Current Electricity

1. Current electricity deals with flow of electric charges. Flow of electric charges constitutes an electric current. By flow we mean of directed motion of charges.

2. The current strength or the current through a given area of a conductor is net charges passing per unit time through that area. Mathematically, instantaneous electric current \[I(t) = \frac{dq}{dt}\]where dq is the charge flown through the given area in time dt. Electric current is a scalar quantity.

3. SI unit of electric current is 1 ampere (1 A). Current is side to be a 1 A if rate of flow of charge is 1Cs^-1.  As in case of metallic conductors current is mainly due to flow of free electrons, hence in terms of electron flow 1 A electric current means flow of 6.25 * 10¹⁸ electrons through a cross section of conductor per second.

4. Conventional direction of electric current is the direction of motion of positive charge. Thus, conventional current is in a direction opposite to that of the direction of flow of electrons.

5. If current flows through a conductor at a steady rate in a given direction, then it is called direct steady current (d.c.). However, if direction of current remains unchanged but its magnitude varies then such a current is called a varying current. If magnitude as well as direction change periodically, the current is called an alternating current.

6. When current (i.e., charge) flows in a conductor, the equilibrium distribution of charges is violated and the surface of the conductor is no longer an equipotential surface. Thus, there is an electric field inside the conductor and a  tangential component of electric field on the surface is also present.

7. For starting and maintaining an electric current following two conditions should be fulfilled :(i) Sufficient number of charge carriers (free electron in metals, ions and ionic solids and electrolytes, electrons and ions in gases, electron and holes in semiconductors etc.) should be available. (ii) An external electric field must exist whose energy is used in starting and maintaining the flow of charge carriers. In other words, a source of electric energy is required.

8. To maintain a steady current, we must have a closed circuit in which an external agency transports electric charge from lower to higher potential energy. The work done per unit charge by the source in taking the charge from lower to higher potential energy (i.e., from one terminal of the source to the other) is called the electromotive force or emf to the source.

9. It should be clearly noted that the emf is not a force, it is the potential difference between the two terminals of a source in an open circuit. SI unit of emf is 1 volt (1 V ).

10. Positive electric charge flow from higher potential to lower potential. External source is needed to push the charge back from lower potential to higher potential.

11. In a conductor through which a current is flowing, Ohm’s law is stated as “physical conditions (temperature etc.) remaining unchanged the current flowing, through a conductor is directly proportional to the potential difference across its ends“ i.e. $ V \propto I $ or \[V = {I}{R}\] Where R is called the resistance of given conductor.

12.Resistance of a conductor is a measure of opposition offered by it for flow of electric current through itself. Mathematically, resistance of a given conductor is equal to the potetial difference being maintained across its ends in order to maintain steady flow of unit amount of current. SI unit of resistance is 1 ohm $ (1 \Omega ).$  

13. The resistance of a given conductor at a given temperature is (i) directly proportional to its length l, (ii) inversely proportional to its cross- section area A, and (iii) depends upon the nature of the material of conductor. Thus, \[R = \rho \frac{l}{A}\]Where p is known as the resistivity of the material of given conductor.

14. Resistivity of the material of a conductor is defined as the resistance offered by a conductor of that material having unite length and unit cross-section area. SI unit of resistivity is ohm- meter.For a given material, resistivity is independent of its dimensions. 

15. Reciprocal of resistance is called conductance. Thus, Conductance \[G = \frac{1}{R} = \frac{I}{V}\]SI unite of conductance is   or mho or siemen (S). 

16. Reciprocal of resistivity is called the conductivity of the given material. Thus, \[\sigma = \frac{1}{\rho} = \frac{l}{RA} = \frac{Gl}{A}\]SI unit of electric conductivity is Sm^-1.

17. A perfect conductor would have zero resistivity and a perfect insulator would have infinite resistivity. Generally, good electric conductors are also good thermal conductors. Pure metals are good conductors having low resistivity in the range of 10^-8 -10^-6 ohm m. Alloys have somewhat higher resistivity then pure metals. Insulators like glass, rubber etc., have extremely high resistivity ranging from 10¹⁰ m to 10¹⁶ m. The semiconductors like germanium, silicon from a class intermediate between the conductors and insulators. Their resistivity may vary from 10^-5 m to 10³m. 

