Electrostatic (Previous year Questions)

Q1. If in the electric field, an electron is kept freely. If the electron is replaced by a proton, what will be the relationship between the forces experienced by them?

Ans. When Electron is replaced by proton then the force on proton will remain the same but the direction of the force gets reverse. 

Q2. How does the Coulombian force between two-point charges depend upon the dielectric constant of the intervening medium?

Ans. If medium has some dielectric constant value, then force will be $ \frac{1}{K} $ times the force when medium is air. K is the dielectric constant.

Q3. Draw electrostatic field lines due to a small conducting sphere having a negative charge on it.

Ans.  Electric fields coming out of the charge.

Q4. An electrostatic field line cannot be discontinuous. Why?

Ans. Electric field lines can not be discontinuous because field is continuous in space and exist at all points in space. Click here

Q5. Two electric field lines never cross each other. Why?

Ans. If electric field lines will intersect then there will be two directions of the electric field at a point which is not possible.

Q6. Sketch the pattern of electric field lines due to an electric dipole.

Ans. Click here    

Q7. Which orientation of an electric dipole in a uniform electric field corresponds to stable equilibrium? 

Ans. When the dipole is placed along the direction of the electric field.

Q8. At what points, is the field due to an electric dipole parallel to the line joining the charges?

Ans. At Equitorial point and axial line, Electric field is parallel to dipole.

Q9. If the radius of the Gaussian surface enclosed a surface is halved, how does the electric flux through the Gaussian surface change?

Ans. There will be no change in electric  flux.  

Q10. Electric field inside a conductor is zero, explain.

Ans. Total charge within the conductor is zero so electric field is zero. 

Q11. The electric field E due to a point charge at any point near it is defined as \[E = \lim_{q_0\to 0 } \frac{F}{q_0}\] Where q₀ is the test charge and F is the force acting on it. What is the physical significance of \[\lim_{q_0\to 0 }\] in this expression? 

Ans. The charge is taken as small as possible so that it's presence does not affect the electric field.

Q12. An electric dipole is free two move in a uniform electric field. Explain its motion.

Ans. In a uniform electric field, total force acting on the dipole is zero but torque acting is not zero. Because of the torque,  dipole rotate in the uniform electric field.

Q13. How much work is done in moving a 500 mC charge between two points on an equipotential surface? 

Ans. Work done on moving a charge over equipotential surface is zero irrespective of the magnitude of the charge.

Q14. In which position, a dipole placed in a uniform electric field is in (i)  stable (ii) unstable equilibrium?

Ans. (i) Dipole is stable when placed along the direction of the electric field.

(ii) Dipole is unstable when placed opposite to the direction of the electric field.

Q15. Why does the electric field inside a dielectric decrease when it is placed in an external electric field?

Ans. Electric field decrease because due to external electric field there is induced dipole moment in the dielectric which gives rise to internal electric field that oppose the external field. That is why field decrease inside the dielectric.

Q16. Can two equipotential surfaces intersect each other? Give reasons.

Ans. No, if they intersect then there will be two directions of the electric field which is not possible.

Q17. A parallel plat capacitor is charged to a potential different V by a d.c. source. The capacitor is then disconnected from the source. If a distance between the plate is doubled, state with reason how the following will change; 

(i) electric field between the plates,

(ii) capacitance, and

(iii)energy stored in the capacitor.

Ans. When plate separation is doubled then capacitance will be half of initial value.

(i) Since, Electric field is given by $ E = \frac{\sigma}{\epsilon_0}$ which is independent of the saparation so electric field will be same.

(ii) Since, Capacitance is given by $ C_i = \frac{\epsilon_0 A}{d} $. When d is increased to 2d then C will be half.

(iii) Energy is given by $U = \frac{Q^2}{2C} $. When C will be half then energy will be double of the initial value of energy.   

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Electrostatics : Potential and Capacitors

1. Electric potential V is a scalar.

2. The electric potential difference between two given points in an electric field is equal to the amount of work done against the electric field in order to bring a unit positive test charge (without acceleration) from one point to the other. Mathematically, the potential difference between the points B and A, i.e., V_B – V_A  is given by \[V_B - V_A= \frac{W_{A \to B}}{q_0}\]  Where $ {W_{A \to B}} $ is the work done in order to carry a test charge q₀ from point A to point B. The value of the test charge should be as small as possible.

