Electrostatics : Charge, Force, and Field

1. Electrostatics: The word “Static” means anything that does not move or change with time i.e. remains at rest. Hence, we can say that Electrostatics deals with the study of forces, fields, and potentials arising from static charges. 

2. Electric Charge: Electric charge is an inherent characteristic of matter. Any piece of matter, even if electrically neutral as the whole, actually consists of elementary charged particles e.g., electrons and protons. It is the fundamental property of the object by which it experiences electric effects. There are two types of charges. Positive and negative. The unit of charge is Coulomb. It is represented by C. Electric charge is of two types (i.e., positive and negative). The charge of the electron is $-1.6*10^{-19}$  C. 

3. There are some properties of electric charges: 

(i). The same charges repel each other and opposite charges attract each other. Here is an example. We are given three charges. A and B are positive charges so these will repel each other while B and C are opposite charges so attract each other. What about A and C? Think about it. 

(ii). Conservation of the charge: It means charge of an isolated system is conserved. It is possible to create or destroy charged particles but it is not possible to create or destroy net charge. 

(iii). Quantization: The Charge on a body is an integral multiple of the charge of the electron. i.e. Q = ne where n is an integer and e is a charge of an electron. 

(iv). Additive Property: Charge is a scalar quantity so it can be added like a scalar. 

(v). Velocity invariant: This property states the fact that the charge of a moving body will not change with velocity or speed. It will remain the same irrespective of the speed of the body. 

4. Charging of the body: A body can get charge in different ways like conduction, Induction, or friction. 

(i). Charging by friction: Suppose I have a comb and I rubbed it with my hairs. When this comb is taken near any paper then it attracts the paper. This means that comb is charged. We all have tried it. This phenomenon of charging of comb is due to rubbing or friction. Because of friction, the comb gets charged and it attracts the paper. 

(ii). Charging by Conduction: It is a very simple method like making a circuit to glow a bulb. Suppose we have an iron rod and we connect it to the circuit then it will get charge. 

(iii). Charging by Induction: Let us review the example of the comb. Comb get charged due to friction but the paper also gets some charge due to which it attracts the comb. This phenomenon is called Induction. So in simple words, Induction is a phenomenon in which if a charged body is taken near the neutral body then it will get appositively charged toward the charged body and the same charge on another side so that the total charge on a neutral body is zero. 

5. Charge Distribution: Charge is distributed over an object in two ways either Discrete or Continuous. Discrete charge distribution means the charge is distributed randomly not continuously. Continuous charge distribution means the charge is distributed continuously without any gap.

(a). Linear charge Density: If the charge is distributed continuously over the length of the object, It is called Linear charge Density. It is represented by lambda (λ).   

$\lambda = \frac{dQ}{dL}$

(b). Surface Charge Density: If the charge is distributed continuously over the surface area of the object, It is called Linear charge Density. It is represented by sigma (σ). 

$\sigma = \frac{dQ}{dS}$

(c). Volume charge Density: If the charge is distributed continuously inside the bulk of objects, it is called Linear charge Density. It is represented by rho (ρ). 

$\rho = \frac{dQ}{dV}$

6. The force between two Charges: Coulomb's Law: This law is given by Coulomb. It states that the force between two charges is proportional to the product of two charges. 

$F \propto Q_1Q_2$                                                                                                      (i) 

It also states that the force between two charges is inversely proportional to the square of the distance between two charges 

$F \propto \frac{1}{ r^2}$                                                                                           (ii)

It is assumed that the medium between these two charges is air/vacuum. Combining these two equations, we get the equation 

$|F| = \frac{Q_1 Q_2}{4 \pi \epsilon_0 r^2}$ 

 where proportionality constant is 

 $\frac{1}{4 \pi \epsilon_0}$ = $9 * 10^9 Nm^{2}C^{-2}$ 

 $\epsilon_0 = 8.85 * 10^{-12} C^{2}N^{-1}m^{-2}$ 

Force between two charges follows "Newton's Third of motion". i.e. $\vec{F_{12}}=-\vec{F_{21}}$

7. If there is any medium between two charges then the force will be given by 

 $|F| = \frac{Q_1 Q_2}{4 \pi \epsilon_0 \epsilon_r r^2}$ 

        = $\frac{Q_1 Q_2}{4 \pi \epsilon_0 k r^2}$ 

Where k is the dielectric constant of the medium. Force between two charges is independent of the presence of other charge.  But if medium is changed between them then force will change.

