Chapter 03 Current Electricity PDF

CHAPTER - 03 CURRENT ELECTRICITY CHAPTER – 03 CURRENT ELECTRICITY Syllabus:- Electric current, Drift velocity, Mobility and their relation with elec...

CHAPTER - 03 CURRENT ELECTRICITY CHAPTER – 03 CURRENT ELECTRICITY Syllabus:- Electric current, Drift velocity, Mobility and their relation with electric current, Ohm’s law, Electrical resistance, Conductance, Resistivity, Conductivity, Effect of temperature on resistance, 𝑽 ~ 𝑰 characteristics (linear and non-linear), Electrical energy and power, Carbon resistors, Colour code of carbon resistors, Combinations of resistors in series and parallel. EMF and potential difference. Internal resistance of a cell, Combination of cells in series and parallel, Kirchhoff’s laws and simple applications, Wheatstone bridge and Meter bridge. Potentiometer - Principle and its applications to measure P.D. and for comparing EMF of two cells, Measurement of internal resistance of a cell. Introduction:- When two ends of a conductor are at different levels of electricity, charge flows from one end of conductor to other end. For example an electric bulb glows when charge flows through it, T.V. will operate on flowing charges. This flow of charge is called as flow of electric current or simply flow of current. A metallic wire contains a large number of electrons move in random directions are called free electrons (not bound to atoms). The flow of free electrons in a definite direction in a conductor form electric current. The force to make free electrons move in a definite direction is called as electromotive force. Thus, the branch of physics which deals with study of charges in motion is called as electric current. Important:- The two major source of electric current are: battery and dynamo (generator). In batteries, chemical energy is converted into electrical energy and in a dynamo, mechanical energy is converted into electrical energy. Electric current (𝑰):- The rate of flow of charge across any cross-section of the conductor per unit time is called as electric current. Let a charge „𝒒‟ passes through any cross-section of conductor in time „𝒕‟. 𝑻𝒐𝒕𝒂𝒍 𝒄𝒉𝒂𝒓𝒈𝒆 𝒇𝒍𝒐𝒘𝒊𝒏𝒈 𝒒 Now current, 𝑰 = =. 𝑻𝒊𝒎𝒆 𝒕 In a non-uniform flow, if „∆𝒒‟ is small amount of charge flow across a conductor in a ∆𝒒 small interval of time „∆𝒕‟ than current is, 𝑰 =. ∆𝒕 ∆𝒒 𝒅𝒒 If the time interval „∆𝒕‟ chosen to be very small, i.e. ∆𝒕 → 𝟎, than, 𝑰 = 𝒍𝒊𝒎∆𝒕→𝟎 =. ∆𝒕 𝒅𝒕 Where, „𝒅𝒒‟ is small amount of charge flowing across any cross-section of conductor in a small interval 𝒅𝒒 of time „𝒅𝒕‟. Here, „ ‟ is the first derivative of „𝒒‟ w.r.t. „𝒕‟. 𝒅𝒕 Thus, “Current through a conductor at any time is defined as the first derivative of charge with respect to time passing through a cross-section of the conductor in a particular direction”. Note:- If no potential difference between two points in a circuit, no current will flow between these points. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 1  CHAPTER - 03 CURRENT ELECTRICITY Important:- 𝒏𝒆 (1) As, 𝒒 = 𝒏𝒆. Thus, electric current may be written as, 𝑰 =. 