3. The electric potential at a point in an electric field is equal to the amount of work done by the external force (against the electric field) in order to bring a unit positive test charge (without acceleration) from infinity to that point. Mathematically, electrostatic potential at a point A is given by \[V_A = \frac{W_{\infty \to A}}{q_0}\]  Where  \[{W_{\infty \to A}}\] is the amount of work done in order to carry a test charge q₀ from to the point, A. value of the test charge should be as small as possible.

4. Electrostatic potential, as well as potential differences, are scalar quantities and their SI unit is volt.

5. Thus, the electric potential at a point is said to be 1 volt, if 1 J of work is being done in order to move a positive test charge of 1 C from infinity to that point.

6. The electric potential at a point situated at a distance r from a point charge q is given by \[V = \frac{q}{4 \pi \epsilon r}\]. Thus, potential due to a free, independent +ve point charge is positive because work is being done against the repulsive force experienced by + ve test charge. However, potential due to a -ve point charge is negative because here work is done by the attractive force acting on the +ve test charge.

7. It is possible to maintain a positively charged body at a negative or zero potential and a negatively charged body at zero or even positive potential.

8. Electric potential of earth is considered to be zero. Thus, electric potential of any body connected to earth will also be zero.

9. If a number of point charges are present then electric potential at a given point is equal to the algebraic sum of potentials due to different charges. Thus, if a number of charges q₁, q₂, q₃ ….  are present at distance r₁, r₂, r_{3 ….}etc. respectively from a given point then total electric at that point is \[V = V_1 + V_2 + V_3 + ......... = \frac{1}{4 \pi \epsilon }[{\frac{q_1}{r_1}+\frac{q_2}{r_2}+\frac{q_3}{r_3}+..........}] = \frac{1}{4 \pi \epsilon } \Sigma_{i=1}^{N} \frac{q_i}{r_i}\].

10. Electrostatic force is a conservative force and electric field is an example of a conservative field. It means that amount of work done in carrying a test charge in an electric field depends only upon the positions of initial point and the final point and is independent of the path followed. Moreover, the work done in carrying a charge in an electric field along a complete cyclic path is zero. Mathematically, \[\oint \vec{E}\vec{dl} = 0\].

11. Electric field at a point due to a given point charge is inversely proportional to the square of distance of given point from the point charge i.e., \[E \propto \frac{1}{r^2}\]. However, electric potential due to a given point charge is inversely proportional to the distance i.e., \[V \propto \frac{1}{r}\]. 

12. Electrostatic potential difference between two points in an electric field may also be defined as negative of the line integral of the electric field between the given points i.e., \[V_B-V_A= - \int_A^B \vec{E}.\vec{dl}\]. The electric potential at a the point in an electric field is equal to negative of the line integral of the electric field from infinity to that point i.e. \[V_A = - \int_{\infty}^A \vec{E}\vec{dl} \]

13. Electric field at a point may, thus, be defined as the negative of the rate of change of electric potential with position (i.e., the negative of the potential gradient) at that point i.e., \[\vec{E} = -\frac{dV}{dl}\]  and \[|E| = \frac{dV}{dl}\]. Moreover, the direction of the electric field is the direction in which electric potential is decreasing at a maximum rate i.e., where the decrease of potential is steepest.

14. The electric potential at a point situated at a distance ‘r’ from the mid-point of a short electric the dipole of dipole moment p inclined at an angle θ from the axial line of dipole is given by \[V = \frac{1}{4 \pi \epsilon} \frac{pcos\theta}{r^2}\].

15. At a point situated on the axial line of electric dipole θ = 0^o  or π and hence \[V = \pm \frac{1}{4 \pi \epsilon}\frac{p}{r^2}\]. Here +ve sign is taken for θ = 0^o and -ve sign is taken for θ = π. Electric potential at any point situated on the equatorial line of an electric dipole ( θ = 90⁰ ) is zero i.e., V = 0.

16. Electric potential due to an electric dipole, in general, is inversely proportional to the square of the distance of the point from mid-point of the dipole, i.e., \[V \propto \frac{1}{r^2}\]. 