8. Dielectric Constant: It is defined as the ratio between the force between two charges in the air to force in any medium.

 $k = \frac {F_{air}}{F_{medium}}$ 

The value of dielectric constant of medium is greater than 1 generally. K = 1 for air or vacuum. For metals, the dielectric value is infinity.

9. Vector Form of Coulomb’s Law: Two charges q1 and q2 with their position vector r1 and r2. Let us assume that F12 is the force on q1 due to q2. Then force F12 is given by equation 

 $\vec{F}$ = $\frac {q_1q_2 }{4 \pi \epsilon_0 r^3}\vec{r}$

The direction is given by Vector law i.e. $\vec{R}$ =  $\vec{a}+ \vec{b}$ .

10. Superposition Principle: It states that the total force on a charged particle is the vector sum of the individual force of each particle on the charged particle. 

 $\vec{F_{1,net}}$ = $\vec{F_{12}} + \vec{F_{13}}+ . . . . . . . + \vec{F_{1n}}$ 

                                            or  

$\vec{F_{1,net}}$ = $\frac{q_1}{4 \pi \epsilon_0 } \Sigma_{i=2}^{n} \frac {q_i}{r_{ij}^{2}}\vec{ r_{1i}}$ 

11. Electric Field: Region in which the effect of any charge can be experienced. It is defined mathematically as Electric Force per unit charge. 

$\vec{E}$ = $\frac{\vec{F}}{q_0}$ 

The unit of Electric Field is Newton/Coulomb (N/C). The field is in the same as the direction of force if the charge is positive, opposite if the charge is negative. 

12. Electric Field Lines: Electric field lines are imaginary lines that tangent to which give the direction of the electric field at point. 

    Properties: 

        1. These lines start from a positive charge and enter into a negative charge. 

        2. Tangent to line gives the direction of Electric field. 

        3. These lines never form a closed loop. 

        4. Lines are imaginary, not real. 

        5. These lines do not intersect. If these intersect then there will be two directions of the electric                 field  which is not possible. 

        6. The density of these lines determines the EF strength at that point. If lines are dense then EF is             stronger than the point where these are not dense. These lines are diverse as going out of the                 charge.

13. The electric field is radially outward if the charge is positive and inward for a negative charge.

14. The electric field also follows the Superposition Principle. 

i.e. $\vec{E_{net}}$ = $\vec{E_{1}} + \vec{E_{2}}+ . . . . . . . + \vec{E_{n}}$

15. Electric Field strength is given by the number of electric field lines crossing unit area normally. More the number of field lines, larger is strength and vice-versa.

16. Electric field lines for uniform field are parallel.

17. Electric Dipole: It is the arrangement of two oppositely charges of the same magnitude placed at some small separation. It is a vector quantity. Dipole Moment is given by $\vec{p} = 2q\vec{a}$  where a is the separation between two charges. SI unit of Dipole Moment is C-m. Its direction is from the negative charge to the positive charge.   

18. Electric field due to a dipole at a point at a distance r from the center of the dipole along its axial line is given by 

$\vec{E} = \frac{1}{4 \pi \epsilon_0 } \frac{2r\vec{P}}{{(r^2-a^2)}^2}$

and for a short dipole or for large distance ( a << r), we have 

$\vec{E} = \frac{1}{4 \pi \epsilon_0 } \frac{2\vec{P}}{r^3}$

Direction of Electric Field  $\vec{E}$ is along the direction of the dipole moment.

19. The Electric field due to Dipole at a point at a distance r from dipole along the equitorial line is given by 

$\vec{E} = - \frac{1}{4 \pi \epsilon_0 } \frac{\vec{P}}{{(r^2+a^2)}^{\frac{3}{2}}}$

and for a short dipole or for large distance ( a << r), we have 

$\vec{E} = - \frac{1}{4 \pi \epsilon_0 } \frac{\vec{P}}{r^3}$

Direction of Electric Field  $\vec{E}$ is opposite the direction of the dipole moment.