𝒕 (2) If an electron is revolved in a circular path of radius „𝒓‟ with velocity „𝒗‟ than the time period of the revolving 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒄𝒐𝒗𝒆𝒓𝒆𝒅 𝟐𝝅𝒓 electron is, 𝑻 = =. 𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒗 𝒒 𝒒 𝒒𝒗 𝒆𝒗 Now, current, 𝑰 = = 𝟐𝝅𝒓 =. Thus, electric current for revolving electron, 𝑰 =. 𝑻 𝟐𝝅𝒓 𝟐𝝅𝒓 𝒗 𝟐𝝅 𝟐𝝅 𝒒 𝒒 𝒒𝝎 𝒒𝝎 (3) Since angular velocity, 𝝎 = ,⟹𝑻=. Now, current, 𝑰 = = 𝟐𝝅 =. Thus, electric current, 𝑰 =. 𝑻 𝝎 𝑻 𝟐𝝅 𝟐𝝅 𝝎 𝒒 𝟐𝝅𝒇 (4) Since, 𝝎 = 𝟐𝝅𝒇, where, 𝒇 = 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚. Now, current, 𝑰 = = 𝒒𝒇. Thus, electric current, 𝑰 = 𝒒𝒇. 𝟐𝝅 (5) If a cross-section of the conductor in a time „𝒕‟, if a total positive charge „𝒒𝟏 ‟ is flowing from „𝑨‟ to „𝑩‟ and 𝒒𝟏 𝒒𝟐 𝒏𝒆+ 𝒆 𝒏𝒆− 𝒆 𝒏𝒆+ + 𝒏𝒆− 𝒆 total negative charge „𝒒𝟐 ‟ is flowing from „𝑩‟ to „𝑨‟ then, 𝑰 = + = + , i.e. 𝑰 =. 𝒕 𝒕 𝒕 𝒕 𝒕 𝒒 𝒏𝒆 𝒆𝒗 𝒒𝝎 𝒏𝒆+ + 𝒏𝒆− 𝒆 (6) In general electric current may be written as, 𝑰 = = = = = 𝒒𝒇 =. 𝒕 𝒕 𝟐𝝅𝒓 𝟐𝝅 𝒕 These relations are satisfied for uniform flow of charge in a conductor. 𝒅𝒒 (7) For a non-uniform flow of charge electric current, 𝑰 =. 𝒅𝒕 Example:- Find current if 𝒒 = 𝟑𝒕𝟐 + 𝟐𝒕 at 𝒕 = 𝟐 𝒔𝒆𝒄. 𝒅𝒒 𝒅 𝟑𝒕𝟐 +𝟐𝒕 𝒅 𝟑𝒕𝟐 𝒅 𝟐𝒕 Now, 𝑰 = = = + = 𝟑 × 𝟐𝒕𝟐−𝟏 + 𝟐 × 𝟏 = 𝟔𝒕 + 𝟐. At, 𝒕 = 𝟐, 𝑰 = 𝟔 × 𝟐 + 𝟐 = 𝟏𝟒𝑨. 𝒅𝒕 𝒅𝒕 𝒅𝒕 𝒅𝒕 Nature of electric current:- Though in electric circuits, the direction of current is specified by arrows, but it does not mean that the electric current is a vector because it does not obey the laws of vector addition. Since the resultant of two vectors depends on angle between the vectors, but this law does not hold in case of currents. Thus, “Electric current is a scalar quantity”. For example, consider three metallic wires „𝑨‟, „𝑩‟ and „𝑪‟ meeting at a point „𝑶‟ as shown the figure. Let 𝟑𝑨 and 𝟒𝑨 currents flow through the wires „𝑨‟ and „𝑩‟ respectively. Now the current in third wire „𝑪‟ is 𝟕𝑨 = (𝟑𝑨 + 𝟒𝑨) whatever be the angle between them. Charge carriers:- The charged particles which constitute an electric current in solids, liquids or gases are known as charge carriers. Important:- Electric current is due to flow of – (i) Electrons in conductors. (ii) Electrons and holes in semi-conductors. (iii) Coherent pairs of electrons in super conductor. (iv) Positive and negative ions in an electrolyte. (v) Positive ions and electrons in gases. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 2  CHAPTER - 03 CURRENT ELECTRICITY Types of electric current:- There are two types of current i.e. Steady current and Variable current. Steady current:- A current is said to be steady if its magnitude does not change with time and direction is always same as shown the figure. Variable current:- A current is said to be variable if its magnitude is changes with time while its direction may or may not be change as shown the figure. (a) (b) In figure (𝒂), magnitude of current changes within 𝟏𝑨 and 𝟑𝑨 while direction is same i.e. positive. Such type of current is called variable direct current. In figure (𝒃), the magnitude of current changes in between +𝟏𝑨 and −𝟏𝑨 while its direction gets reversed after equal interval of time. Such type of current is called as alternating current. Units of electric current:- (a) S.I. unit of electric current:- The S.I. unit of electric current is Ampere. The current flowing through a conductor is said to be one ampere if a charge of one coulomb flows across any 𝟏 𝑪𝒐𝒖𝒍𝒐𝒎𝒃 of its cross-section in one second. 