Electrostatics

Electrostatics pertains to the study of stationary electric charges and their associated phenomena and properties in physics.. It primarily focuses on the behavior of these charges and their interactions at rest without considering the effects of their motion (which falls under electrodynamics). This may include the study of electric Charge, coulomb's law, electric field, electric Potential, gauss law, electrostatics potential energy, capacitors.

1.0Electric charge

Charge is an inherent characteristic of matter that generates and interacts with electric and magnetic phenomena..

Types of Charges

(1). Positive charge- Due to deficiency of electrons.

(2). Negative charge-Due to excess of electrons

Units of Charge

• SI - Coulomb(C)
• CGS-Stat coulomb (stat C) 
• Dimensional Formula -[AT]

2.0Basic Properties of Electric charge

(1). Additivity of electric Charge.

(2). Conservation of electric Charge

(3). Quantization of Electric Charge

(4). Charge can be transferred

(5). Charge is invariant.

(6). Charge is a scalar quantity

(7). Charge is always associated with mass

Methods of charging

(1). Charging by Friction

(2). Charging by Induction

(3). Charging by Conduction

3.0Coulomb’s Law

The electrostatic force between two point charges is proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. This force acts along the line connecting the two charges at all times.

$\mathrm{F}=\mathrm{K}\frac{q_1{q}_2}{r^2}$

$\mathrm{k}=\frac{1}{4\pi {ϵ}_0}=9\times 10^9{\mathrm{Nm}}^2{C}^{-2}$ = Electrostatic constant or Coulomb’s Constant

Coulomb's Law in Vector Form

$\overrightarrow{F_{21}}=\frac{1}{4\pi {ϵ}_0}{q}_1{q}_2\frac{\left(\overrightarrow{r_2}-{\vec{r}}_1\right)}{\mid {\vec{r}}_2-{\vec{r}}_{1}3}$ or 

$|F_{12}=\frac{1}{4\pi {ϵ}_0}{q}_1{q}_2\frac{\overrightarrow{r_1}-\overrightarrow{r_2}}{|\overrightarrow{r_1}-\overrightarrow{r_2}|3}$

4.0Superposition Principle

When several charges interact, the total force on a given charge is the vector sum of all the individual forces exerted on it by all other charges.

$\vec{F}=\overrightarrow{F_{01}}+\overrightarrow{F_{02}}+\overrightarrow{F_{03}}+\dots ..\overrightarrow{F_{0n}}$

$\vec{F}=\frac{k{q}_0{q}_1}{r_1^2}\widehat{r_1}+\frac{k{q}_0{q}_2}{r_2^2}\widehat{r_2}+\dots \dots +\frac{k{q}_0{q}_i}{r_i^2}\widehat{r_i}+\dots .\frac{k{q}_0{q}_n}{r_n^2}\widehat{r_n}$

Relative Permittivity $\left({ϵ}_r\right)$ or Dielectric Constant (K)

Dielectric Constant (K) = $\frac{\text{ Permittivity of a Medium }\left({ϵ}_r\right)}{\text{ Permittivity of }\mathrm{Vacuum}\left({ϵ}_0\right)}$ 

$K=\frac{{ϵ}_r}{{ϵ}_0}$

$ϵ={ϵ}_0{ϵ}_r$

The value of ${ϵ}_r\text{ or }\mathrm{K}\ge 1.$

5.0Electric Field Intensity

• The space around a charge or charge distribution, where another charge experiences an electric force, is called an electric field.
• Mathematically, $\vec{E}=\frac{\vec{F}}{q_0}$ and SI Unit : N/C or V/m
• $\vec{E}={\lim }_{q_0\to 0}\frac{\vec{F}}{q_0}$
• Dimensional Formula: $\left[M^1{L}^1{T}^{-3}{A}^{-1}\right]$
• It is a vector quantity.

6.0Electric Field Lines 

• Electric field lines are conceptual lines, either straight or curved, used to represent the direction and strength of the electric field surrounding an electrically charged object. a region such that the tangent at any point on the field lines gives the direction of the electric field lines at that point in the area.
• Two electric field lines can never intersect each other.

• Electrostatic field lines can never form any closed loop

Electric field lines due to positive or negative charges

7.0Electric line of force due to an electric dipole

8.0Continuous Charge Distribution

A collection of closely positioned electric charges constitutes a continuous charge distribution.