17. An equipotential surface is a surface with a constant value of the potential at all points on the surface. For any charge configuration, the equipotential surface through a point is normal to the electric field at the point.

18. For a point charge, equipotential surfaces are concentric spheres with a given charge point as the center. For a uniform electric field, equipotential surfaces are planes perpendicular to the direction of the electric field. 

19. As potential at all points of an equipotential surface is the same, hence, work done in moving a charge from one point to another along the equipotential surface is always zero.

20. The electric potential the energy of a system of point charges is equal to the amount of work done in assembling the given system of charge by bringing them to their respective positions from infinity. A point to be noted is that the potential energy is characteristic of the present state of assembly (or configuration ) and not the way the state is achieved. SI unit of electric potential energy is joule ( J ). For atomic and sub-atomic particals a unit “electron volt” (1 ev) is frequently used, where 1 eV = 1.60 *10^-19J.

21. For a system of two point charges q₁ and q₂ separated by a distance r the potential energy is given by \[U = \frac{q_1 q_2}{4 \pi \epsilon r}\]. If two charges are like charges then the force between them is repulsive. Work is being done against this repulsive force while bringing the charges to their present position and hence electric potential energy of the system will be positive. If two charges are unlike one, the force between them is attractive and work is being done by the attractive force. Consequently, the potential energy of the system will be negative.

22. For a system of n point charges the total electric potential energy of the system is given by \[U = \frac{1}{2}[ \frac{1}{4 \pi \epsilon } \Sigma_{i=1}^{n} \Sigma_{j = 1, i \neq j}\frac{q_i q_j}{r_{ij}}]\]. Here the factor $ \frac{1}{2} $  has been incorporated on account of the that in the summation each term has been counted twice ij and ji in the above expression.

23. The electrostatic potential energy of a single charge q in an external electric field E is given by U(r) = qV(r), Where V(r) is the potential at the given point due to the external electric field.

24. Electrostatic potential energy of a system of two charges q₁ and q₂ located at $ \vec{r_1} $ and $ \vec{r_2} $ respectively in an external electric field is given by \[U = q_1V(r_1) + q_2V(r_2) + \frac {q_1 q_2}{4\pi \epsilon r_{12}}\].

25. Work done for rotating an electric dipole of dipole moment p in a uniform electric field E from orientation θ₁ to orientation θ₂ is given by \[W = - pE(cos \theta_2 - cos \theta_1 ) = pE(cos \theta_1 - cos \theta_2 )\]. The electrostatic potetial energy of an electric dipole in a uniform electric field E is given by \[W = - \vec{p}.\vec{E} = - pEcos\theta\] Where $ \theta $ is the angle which axis of given dipole makes with the direction of electric field E.

26. An electric dipole is in a state of stable equilibrium when dipole is placed along the direction of external electric field (i.e., p and E are in same direction) because in that orientation torque on dipole is zero and potential energy of dipole in minimum having a value -pE. On the other hand electric dipole is in a state of unstable equilibrium when p and E are in mutually opposite directions because in that orientation torque on dipole is zero and potential energy of dipole is maximum having a value +pE.

27. Conductors are the materials which possess large number of free electron and, therefore, allow flow of electric charge through them easily. Silver, copper, aluminium and other metals, mercury etc., are example of good conductors of electricity. 

28. When a conductor is placed in an electric field, it exhibits the following properties :               

(i). Net electric field inside the conductor is zero.  

(ii). Electric charge always reside on the outer surface of the conductor only.                                     

(iii). Net electric charge in the interior of the conductor is zero in equilibrium state.

(iv). Just outside the surface of a conductor, the electric field is perpendicular (normal) to the surface at every point.                                                                                                                            

(v). Electric field lines do not pass through the interior of the conductor.

(vi). Electric potential at all points of the conductor, situated inside as well as on its surface, is uniform. Moreover, it has the same value as on its surface.                                                                 

(vii). Electric field at the surface of a charged conductor is given by \[\vec{E} = \frac{\sigma}{\epsilon_0}\widehat{n}\], Where σ is the surface charge density and n is a unit vector normal to the surface. For σ > 0, electric field is normal to the surface outward but for σ < 0, electric field is normal to the surface inward.