𝒊. 𝒆. 𝟏 𝑨𝒎𝒑𝒆𝒓𝒆 =. 𝟏 𝑺𝒆𝒄𝒐𝒏𝒅 (b) C.G.S. unit of electric current:- The C.G.S. unit may be electrostatic unit or electromagnetic unit. (1) C.G.S. electro-static unit (e.s.u.):- The e.s.u. of electric current is Stat-ampere. The current flowing through a conductor is said to be one stat-ampere if a charge of one stat-coulomb flows 𝟏 𝑺𝒕𝒂𝒕−𝒄𝒐𝒖𝒍𝒐𝒎𝒃 across any of its cross-section in one second. 𝒊. 𝒆. 𝟏 𝑺𝒕𝒂𝒕 − 𝒂𝒎𝒑𝒆𝒓𝒆 =. 𝟏 𝑺𝒆𝒄𝒐𝒏𝒅 (2) C.G.S. electro-magnetic unit (e.m.u.):- The e.m.u. of electric current is Ab-ampere. The current flowing through a conductor is said to be one ab-ampere if a charge of one ab-coulomb flows 𝟏 𝑨𝒃−𝒄𝒐𝒖𝒍𝒐𝒎𝒃 across any of its cross-section in one second. 𝒊. 𝒆. 𝟏 𝑨𝒃 − 𝒂𝒎𝒑𝒆𝒓𝒆 =. 𝟏 𝑺𝒆𝒄𝒐𝒏𝒅 𝟏 𝑪𝒐𝒖𝒍𝒐𝒎𝒃 𝟑 × 𝟏𝟎𝟗 𝒔𝒕𝒂𝒕−𝒄𝒐𝒖𝒍𝒐𝒎𝒃 Relation between Ampere and Stat-ampere:- As, 𝟏 𝑨𝒎𝒑𝒆𝒓𝒆 = =. 𝟏 𝑺𝒆𝒄𝒐𝒏𝒅 𝟏 𝒔𝒆𝒄𝒐𝒏𝒅 ⟹ 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝟑 × 𝟏𝟎𝟗 𝒔𝒕𝒂𝒕 − 𝒂𝒎𝒑𝒆𝒓𝒆. Relation between Ampere and Ab-ampere:- 𝟏 𝟏 𝑪𝒐𝒖𝒍𝒐𝒎𝒃 𝒂𝒃−𝒄𝒐𝒖𝒍𝒐𝒎𝒃 𝟏 𝟏𝟎 As, 𝟏 𝑨𝒎𝒑𝒆𝒓𝒆 = = , ⟹ 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝒂𝒃 − 𝒂𝒎𝒑𝒆𝒓𝒆. 𝟏 𝑺𝒆𝒄𝒐𝒏𝒅 𝟏 𝒔𝒆𝒄𝒐𝒏𝒅 𝟏𝟎 Relation between Ab-ampere and Stat-ampere:- 𝟏 As, 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝒂𝒃 − 𝒂𝒎𝒑𝒆𝒓𝒆 ……...… (1) and 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝟑 × 𝟏𝟎𝟗 𝒔𝒕𝒂𝒕 − 𝒂𝒎𝒑𝒆𝒓𝒆 ……...… (2) 𝟏𝟎 𝟏 Now, equating (1) and (2) we get, ⟹ 𝒂𝒃 − 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝟑 × 𝟏𝟎𝟗 𝒔𝒕𝒂𝒕 − 𝒂𝒎𝒑𝒆𝒓𝒆. 𝟏𝟎 ⟹ 𝟏 𝒂𝒃 − 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝟑 × 𝟏𝟎𝟏𝟎 𝒔𝒕𝒂𝒕 − 𝒂𝒎𝒑𝒆𝒓𝒆. Dimension of electric current:- The dimension of electric current is 𝑴𝟎 𝑳𝟎 𝑻𝟎 𝑨𝟏 because it is a fundamental quantity. So, dimension of current are 𝟎, 𝟎, 𝟎 and 𝟏 in mass, length, time and current respectively. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 3  CHAPTER - 03 CURRENT ELECTRICITY Important:- The smaller currents are expressed in the following units. 𝟏 𝑴𝒊𝒍𝒍𝒊 − 𝒂𝒎𝒑𝒆𝒓𝒆 (𝟏𝒎𝑨) = 𝟏𝟎−𝟑 𝑨 and 𝟏 𝑴𝒊𝒄𝒓𝒐 − 𝒂𝒎𝒑𝒆𝒓𝒆 𝟏𝝁𝑨 = 𝟏𝟎−𝟔 𝑨. Conventional current:- If the direction of flow of current is taken as opposite to direction of motion of electrons i.e. along the direction of proton than it is called conventional current. Electronic current:- If the direction of flow of current is taken to the direction of flow of electrons than it is called as the electronic current. Direction of electric current:- The direction of flow of positive charge or direction in which electric field is applied gives the direction of electric current. But in a conductor, the flows of electrons i.e. negative charges constitute electric current. So, the direction of conventional current is opposite