18. Ohm’s law is not a fundamental law of nature. Substance following Ohm’s law have a linear V – I characteristics and are known as ohmic resistors. Metallic conductors are ohmic resistors. Non-ohmic resistors are those, V – I characteristic for which may have any shape other then a straight line passing through the origin. Electrolytes, semiconductor, vacuum tubes, solar cells, transistors, diodes etc., are some examples of non-ohmic resistors. For resistors in the high range from few kilo ohms to a mega ohm, generally, carbon resistors are used. Carbon resistors are compact, inexpensive and, thus, find extensive use in electronic circuits.

19. For carbon resistors a colour code has been provided. According to it, generally four bands are provided on the body of a given resistor. The first-two bands indicate the first-two significant figure of the resistance. Third band indicate the decimal multiplier. Fourth band stands for tolerance. 

Color	Number	Multiplier	Tolerance ( % )
Black	0	$ 10^0 $	-
Brown	1	$ 10^1 $	-
Red	2	$ 10^2 $	-
Orange	3	$ 10^3 $	-
Yellow	4	$ 10^4 $	-
Green	5	$ 10^5 $	-
Blue	6	$ 10^6 $	-
Violet	7	$ 10^7 $	-
Gray	8	$ 10^8 $	-
White	9	$ 10^9 $	-
Gold	-	$ 10^{-1} $	5
Silver	-	$ 10^{-2} $	10
No color	-	-	20

 Method to remember the color codes is " BB ROY of Great Bharat has a Very Good Wife wearing Gold Silver Necklace".

20. To understand the electric conduction in conductor, free electrons are treated as electron gas. In the absence of an electric field, the average velocity of free electrons is zero because their direction are random. On applying an external electric field, the electrons move on an average with a drift speed $ v_d $ in a direction opposite of the electric field. The drift speed is given by \[v_d = \frac{eE}{m}\tau \implies \vec{v_d} = - \frac{e\vec{E}}{m}\tau\]Where e = electronic charge, m = mass of electron, E = external field and τ = the average time between successive collisions of electrons with the atoms or ions of the conductor and is knows as the relaxation period.

21. In terms of drift speed electric current is given by \[I = neAv_d \implies I = -ne(\vec{A}.\vec{v_d})\]Where n = number density of free electrons, A = normal area of cross-section of the conductor. Direction of conventional current is opposite to that of drift velocity of electrons.

22. Current density J is a vector quantity and the magnitude of  J is the amount of charge flowing per unit cross-section area per second. Alternately, current per unit area ( taken normal to the current ) is called current density. Its unit is A m^-2.  Current is given by \[I = \vec{J}.\vec{A}\]In terms of drift velocity,\[\vec{J} = -ne\vec{v_d}\]Here – ve sign implies that direction of current density is opposite to that of drift velocity. 

23. On the basis of the concept of dirft speed of electron, the resistance of a conductor of length l and cross-section area A is given by \[R = \frac{m}{ne^2\tau} \frac{l}{A}\]And the resistivity of the material of conductor is given by \[\rho = \frac{m}{ne^2\tau}\]Thus, resistivity of the material of a conductor is (i) inversely propositional to the number density of free electrons, and (ii) inversely proportional to the relaxation time τ. In term of current density Ohm’s law may be expressed as \[\vec{J} = \frac{\vec{E}}{\rho} = \sigma \vec{E}\]Where $\vec{E} $ is the external electric field. Thus, current density $\vec{J} $ is in the direction of $\vec{E} $.

24. Magnitude of drift velocity per unit electric field is known as the mobility. Thus, mobility\[\mu = \frac{v_d}{E} = \frac{e}{m} \tau\]The SI unit of mobility is m²/V-s.

25. For good conductors drift speed is of the order of 10^-4 -10^-5 m s^-1, whereas relaxation period is of the order of 10^-14s or even less.