(1). Linear Charge Density ($\lambda$): $\lambda =\frac{Q}{l}$

$\text{ SI Unit }\frac{C}{m}$

(2). Surface Charge Density ($\sigma$): $\sigma =\frac{Q}{S}$

$\text{ SI Unit }\frac{C}{m^2}$

(3). Volume Charge Density $\text{ ( }\rho \text{ ) }$

$\rho =\frac{Q}{V}$

$\text{ SI Unit }\frac{C}{m^3}$

9.0Electric Dipole

It is a pair  of two same  and opposite charges placed at small separation.

• The dipole moment equals the product of the magnitude of either Charge and the separation between the charges: $\vec{p}=q(\overrightarrow{2l})$
• S.I. unit- Cm

Electric Field Due to a Dipole

(1) At Axial / End on position

${\mathbf{E}}_{\text{Axial }}=\frac{12p}{4\pi {ϵ}_0{r}^3}$

(2) At Equator/Broadside on position

${\mathbf{E}}_{\text{Equi }}=\frac{1p}{4\pi {ϵ}_0{r}^3}$

(3) At general position

$\mathbf{E}=\frac{p}{4\pi {ϵ}_0{r}^3}\sqrt{3{\cos }^2\vartheta +1}$

$\mathrm{Tan}\alpha =\frac{1}{2}\mathrm{Tan}\theta$

Electric Field Intensity due to a charged wire

Special Cases:

1. For Infinite Wire, ( both ends goes to infinite)

${\mathrm{E}}_{\mathrm{X}}=E_{\perp }=\frac{2K\lambda }{r}$

$E_Y=E_{\Vert }=0$

2. For Semi-infinite wire

${\mathrm{E}}_{\mathrm{X}}={\mathrm{E}}_{\perp }=\frac{K\lambda }{r}$

${\mathrm{E}}_Y=E_{\Vert }=\frac{K\lambda }{r}$

${\mathrm{E}}_{\text{Resultant }}=\sqrt{2}\frac{K\lambda }{r}$

3. Electric field due to finite wire at symmetric point:

${\mathrm{E}}_{\mathrm{x}}={\mathrm{E}}_{\perp }=\frac{2K\lambda }{r}\sin \theta$

$E_Y=E_{\Vert }=0$

4. Electric Field due to a uniformly charged Arc

$\mathrm{E}=\frac{2K\lambda }{R}\mathrm{Sin}\left(\frac{\theta }{2}\right)$

5. Electric field at centre of uniformly charged Ring

$\mathrm{E}=\frac{2K\lambda }{R}\mathrm{Sin}\left(\frac{\theta }{2}\right),\theta \to \text{ angle of arc }$

$\theta =360^{\circ }$

$\mathrm{E}=\frac{2K\lambda }{R}\mathrm{Sin}\left(\frac{360}{2}\right)$

E = 0

6. Electric Field due to a Uniformly charged Ring at its Axis

$\mathbf{E}=\frac{KQx}{{\left(R^2+x^2\right)}^{3/2}}$

Special cases:

• Electric field on the axis for small values of x: $\mathrm{E}=\frac{KQ}{R^3}\cdot \mathrm{X}$
• Electric field at the centre of the ring is zero because x = 0
• Electric field at the axis for larger values of x: $\mathrm{E}\simeq \frac{kQ}{X^2}$
• Maximum value of electric field

$\frac{dE}{dx}=0$

$\mathrm{x}=\mp \frac{R}{\sqrt{2}}$

$E_{\text{max }}=\frac{2kQ}{3\sqrt{3}{R}^2}$

10.0Electrostatic Potential Energy(EPE)

• Electrostatic Potential Energy, is the amount of work done in bringing the charge from infinity to the present position without changing its kinetic energy.

Electrostatic Potential Energy (EPE) of two point charge system 

$\Delta U={\left(W_{\infty \to r}\right)}_{\text{ext }}=-{\int }_{\infty }^r{\vec{F}}_{\text{ext }}\cdot \overrightarrow{dr}$

$\Rightarrow U_r-U_{\infty }=-{\int }_{\infty }^r{\vec{F}}_{\text{ext }}\cdot \overrightarrow{dr}=\frac{kQq}{r}$

$\left(\therefore U_{\infty }=0\right)$

$\Rightarrow U-0=\frac{kQq}{r}$

$\Rightarrow U=\frac{kQq}{r}$

Electrostatic Potential energy of a three system of charges

$U_S=U_{12}+U_{23}+U_{13}$

$U_S=\frac{k{q}_1{q}_2}{r_{12}}+\frac{k{q}_2{q}_3}{r_{23}}+\frac{k{q}_1{q}_3}{r_{13}}$

Note: For n point charges system 

No. of pairs $=\frac{n(n-1)}{2}$

11.0Electric potential

It is defined as the work done by an external agent in taking a unit positive charge from a reference point  to that point without changing its Kinetic Energy.