to the direction of flow of negative charge i.e. electrons. Transient current:- The current which remains for short duration are called as transient current. Lighting, which is the flow of electric charge between two clouds or from a cloud to the earth is an example of transient current i.e. a current of short duration. Drift velocity 𝑽𝒅 :- In a good conductor like metal there are a large number of free electrons (It is about 𝟏𝟎𝟐𝟖 per 𝒎𝟑 ) and these free electrons move at random in all possible directions inside the metal colliding with one another and with atoms. Thus, the average velocity of electrons at any time is zero in any direction at ordinary room temperature although the velocity of individual electrons may be as high as 𝟏𝟎𝟓 𝒎/𝒔𝒆𝒄. On applying electric field across the conductor, all free electrons start to move towards the positive end of applied electric field. Now the average velocity is not zero, so current flows through the conductor as shown the figure. The average velocity with which the free electrons are drifted towards the positive terminal under the action of the applied electric field is called as drift velocity of the free electrons. Let 𝑼𝟏 , 𝑼𝟐 , 𝑼𝟑 … … … … … … 𝑼𝒏 be the initial velocities of various electrons present in the conductor. 𝑼𝟏 + 𝑼𝟐 + 𝑼𝟑 + ………+ 𝑼𝒏 Now, initial average velocity, 𝑼 = = 𝟎. 𝒏 Where, 𝒏 = total number of electron in the conductor. If „𝑽‟ be the potential difference between the two ends of the conductor of length „𝒍‟, than electric field 𝑽 𝒅𝑽 „𝑬‟ is given by, 𝑬 = − ∵𝑬=−. 𝒍 𝒅𝒓 When an electric field 𝑬 is established in the conductor, than each free electron experiences a force 𝑭 is given by, 𝑭 = 𝒒𝑬 = −𝒆 × 𝑬. Here the negative sign indicate that, the force is directed opposite to 𝑬. 𝑭 −𝒆 × 𝑬 𝒆 𝑽 𝒆𝑽 But force, 𝑭 = 𝒎 × 𝒂, where „𝒎‟ is mass of electron. ⟹ 𝒂 = = = − – ,⟹𝒂=. 𝒎 𝒎 𝒎 𝒍 𝒎𝒍 This above equation gives the acceleration of the free electron. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 4  CHAPTER - 03 CURRENT ELECTRICITY Let 𝒕𝟏 , 𝒕𝟐 , 𝒕𝟑 … … … …. 𝒕𝒏 be the average time interval between two consecutive collisions of electrons numbered as 𝟏, 𝟐, 𝟑 … … … …. 𝒏 respectively. Now, the velocity of various electrons are given by, 𝑽𝟏 = 𝑼𝟏 + 𝒂 𝒕𝟏 ∵ 𝒗 = 𝒖 + 𝒂𝒕. 𝑽𝟐 = 𝑼𝟐 + 𝒂 𝒕𝟐 𝑽𝟑 = 𝑼𝟑 + 𝒂 𝒕𝟑 ………………... ………………... 𝑽𝒏 = 𝑼𝒏 + 𝒂 𝒕𝒏 𝑽𝟏 + 𝑽𝟐 + 𝑽𝟑 + ……………… + 𝑽𝒏 Now, the final average velocity called as drift velocity 𝑽𝒅 is given by, 𝑽𝒅 =. 𝒏 𝑼𝟏 + 𝒂 𝒕𝟏 + 𝑼𝟐 + 𝒂 𝒕𝟐 + 𝑼𝟑 + 𝒂 𝒕𝟑 + ……………… + 𝑼𝒏 + 𝒂 𝒕𝒏 𝑼𝟏 + 𝑼𝟐 + 𝑼𝟑 + ………… + 𝑼𝒏 𝒂 𝒕𝟏 +𝒕𝟐 +𝒕𝟑 + ………… +𝒕𝒏 ⟹ 𝑽𝒅 = = +. 𝒏 𝒏 𝒏 𝒕𝟏 +𝒕𝟐 + 𝒕𝟑+ ……….. + 𝒕𝒏 ⟹ 𝑽𝒅 = 𝟎 + 𝒂 × 𝝉. Where, 𝝉 = , which is average time and is called as relaxation time. 𝒏 Thus, “Relaxation of time is defined as the average time intervals between two consecutive collisions of different electrons”. The order of relaxation of time is 𝟏𝟎−𝟏𝟒 second. 𝒆𝑽 Now, putting the value of „𝒂‟ in the above equation we get, ⟹ 𝑽𝒅 = 𝝉. 