29. As electric charge, as well as electric field inside a cavity of any conductor, is zero, the cavity remains shielded from outside electric influence. It is known as electrostatic shielding. Thus, electrostatic shielding (or screening) is the phenomenon of maintaining a certain region in space completely free from external electric fields. Property of electric shielding is made use of in protecting sensitive instruments from outside electric influences.

30. Insulators are those materials that cannot conduct electricity. Insulators possess a negligibly small number of free electrons.

31. Dielectrics are non-conducting substances. When a dielectric is held in an electric field. Small induced charges appear on the surface of the dielectric. However, there is no free movement of charges inside a dielectric. Dielectrics are of two types: polar and non-polar dielectrics. Non-polar dielectrics e.g., O₂, N₂, H₂, CO₄, etc., consist of non-polar molecules in which the centre of positive charge exactly coincides with the centre of negative charge and dipole moment of a molecule is zero. Polar dielectrics e.g.  H₂O, HCl, NH₃, alcohol are made of polar molecules in which centre of positive charge does not coincides with the centre of negative charge and each molecule has some intrinsic electric dipole moment.

32. When a dielectric is placed in an external electric field, the field induces dipole moment by stretching or re-orienting molecules of the dielectric. As a collective effect of these molecular dipole moments, some net charge is developed on the surface of the dielectric which produces a field that opposes the external electric field. The opposing field so induced reduce the external field. The electric dipole moment developed per unit volume in a dielectric when placed in an external electric field E is called “polarisation” or polarisation vector P. For linear isotropic dielectrics, \[\vec{p} = \chi_e \vec{E}\]  where $ \chi_e $ is a constant, known as the electric susceptibility of the given dielectric, whose value depends on the nature of the dielectric and is a characteristic of the dielectric.

33. The potential of a given charged conductor is directly propositional to its charge. The ratio of charge Q of an isolated conductor to its potential V is called a capacitance C of the given conductor. Thus, \[C = \frac{Q}{V}\]. Alternately capacitance of a given conductor is equal to the amount of charge given to the conductor in order to raise its electric potential by unity. SI unit of capacitance is farad (F). 

34. Capacitance of a conductor depends upon its dimensions and shape. Capacitance of an isolated spherical conductor is given by \[C = 4 \pi \epsilon_0 R\] in free space, where R is the radius of conductor. Capacitance of a conductor is independent of the amount of charge given to it. However, capacitance of a conductor depends on the nature of the surroundings.

35. Capacitance of a system of two-conductor, besides their geometrical configuration (shape and size), also depends on (i) the separation between the two conductors, and (ii) the nature of the dielectric separating the two conductor.

36. A parallel plate capacitor is simplest type of capacitor.it consists of two parallel metal plates separated by a thin layer of dielectric. Capacitance of a parallel plate capacitor is \[C = \frac{\epsilon_0 A}{d}\] if free space is the intervening dielectric. Here, A = surface area of either plate and d = separation between the two plates of capacitor.

37. When capacitors is connected in series then the resultant capacitance C_s is given by \[\frac{1}{C_S} = \frac{1}{C_1} +\frac{1}{C_2}+\frac{1}{C_3}+............\] where C₁, C₂, C₃… etc. are the capacitances of individual capacitors.In series combination charge on all the capacitors is same but potential difference between the plates of difference capacitors is inversely proportional to their capacitances. In a series combination of capacitors, resultant capacitance is less then the capacitance of anyone capacitor. However, the combination may withstand a higher potential difference (voltage).

38. In parallel grouping of capacitors resultant capacitance is the sum of individual capacitance of capacitors joined in parallel i.e., \[C_P = C_1+C_2+C_3+........\]. In parallel arrangement potential difference across all the capacitors is same but charges on individual capacitors are directly proportional to their capacitances.

40. On filling a dielectric medium of dielectric constant K between the plates of a parallel plate capacitor, due to polarisation of the dielectric, the net electric field and hence the potential difference between the plates of capacitor is reduced to $ \frac{1}{K} $ times its precious value. Consequently, the capacitance of the capacitor increases K times. Thus, $  K = \frac{Capacitance in presence of dielectric medium C_m­­}{Capacitance when free space is the medium C_0 }$.

