric field intensity at that point.

Electric field intensity $E=\underset{q_0\to 0}{\lim }\frac{F}{q_0}$

Where F = force experienced by the test charge q₀.

Its SI Unit is NC^–1 or V/m and its dimension is [MLT^–3A^–1].

It is a vector quantity and its direction is in the direction of electrostatic force acting on positive charge.

Electric field intensity due to a point charge q at a distance r is given by

$E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}$

Electric field due to system of Charges

If E is electric field at point P due to the systems of charges is given by

\boxed{E\, = \,\dfrac{1}{{4\pi {\varepsilon _0}}}\,\sum\limits_{i = 1}^n {\dfrac{q}{{r_i^2}}{{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{r} }_i}} }