𝒎𝒍 This above equation gives the mathematical expression for drift velocity of free electrons in a conductor. This equation also gives the relation between drift velocity, electric field and relaxation time. Important:- The drift velocity of free electrons is – (i) Directly proportional to applied voltage. (ii) Inversely proportional to the length of conductor. (iii) Independent of area of cross-section. (iv) Directly proportional to the relaxation time. At a given temperature, the relaxation time remains constant, so drift velocity remains constant. Note-(1):- The drift velocity is lies in between 𝟎. 𝟏 𝒄𝒎/𝒔𝒆𝒄 𝒕𝒐 𝟏. 𝟎 𝒄𝒎/𝒔𝒆𝒄 i.e. 𝟏𝟎−𝟒 𝒎 𝒔𝒆𝒄 𝒕𝒐 𝟏𝟎−𝟑 𝒎 𝒔𝒆𝒄. Note-(2):- The order of drift velocity is very small i.e. approximately, 𝟏𝒎𝒎/𝒔𝒆𝒄. Note-(3):- When the temperature increases, relaxation of time decreases, so drift velocity decreases. Mean free path:- The distance travelled by a conduction electron during relaxation time is called as mean free path. 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 Mathematically, 𝑴𝒆𝒂𝒏 𝒇𝒓𝒆𝒆 𝒑𝒂𝒕𝒉 𝝀 = 𝑫𝒓𝒊𝒇𝒕 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝑽𝒅 × 𝑹𝒆𝒍𝒂𝒙𝒂𝒕𝒊𝒐𝒏 𝒕𝒊𝒎𝒆 𝝉. ∵ 𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 = 𝑻𝒊𝒎𝒆 This above equation gives the relation between mean free path and drift velocity. Relation between electric current (𝑰) and drift velocity (𝑽𝒅 ):- Let us consider a conductor of length „𝒍‟ and uniform area of cross-section „𝑨‟ as shown below. Let „𝑽‟ be the applied potential difference across the two ends of the conductor. Now, the magnitude of the electric field set 𝑽 𝒅𝑽 up across the conductor is, 𝑬 =. ∵ 𝑬 = 𝒍 𝒅𝒓 Let „𝒏‟ be the number of free electrons per unit volume of conductor. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 5  CHAPTER - 03 CURRENT ELECTRICITY 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒇𝒓𝒆𝒆 𝒆𝒍𝒆𝒄𝒕𝒓𝒐𝒏𝒔 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 Total number of electrons in the conductor = × = 𝒏 × 𝑨𝒍. 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒗𝒐𝒍𝒖𝒎𝒆 𝒕𝒉𝒆 𝒄𝒚𝒍𝒊𝒏𝒅𝒆𝒓 If „𝒆‟ is the charge of each electron than total charge in the conductor is, 𝑸 = 𝒏𝑨𝒍 𝒆. ∵ 𝒒 = 𝒏𝒆 𝒍 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 Now, time taken by the charge to cross the conductor of length „𝒍‟ is, 𝒕 =. ∵ 𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 = 𝑽𝒅 𝑻𝒊𝒎𝒆 Where, „𝑽𝒅 ‟ is the drift velocity of free electrons. 𝑸 𝒏𝑨𝒍 𝒆 By the definition of electric current, 𝑰 = = , ⟹ 𝑰 = 𝒏𝑨𝑽𝒅 𝒆. 𝒕 𝒍 𝑽𝒅 This above equation gives the relation between electric current (𝑰) and drift velocity (𝑽𝒅 ). From the above equation it is clear that, „𝒏‟, „𝑨‟ and „𝒆‟ are constant and the equation becomes, ⟹ 𝑰 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 × 𝑽𝒅. ⟹ 𝑰 ∝ 𝑽𝒅. Thus, “The electric current flowing through a conductor is directly proportional to the drift velocity of the free electrons”. From the above equation it is clear that, the electric current in a conductor is – (i) Directly proportional to the number of free electrons per unit volume. (ii) Directly proportional to the area of cross-section of the conductor. (iii) Directly proportional to the drift velocity of free electrons. Important:- (1) Always electrons are drifted in the direction of increasing potential. 