26. The resistivity (and hence the resistance) of all metallic conductors increase with increase in temperature. Over a limited temperature range the relationship is linear and is given by : \[\rho_T = \rho_0[1+\alpha (T - T_0)]\]Where ρ₀ is the resistivity at a reference temperature T₀ and ρ_T its value of temperature T. Factor a is called the temperature coefficient of resistivity of given metal. In terms of resistance of a metallic conductor, we can write \[R_T = R_0[1+\alpha (T - T_0)]\]For elemental metals e.g., Cu, Al, etc., value of $\alpha $ is of the order of 10^-3K^-1. However, for alloy like manganin (an alloy consisting of 83% Cu, 4% Ni and 13% Mn) and constantan (an alloy of copper and nickel) value of a is extremely small (of the order of 10^-5 K^-1 ). Therefore, for preparing standard resistor generally manganin or constantan wire is preferred. In case of insulators as well as semiconductors the electrical resistivity decreases with increase in temperature (or conductivity increases with increase in temperature).

27. The equivalent resistance of the resistors in series arrangement is equal to the sum of the individual resistance i.e., \[R_{eq} = R_1 + R_2 + R_3 +....... +R_n\]. In series arrangement same current flows through all the resistors and potential difference across various resister are in the ratio of individual resistances. 

28. In parallel grouping of resistances the reciprocal of the resultant (equivalent) resistance is equal to the sum of the reciprocal of the individual resistances. i.e. \[\frac{1}{R_P} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\frac{1}{R_4 }+.......\]. In parallel arrangement of resistors potential difference across all the resistor is same but individual Currents are in the inverse ratio of respective resistances.

29. Resistance offered by the electrolyte of a cell is known as the internal resistance of the given cell. Internal resistance of a cell ‘r’ depends upon

(i)                  Nature and concentration of electrolyte,

(ii)                Nature of electrodes,

(iii)               Surface area of electrodes immersed into the electrolyte, and

(iv)               Separation between the electrodes.

30. Current through a resistor R when connected to a cell of emf e and internal resistance r is given by \[I = \frac{\epsilon}{R+r}\] 

31. Emf of a cell is the potential difference between its terminals when the cell is in an open circuit i.e., when no current is being drown from the cell. Internal resistance of the cell can be calculated by using the formula \[r = \frac{\epsilon - V}{V}R\]

32. If a current I flows through a conductor for time t such that potential drop across the ends of conductor be V (or resistance of conductor be R), then amount of electric energy lost or heat energy produced is given by \[H = I^2Rt = VIt = \frac{V^2}{R}t\]This relation is known as Joule’s law of heating.

33. Electric power is defined as the rate of electric energy supplied per unit time to maintain flow of electric current through a conductor. Mathematically,\[P = VI = I^2R = \frac{V^2}{R}\] SI unit of electric power is 1 watt (1 W), where 1 watt =1 volt * 1 ampere.

34. When n cell, each of emf e and internal resistance r, are joined in series the net emf of combination =$ n\epsilon $, net internal resistance of combination = $ nr $ and the current flowing through an external resistance R is given by  \[I = \frac{n\epsilon}{nr + R}\]

36. If a cells of respective emfs $ \epsilon_1 , \epsilon_2 ,\epsilon_3, ...... $ and internal resistance r₁, r_{2 …….}etc., are joined in series such that current leaves each cell from positive electrode, then resultant emf of combination\[\epsilon_{eq} = \epsilon_1 + \epsilon_2 +\epsilon_3 + .......+\epsilon_n\] and the internal resistance of the combination \[r_{eq} = r_1 + r_2 + r_3 + .......+r_n\]

37. When n cells, each of emf $ \epsilon $ and internal resistance r, are joined in parallel, the net emf of the combination = $ \epsilon $, net internal resistance of the combination = $ \frac{r}{n} $ and the current flowing through an external resistor R is given by \[I = \frac{n\epsilon}{nR + r}\]

38. If n cells of respective emfs $ \epsilon_1 , \epsilon_2 ,\epsilon_3, ...... $ and respective internal resistance r₁, r₂ ……….  are joined in parallel, then equivalent internal resistance r_eq and equivalent emf e_eq of the combination are given by \[\frac{1}{r_{eq}} = \frac{1}{r_1} +\frac{1}{r_2}+\frac{1}{r_3}+.........+\frac{1}{r_n}\]   and \[\frac{\epsilon_{eq}}{r_{eq}} = \frac{\epsilon_1}{r_1}+\frac{\epsilon_2}{r_2}+\frac{\epsilon_3}{r_3}+...........+\frac{\epsilon_n}{r_n}\]. 