41. The capacitance of a parallel plate capacitor with a dielectric medium introduced between the plates is given by \[C = \frac{K\epsilon_0 A}{d} \]

42. If a dielectric medium of dielectric constant K and thickness t (t < d) is filled between the plates of a capacitor then its capacity is given by \[C = \frac{\epsilon_0 A}{d - t - \frac{t}{K}}\].

43. If a conducting sheet of thickness t (t < d) is introduced between the plates of a capacitor, without touching either plate of a capacitor, then the capacitance of the arrangement is given by \[C = \frac{\epsilon_0 A}{d-t} .\]

44. The energy stored in a charged capacitance is \[U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\] where Q = charge given to capacitance C, and V = potential difference between the plates of capacitor.

46. The energy density (energy per unit volume) of electric field in a capacitor is \[E' = \frac{1}{2} \epsilon_0 E^2\] electric field between the plates of capacitor.

47. When two charged capacitor are joined together, they share their charges till they acquire same “common potential” V, which is given by \[V = \frac{Total Charge}{Total Capacitance} = \frac{C_1V_1+C_2V_2}{C_1+C_2}\]. Total charge remains the same during this process.

48. The sharing of charges between two capacitors is always accompanied by some loss of electrical energy. Loss of electrical energy is given by \[\Delta U = U_2 - U_1 = \frac{C_1C_2(V_1-V_2)^2}{2(C_1+C_2)}\]. 

49. Dielectric strength of a dielectric is the maximum value of electric field (or potential gradient) which it can tolerate without its electric break-down. For reasons of safety a maximum electric field equal to 10% of dielectric strength of the material is actually applied.

Video Lectures:

1. Electric Potential energy, Electric potential definition, Electric potential unit, Electric force conservative nature watch video

2. Electric potential due to group of the charges, Electric potential due to electric dipole watch video

3. Equipotential surfaces and relation between Electric field and electric potential watch video

4. Conductor properties, Capacitor and Capacitance, spherical capacitor watch video

5 Parallel plate capacitor with conducting slab and dielectric slab watch video

6 Grouping of the capacitors watch video

7 Energy stored in capacitor, Common potential and Energy loss in charge sharing watch video

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Unit and Dimensions

1. Measurement of any physical quantity involves comparison with a certain basic, arbitrary chosen, widely accepted reference standard called Unit. 

Mathematically, a measure of a quantity Q = nu, where u is the size of the unit, and n is the numerical value of the given measure.  

2. Fundamental quantities: Fundamental quantities are the base quantities. There are 7 fundamental quantities: 

(i) Length 

(ii) Mass

(iii) Time

(iv) Electric Current

(v) Thermodynamic Temperature

(vi) Amount of substance

(vii) Luminous Intensity.

3. Derived quantities: These quantities are formed using fundamental quantities like density, volume, force, etc.

4. Length: Unit is metre (m). Meter is defined as the length of the path traveled by light in vacuum during a time interval of $ \frac{1}{299792458} $ part of a second. 

 1 fermi  = $1f =   10^{-15} m $

 1 angstrom =  $ 1A = 10^{-10} m$

 1 nano-metre = $1nm = 10^{-10}m$

 1 micro-metre = $ 1\mu m = 10^{-6}m$

 1 mili-metre = $1mm = 10^{-3} m$

 1 Astronomical unit = $ 1AU  = 1.496 \times 10^{11}m$

 1 light-year = $ 1ly = 9.46 \times 10^{11} m$

 1 parsec = $ 3.08 \times 10^{16}m $

5. Mass: Unit is Kilogram(kg). The mass of a cylinder made of platinum-iridium alloy kept at the International Bureau of Weights and Measures is defined as 1 kg.

6. Time: Unit is second(s). One second is the time taken by 9 192 631 770 oscillations of the light (of a specified wavelength) emitted by a cesium-133 atom.

7. Electric Current: Unit is Ampere. If equal currents are maintained in the two wires so that the force between them is $ 2 x 10^{-7} $ newton per meter of the wires, the current in any of the wires is called 1 A

8. Thermodynamic Temperature: Unit is Kelvin(K). The fraction $ \frac{1}{273.16} $ of the thermodynamic temperature of the triple point of water is called 1 K.