𝒆𝑽 𝒆 𝑽 𝒆 𝒆𝑬 𝑽 (2) As drift velocity, 𝑽𝒅 = 𝝉= 𝝉= 𝑬 𝝉, ⟹ 𝑽𝒅 = 𝝉. ∵𝑬= 𝒎𝒍 𝒎 𝒍 𝒎 𝒎 𝒍 This above equation gives the relation between drift velocity and electric field intensity. 𝒆𝑬 𝒏𝑨𝒆𝟐 𝑬𝝉 (3) As electric current, 𝑰 = 𝒏𝑨𝑽𝒅 𝒆 = 𝒏𝑨 𝝉 × 𝒆, ⟹ 𝑰 =. 𝒎 𝒎 Electron mobility (𝝁):- The drift velocity acquired per unit electric field strength is called as mobility of the electron. 𝑽𝒅 Mathematically, electron mobility, 𝝁 =. If 𝑬 = 𝟏, than, 𝝁 = 𝑽𝒅. 𝑬 Thus, “mobility of electrons may be defined as the drift velocity acquired by the electron due to unit strength of electric field”. 𝑽𝒅 Since, 𝝁 = , ⟹ 𝑽𝒅 = 𝝁 × 𝑬. But electric current, 𝑰 = 𝒏𝑨𝑽𝒅 𝒆. ⟹ 𝑰 = 𝒏 × 𝑨 × 𝝁 × 𝑬 × 𝒆 ………… (1) 𝑬 This above equation gives the relation between electric current and electron mobility. 𝑽 𝒅𝑽 If „𝑽‟ is the potential difference across a conductor of length „𝒍‟ than, 𝑬 =. ∵𝑬= 𝒍 𝒅𝒓 𝑽 𝒏𝑨𝝁𝑽𝒆 ⟹ 𝑰 = 𝒏 × 𝑨 × 𝝁 × × 𝒆, ⟹𝑰= ………… (2) 𝒍 𝒍 From equation (1) and (2) it is clear that, for certain strength of electric field, “The electric current is directly proportional to mobility of the electron”. i.e. 𝑰 ∝ 𝝁. (∵ 𝒏, 𝑨, 𝒆, 𝑽 and 𝒍 all are constants). 𝑽𝒅 𝒆𝑽𝝉 𝑽 𝒆𝑽𝝉 𝒆 Note-(1):- As we know, 𝝁 =. Since, 𝑽𝒅 = and 𝑬 =. ⟹𝝁= 𝑽 𝒎𝒍 ,⟹𝝁= 𝝉. 𝑬 𝒎𝒍 𝒍 𝒍 𝒎 This above equation gives the relation in between mobility and relaxation time. Note-(2):- When the temperature of a metallic conductor increases the speed of electrons increases causes an increase in number of collisions of electrons and hence decrease in relaxation time occurs. Thus, “Mobility decreases with increase in temperature”. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 6  CHAPTER - 03 CURRENT ELECTRICITY 𝑽𝒅 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝑻𝒊𝒎𝒆 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝑪𝒉𝒂𝒓𝒈𝒆 𝑽𝒅 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝑻𝒊𝒎𝒆 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝑳𝒆𝒏𝒈𝒕𝒉 Unit of mobility:- Mobility, 𝝁 = = = × (Or) 𝝁 = = = ×. 𝑬 𝑭𝒐𝒓𝒄𝒆 𝑪𝒉𝒂𝒓𝒈𝒆 𝑻𝒊𝒎𝒆 𝑭𝒐𝒓𝒄𝒆 𝑬 𝑷𝒐𝒕𝒆𝒏𝒕𝒊𝒂𝒍 𝑳𝒆𝒏𝒈𝒕𝒉 𝑻𝒊𝒎𝒆 𝑷𝒐𝒕𝒆𝒏𝒕𝒊𝒂𝒍 𝟐 In M.K.S. ⟶ 𝑪𝒐𝒖𝒍𝒐𝒎𝒃 × 𝑴𝒆𝒕𝒆𝒓 𝑵𝒆𝒘𝒕𝒐𝒏 × 𝑺𝒆𝒄𝒐𝒏𝒅 (Or) 𝑴𝒆𝒕𝒆𝒓 𝑽𝒐𝒍𝒕 × 𝑺𝒆𝒄𝒐𝒏𝒅. 𝟐 In C.G.S. ⟶ 𝑺𝒕𝒂𝒕 − 𝒄𝒐𝒖𝒍𝒐𝒎𝒃 × 𝑪𝒎 𝑫𝒚𝒏𝒆 × 𝑺𝒆𝒄𝒐𝒏𝒅 (Or) 𝑪𝒎 𝑺𝒕𝒂𝒕 − 𝒗𝒐𝒍𝒕 × 𝑺𝒆𝒄𝒐𝒏𝒅. 𝑽𝒅 𝑳𝟏 𝑻−𝟏 Dimension of mobility:- Mobility, 𝝁 = = = 𝑴−𝟏 𝑳𝟎 𝑻𝟐 𝑨𝟏. 𝑬 𝑴𝟏 𝑳𝟏 𝑻−𝟑 𝑨−𝟏 Thus, the dimension of mobility are −𝟏, 𝟎, 𝟐 and 𝟏 in mass, length, time and current respectively. Current density vector (𝑱):- The electric current density at any point inside a conductor is defined as the amount of charge flowing per second through a unit area held normal to the direction of flow of charge. 𝑪𝒖𝒓𝒓𝒆𝒏𝒕 (𝑰) Mathematically, 𝑪𝒖𝒓𝒓𝒆𝒏𝒕 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝑱 =. 𝑨𝒓𝒆𝒂 (𝑨) 𝑰 Current density is a vector quantity whose magnitude is 𝑱 = and its direction is same as that of the 𝑨 direction of the conventional current i.e. in the direction of flow of positive charge or along the direction of applied electric field across the conductor as show in figure (a). If the area (𝑨) is not perpendicular to the direction of current and normal to this area makes an angle 𝜽 with the direction of current, then the component of (𝑨) normal to the direction of flow will be, 𝑨𝒏 = 𝑨 𝑪𝒐𝒔𝜽 as show in figure (b). 