39. For drawing maximum power from a cell or cell combination the total internal resistance of cell combination should be exactly equal to the external resistance joined in the circuit.

40. To analyse complicated electrical networks Kirchhoff gave two laws :

(i) Junction rule or current rule – the algebraic sum of all currents flowing into a junction is zero. In the order words, at any junction, the sum of currents entering the junction must be equal to the sum of current leaving it. Kirchhoff’s first law of the principle of conservation of electric charge. Mathematically, \[\Sigma I = 0\]While applying Kirchhoff’s current rule the current flowing towards the junction is taken + ve and the current flowing away from the junction is taken -ve

(ii) Loop rule or voltage rule – the algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop must be zero. Mathematically, \[\Sigma V = 0 \implies \Sigma(\epsilon + RI) = 0\]Kirchhoff’s second law is based on the principal of conservation of energy.

While applying Kirchhoff’s voltage rule the change in potential in traversing a resistance in the direction of current is -IR while in the opposite direction + IR. Again the change in potential in traversing an emf source from negative to positive terminal is +e while in the opposite direction -e.

41. A wheatstone bridge is an arrangement of four resistances used to measure one of them in terms of other three. The bridge is said to be balanced when the galvanometer gives null deflection. For a balanced wheatstone  bridge  \[\frac{P}{Q} = \frac{R}{S}\]

Unknown resistance,S is given by\[S = \frac{Q} {P} . R\]Generally, P and Q are called the ratio arms and R the variable arm of bridge. In a Wheatstone bridge cell and galvanometer can be mutually interchanged without affecting the balance condition. Moreover, resistance of diagonally opposite arms (i.e., P and S and Q and R) may be mutually interchanged. For maximum sensitivity, all the four resistance used in a Wheatstone bridge should be equal or nearly equal. 

42. A metre bridge (also known as a slide wire bridge) is a practical form of wheatstone bridge. It is used to find the value of an unknown resistance X by using the formula  \[X = R \frac{(100-L)}{L}\]Where R is the known resistance and L is distance of null point from the side of resistance R. 

43. A potentiometer is an extremely sensitive and precise device to compare emfs of cells, to measure small potential difference and to measure internal resistance of a cell. A potentiometer works on the principle that for a constant current flowing through the potentiometer wire of uniform cross-section the fall in potential is directly proportional to the length. While measuring an unknown potential/ emf of a cell we balance the unknown potential against an adjustable and measuring potential difference created along the potentiometer wire.

44. The potential gradient of the potentiometer is defined as the fall in potential per unit length along the potentiometer wire. Mathematically, potential gradient k = I . σ, where I is the current flowing through the potentiometer wire and σ is the resistance per unit length of wire. For greater sensitivity potential gradient along the potentiometer wire should have a small value. This is achieved either by increasing the length of potentiometer wire or by joining a high resistance in series with the potentiometer.

45. For comparing emfs of two cells using a potentiometer, we use the formula \[\frac{\epsilon_1}{\epsilon_2} = \frac{l_1}{l_2}\] Where l₁ and l₂ are the balancing length of potentiometer wire for two cells respectively.

46. A potentiometer is superior than a voltmeter for measuring potential difference and cells because measurement is done here in balance condition, when no current is being drawn from given cell.

47. Internal resistance of a cell ‘r’ can also be measured by the use of a potentiometer by using the formula  \[r = \frac{l_1 - l_2}{l_2} R\]Where l₁ = balancing length in open circuit of cell and l₂ = balancing length when circuit of given cell is closed through an external resistance R.

Video Lectures:

1. Electric Current, Current carriers and Drift velocity watch video

2 Mobility, Relation between current and drift velocity and ohm's law watch video

3 Resistance, resistivity and their temperature dependence watch video

4 Resistance combination watch video

5 Cells, terminal potential and grouping watch video 

6 Kirchhoff's Law watch video

7 Wheatstone and meter bridge watch video

8 Potentiometer and it's application watch video

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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

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