$V_p=\frac{{\left(w_{\infty \to p}\right)}_{ext}}{q}(\Delta K=0)$

• It is scalar quantity
• Potential can be positive, negative and even zero
• SI unit is joule/coulomb

Electrostatic potential due to a point charge

• $\mathrm{V}=\frac{kQ}{r}$

Expression for potential due to multiple Charges

$\mathrm{V}=V_1+V_2+V_3+\dots \dots ..V_n$

$V_1=\frac{1}{4\pi {ϵ}_0}\frac{q_1}{r_{1P}},V_2=\frac{1}{4\pi {ϵ}_0}\frac{q_2}{r_{2P}},V_3=\frac{1}{4\pi {ϵ}_0}\frac{q_3}{r_{3P}}$

12.0Equipotential Surface(E.P.S)

Locus of all such points which are at the same potential is called equipotential surface.

Point charge

Plane Charge Sheet

Unlike Charges(Electric Dipole)

Like Charges(Similar Charges)

13.0Solved Questions on Electrostatics

Q-1 Two spheres having equal charges are kept at long separation. If the gravitational force between two spheres is equal to electrostatic force between them. Find the ratio of specific charge $\frac{q}{m}$.

Solution:

$\frac{k{q}^2}{r^2}=\frac{G{m}^2}{r^2}$

$\frac{q}{m}=\sqrt{\frac{G}{k}}=\sqrt{\frac{6.67\times 10^{-11}}{9\times 10^9}}=0.86\times 10^{-10}$

Q-2 Two thin spherical shells made of metal are at a larger distance apart . One of radius 10 cm carries a charge of +0.5 mC and the other, of radius 20 cm , carries a charge of + 0.7 mC. The charge on each, when they are connected by a suitable conducting wire is respectively.

Solution:

$Q_1=\left(\frac{R_1}{R_1+R_2}\right)\mathrm{Q}=\left(\frac{10\mathrm{cm}}{30\mathrm{cm}}\right)1.2\mathrm{mC}\Rightarrow Q_1=0.4\mathrm{mC}$               

$Q_1=\left(\frac{R_2}{R_1+R_2}\right)\mathrm{Q}=\left(\frac{20\mathrm{cm}}{30\mathrm{cm}}\right)1.2\mathrm{mC}\Rightarrow Q_2=0.8\mathrm{mC}$

Q-3. The electric field in a certain region is given by $\vec{E}=\left(\frac{K}{x^3}\right)\hat{i}$. The dimensions of K are.

Solution:

Dimension of electric field = [M¹L¹T^--3A^-1]

$\vec{E}=\left(\frac{K}{x^3}\right)\hat{i}$

$\mathrm{E}=\frac{K}{x^3}$

$\mathrm{K}=\mathrm{E}{x}^3$

( ∴ x representing  the distance so the dimensional formula is [L]

K=[M¹L¹T^--3A^-1] [L]³

K=[M¹L⁴T^-3A^-1]

Q-4. If the electric flux entering and leaving a closed surface is ${\varphi }_1$ and ${\varphi }_2$ respectively then electric charge inside the surface will be.

Solution:

According to Gauss Theorem, $\varphi =\frac{q}{{ϵ}_0}$

Net Flux $(\varphi )={\varphi }_2-{\varphi }_1$                    

${\varphi }_2-{\varphi }_1=\frac{q}{{ϵ}_0}$           

$\mathrm{q}=\left({\varphi }_2-{\varphi }_1\right){ϵ}_0$                 

Q-5. The electric potential and electric field at a point due to a point charge are 600 V and 200 N/C respectively. Calculate the magnitude of point Charge.

Solution:

Electric Potential $(\mathrm{V})=\frac{kq}{r}=600$ ……..(1)

Electric Field $(\mathrm{E})=\frac{kq}{r^2}=200$ ……..(2)

Squaring both the sides in eq.(1)

$\frac{k^2{q}^2}{r^2}=360000$ ………(3) 

From eq (2) and (3)

 $kq$= 1800

$\mathrm{q}=\frac{1800}{k}=\frac{1800}{9\times 10^9}=200\times 10^{-9}c$