𝑰 𝑰 Now, current density is given by, 𝑱 = = , ⟹ 𝑰 = 𝑱𝑨 𝑪𝒐𝒔𝜽 = 𝑱. 𝑨. 𝑨𝒏 𝑨 𝑪𝒐𝒔𝜽 Thus, electric current may be defined as the dot product of current density and area vector. Important:- 𝑰 𝒏𝑨𝑽𝒅 𝒆 (1) As electric current, 𝑰 = 𝒏𝑨𝑽𝒅 𝒆. ⟹ 𝑱 = = , ⟹ 𝑱 = 𝒏𝑽𝒅 𝒆. 𝑨 𝑨 This above equation gives the relation between current density and drift velocity of free electrons. This above equation is applicable when the flow of current is uniform. Again from the above equation we have current density is directly proportional to the drift velocity. (2) When the flow of current is non-uniform, than current through any cross-section is, 𝑰 = 𝑱. 𝒅𝑨. Where is surface integral over whole cross-section of conductor. Here, 𝑱. 𝒅𝑨 is regarded as flux of current density over a given area. Thus, “Electric current is defined as the flux of current density over a given area”. (3) As 𝑰 = 𝑱. 𝑨. Thus electric current (𝑰) is a scalar quantity because it is the dot product of „𝑱‟ and „𝑨‟. 𝑰 𝑪𝒖𝒓𝒓𝒆𝒏𝒕 Unit of current density vector:- As, 𝑱 = =. 𝑨 𝑨𝒓𝒆𝒂 In M.K.S. ⟶ 𝑨𝒎𝒑𝒆𝒓𝒆 / (𝒎𝒆𝒕𝒆𝒓)𝟐 and In C.G.S. ⟶ 𝑨𝒃 − 𝒂𝒎𝒑𝒆𝒓𝒆/𝒄𝒎𝟐 (or) 𝑺𝒕𝒂𝒕 − 𝒂𝒎𝒑𝒆𝒓𝒆/𝒄𝒎𝟐. 𝑰 𝑪𝒖𝒓𝒓𝒆𝒏𝒕 𝑨 Dimension of current density vector:- As, 𝑱 = = = = [𝑴𝟎 𝑳−𝟐 𝑻𝟎 𝑨𝟏 ]. 𝑨 𝑨𝒓𝒆𝒂 𝑳𝟐 Thus, dimension of current density are 𝟎, −𝟐, 𝟎 and 𝟏 in mass, length, time and current respectively. Relation between current density (𝑱) and electric field (𝑬):- 𝑽 𝑽 𝑽𝑨 From Ohm‟s law, 𝑰 = = 𝒍 =. Where, 𝝆 = Resistivity of the conductor. 𝑹 𝝆 𝝆𝒍 𝑨 Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 7  CHAPTER - 03 CURRENT ELECTRICITY 𝑰 𝑽 𝟏 𝑽 𝑱 𝑰 𝟏 𝑽 ⟹ = = , ⟹ 𝑱 = 𝝇 × 𝑬 (or) 𝝇 =. ∵ 𝑱 = ,𝝇 = 𝒂𝒏𝒅 𝑬 = Where, 𝝇 = Conductivity of material. 𝑨 𝝆𝒍 𝝆 𝒍 𝑬 𝑨 𝝆 𝒍 This above equation gives the relation between electric field and current density. Thus, “Conductivity may be defined as electric current density per unit electric field strength”. Important:- 𝒆𝑽 𝒏𝒆𝟐 𝑽 𝒏𝒆𝟐 𝑬𝝉 𝑽 (1) As, 𝑱 = 𝒏𝒆𝑽𝒅 = 𝒏𝒆 𝝉, ⟹ 𝑱 = 𝝉, ⟹ 𝑱 =. ∵𝑬= 𝒎𝒍 𝒎 𝒍 𝒎 𝒍 This equation gives the relation between current density and relaxation time. 𝑱 𝝇𝑬 (2) As, 𝑽𝒅 = , 𝑽𝒅 =. ∵ 𝑱 = 𝝇 × 𝑬 This gives the relation between drift velocity and conductivity of a material. 𝒏𝒆 𝒏𝒆 𝑰 𝒒 𝒕 𝒒 𝒏𝒆 (3) As, 𝑱 = = = ,⟹𝑱=. 𝑨 𝑨 𝑨𝒕 𝑨𝒕 This equation gives the relation between current density and number of charge carriers crossing per unit time. Difference between electric current and current density vector:- Electric Current Current Density Vector 1) It is the rate of flow of charge carriers. 1) It is the current per unit area. 2) It is a scalar and macroscopic quantity. 2) It is a vector and microscopic quantity. 3) It is constant throughout the conductor. 3) It is varies throughout the conductor. Ohm’s law:- In 1826 George Simon Ohm stated a law relating the electric current in a conductor having a potential difference across it. This law is known as Ohm‟s law. It states that, “At a constant temperature the current (𝑰) flowing through a metallic conductor is directly proportional to the potential difference (𝑽) between the two ends of the conductor”. Mathematically, 𝑰 ∝ 𝑽 or 𝑽 ∝ 𝑰. Where, 𝑽 = P.D. between two ends of conductor and 𝑰 = current flowing through the conductor. 