20. The net force experienced by a dipole in uniform Electric Field is zero.

21. Electric Dipole experience a torque in the uniform electric field but no net force

i.e  $\vec{\tau} = \vec{p} * \vec{E}$                                   * indicate cross product.

so $\tau = pEsin{\theta}$                     and  $\vec{F} = 0$ 

Due to this torque, dipole will try to rend along the Electric field. Maximum torque experienced by the dipole is $\tau_{max} = pE$  i.e. when dipole is perpendicular to Electric field. Torque is zero when dipole is either parallel or anti-parallel to the direction of Electric field.

22. If Dipole is placed in the non-uniform electric field then it will experience force as well as torque.

23. If charge distribution is discrete then we find the Electric field due to each charge and add them according to vector law of addition to find the total Electric Field. 

24. If charge distribution is continuous then first find the electric field due to the small charge element and integrate it over the total charge to get the total electric field. 

25. Electric field due to linear charge density is given by 

$E = \frac{1}{4 \pi \epsilon_0 } \int_l   \frac{\lambda dl}{r^2}$  

where $\lambda$  is linear charge density  r is separation of  point from element of length dl.

26. Electric field due to surface charge density is given by 

$E = \frac{1}{4 \pi \epsilon_0 } \int_S   \frac{\sigma dS}{r^2}$  

where $\sigma$  is surface charge density  r is separation of  point from element of length dS.

27. Electric field due to linear charge density is given by 

$E = \frac{1}{4 \pi \epsilon_0 } \int_V   \frac{\rho dV}{r^2}$  

where $\rho$  is volume charge density  r is separation of  point from element of length dV.   

 28. Electric Flux    $\Phi_E$ passing through an area  $\vec{S}$ placed in a uniform electric field  $\vec{E}$ is given by 

$\Phi_E = \vec{E}.\vec{S}  = EScos\theta$  

where $\theta$ is the angle which electric field makes with the normal of the surface area. Unit of Electric flus is $Nm^2C^{-1}$ or V-m. If $\theta$ < 90 then flux will be positive. If $\theta$ > 90 then flux will be negative. Electric flux is maximum when $\theta$ = 0 i.e. normal vector is parallel to the electric field vector. 

29. Gauss' Law states that total flux over a closed surface in free space is $\frac{1}{\epsilon_0}$ times the total charge enclosed with in the surface. 

Mathematically,   $\Phi_E = \int \vec{E}.\vec{dS} = \frac{1}{\epsilon_0}Q$ where Q is the total charge  enclosed with in closed surface.  It follows the inverse square law strictly.

30. Total flux of the dipole with in a closed surface is zero as total charge of dipole is zero. 

31. Guassian surface is a real or imaginary surface constructed to apply Gauss law.

32. Gauss Theorem is applied to find the Electric field, charge with in closed surface.

33. Electric field due to linear charged wire with linear charge density $\lambda$ at a distance r is given by,

$E = \frac {\lambda}{2 \pi \epsilon_0 r}$ 

34. Electric field due to infinite charged plane sheet is  $E = \frac {\sigma}{2 \epsilon_0}$ where $\sigma$  is surface charge density. Electric field due to infinite charged thick sheet is  $E = \frac {\sigma}{ \epsilon_0}$ where $\sigma$  is surface charge density.

35. Electric field due to spherical shell with radius R is given by 

$E = \frac{Q}{4 \pi \epsilon_0  r^2 }$                      outside the shell

$E = \frac{Q}{4 \pi \epsilon_0  R^2 }$                     on the surface

E = 0                                                      inside the shell

36. Electric field due to solid sphere with radius R is given by 

$E = \frac{Q}{4 \pi \epsilon_0  r^2 }$                       outside the sphere

$E = \frac{Q}{4 \pi \epsilon_0  R^2 }$                      on the surface

$E = \frac{Qr}{4 \pi \epsilon_0  R^3 }$                    inside the sphere

Video Lectures:

1. Electric Charge and it's properties watch video

2. Electric force: Coulomb's law, vector form of the coulomb's law watch video

3. Principle of super-position, Continuous charge distribution watch video

4 Electric field, electric field due to group of charges, electric field lines watch video

5 Electric Dipole, Dipole moment, Electric field at axial line and equatorial point watch video

6 Electric field at any point due to dipole, Torque on dipole and Electric field due to ring watch video

7 Area vector, Electric flux and Gauss theorem watch video

8 Electric field due to infinite long wire and spherical shell watch video

9 Electric field due to solid sphere and thin plane sheet watch video