9. Amount of the Substance: Unit is mole(mole). The amount of a substance that contains as many

elementary entities as there is the number of atoms in 0.012 kg of carbon-12 is called a mole. 

10. Luminous Intensity: Unit is Candela(cd). The SI unit of luminous intensity is 1 cd which is the luminous intensity of a blackbody of surface area $ \frac{1}{600 000} m^{2} $ placed at the temperature of freezing, platinum, and at a pressure of 101,325 $ {N/m^{2}} $, in the direction perpendicular to its surface. 

11. Dimensions: Dimensions are the powers to which fundamental quantities are raised to represent that quantity. It is represented by using a square bracket. 

Physical Quantities	Dimensions
Distance, Length, Displacement	$[M^0LT^0]$
Velocity, Speed	$[M^0LT^{-1}]$
Acceleration	$[M^0LT^{-2}]$
Force	$[MLT^{-2}]$
Linear momentum, Impulse	$[MLT^{-1}]$
Torque, Work, Kinetic Energy, Potential Energy, Energy,	$[ML^2T^2]$
Power	$[ML^2T^{-3}]$
Pressure, Stress, Modulus of Elasticity	$[ML^{-1}T^{-2}]$

12. Principle of homogeneity of Dimensions: A correct dimensional equation must be homogeneous i.e. dimensions on both sides are the same. 

13. Use of Dimension: To convert a unit from one system to another system, To find the relation between various physical parameters and to check whether the formula is dimensionally correct or not.

Example1:  Find the dimension of the constants a and b in Van Der Wall Equation

i.e. $ (P + \frac{a}{V^2})(V-b) = RT $         

Solution:  Using principle of homogeneity,  

Dimension of b  = Dimension of V (volume) = $ [{ L^3 }] $

Dimension of P (pressure) = Dimension of $ (\frac{a}{V^{2}}) $                               

Dimension of a = dimension of $ PV{^2} $  = $[ML^{-1}T^{-2}] [L^{3}]^{2} $= $ [ML^{5}T^{-2}] $          

 

Example 2: The value of the gravitational constant is G = $ 6.67 * 10^{-11} $ $ Nm{^2}kg^{-2} $. Convert it into a system based on kilometer, tonne and hour as base units.  

Solution: Dimnsional formula of  G is $ [M^{-1}L^{3}T^{-2}] $

$ n_2 = n_1 [\frac{M_1}{M_2}]^{-1}[\frac{L_2}{L_1}]^{3} [\frac{T_2}{T_1}]^{-2} $  

$ n_1 = 6.67 * 10^{-11}, M_1 = 1 kg, M_2 = 1 tonne = 1000kg, $

$T_1 = 1s, T_2 = 3600s, L_1 = 1m and L_2 = 1000m $

$ n_2 = 6.67*10^{-11}[\frac{1}{1000}]^{-1}[\frac{1}{1000}]^{3} [\frac{1}{3600}]^{-2} = 8.64 * 10^{10} $

Example 3: The frequency f of a stretched string depends upon the Tension (T), length (l) and the linear mass density $ /mu $. Find the relation for frequency. 

Solution: Let frequency depends on T, l, and $ \mu $ as follow:

                           $ f = kT^{a}l^{b}{\mu ^{c}} $                      where k is constant.

            

writing dimension formula of both sides,

$ [M^{0}L^{0}T^{-1}] $ = $ [MLT^{-2}]^{a}[L]^{b}[ML^{-1}]^{c} $  =  $ [M^{a+c}L^{a+b-c}T^{-2a}] $

Comparing dimensions on both sides, 

                                     a + c  =  0

                                a + b - c  =  0

                                        -2a  =  -1

solving these we get,  a = $\frac{1}{2} $, b = -1 and c = $ \frac{-1}{2} $

so relation will be,   $f = \frac{k}{l} \sqrt {\frac{T}{\mu}} $ 

Video Lecture:

Fundamental and Derived Quantities, Dimensions, How to find dimension of any physical Quantity, Formula validation by dimensions, Deriving relation between physical quantities Unit conversion Watch video

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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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