𝑽 ⟹ = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝑹). ⟹ 𝑽 = 𝑰 × 𝑹, Where, 𝑹 = Resistance of the conductor. 𝑰 The value of „𝑹‟ depends upon the nature of the material of the conductor, dimension and temperature of the conductor. It does not depend on the values of „𝑽‟ and „𝑰‟. 𝑽 − 𝑰 Characteristics:- The variation of electric current (I) through a conductor with the variation of potential difference (V) across the ends of the conductor at constant temperature is known as V-I characteristics of the conductor. According to Ohm‟s law, 𝑽 ∝ 𝑰, thus the 𝑽 − 𝑰 characteristics of a conductor is as shown the figure. 𝑽 From diagram, Slope of 𝑽 − 𝑰 graph = = 𝑹. (V - I Characteristics) (I - V Characteristics) 𝑰 Thus, resistance of a conductor is equal to the slope of the 𝑽 − 𝑰 graph. Higher is the slope of 𝑽 − 𝑰 graph higher is the resistance of the conductor and vice-versa. 𝑰 𝟏 𝟏 𝟏 Now, slope of 𝑰 − 𝑽 graph = = 𝑽 = i.e. Resistance, 𝑹 =. 𝑽 𝑰 𝑹 𝑺𝒍𝒐𝒑𝒆 𝒐𝒇 𝑰−𝑽 𝒈𝒓𝒂𝒑𝒉 Thus, higher is the slope of 𝑰 − 𝑽 graph smaller is the resistance of the conductor. Smaller is the slope of 𝑰 − 𝑽 graph, greater is the resistance of the conductor. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 8  CHAPTER - 03 CURRENT ELECTRICITY Limitations of Ohm’s law:- (1) It cannot be applied for complicated electrical network. (2) It cannot be applied when temperature and other factors like mechanical strain are not constant. (3) It holds well when the area of cross-section of the conductor should be uniform and current flowing through the conductor should remain constant. Derivation of Ohm’s law using drift velocity:- 𝒆𝑽𝝉 As drift velocity, 𝑽𝒅 = …………… (1) 𝒎𝒍 Where, 𝒆 = Charge of electron, 𝑽 = Potential difference of conductor, 𝝉 = Time taken for electron for moving one section to another section, 𝒎 = Mass of electron and 𝒍 = Length of conductor. Again we know, 𝑰 = 𝒏𝑨𝑽𝒅 𝒆 …………… (2) Where, 𝒏 = Number of free electrons per unit volume and 𝑨 = Area of cross-section of conductor. 𝒆𝑽𝝉 𝒏𝑨𝒆𝟐 𝝉 Now, equation (2), becomes, ⟹ 𝑰 = 𝒏𝑨 𝒆= 𝑽. 𝒎𝒍 𝒎𝒍 𝟏 𝟏 𝒏𝑨𝒆𝟐 𝝉 𝑽 𝑽 𝑽 𝒎𝒍 𝟏 𝒏𝑨𝒆𝟐 𝝉 ⟹ 𝑰 = × 𝑽, Where, =. ∵𝑹= = = 𝒆𝑽𝝉 = 𝒊. 𝒆. = 𝑹 𝑹 𝒎𝒍 𝑰 𝒏𝑨𝑽𝒅 𝒆 𝒏𝑨 𝒆 𝒏𝑨𝒆𝟐 𝝉 𝑹 𝒎𝒍 𝒎𝒍 𝟏 Since „𝒏‟, „𝑨‟, „𝒆‟, „𝝉‟, „𝒎‟ and „𝒍‟ all are constant thus „ ‟ is also taken as constant. 𝑹 ⟹ 𝑰 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 × 𝑽. ⟹ 𝑰 ∝ 𝑽. This verifies Ohm‟s law. Ohmic conductor:- Those conductors which obey Ohm‟s law are called Ohmic conductor. Example- Aluminium, Copper, Mercury, Iron etc are called as Ohmic conductor. If we plot a graph in between current (𝑰) flowing through a conductor along 𝑿 − axis and potential difference (𝑽) along 𝒀 −axis than graph will be a straight line passing through the origin as 𝑽 shown the figure. Here, the slope of the graph is, 𝑻𝒂𝒏𝜽 = = 𝑹. 𝑰 Non-ohmic conductor:- Those conductors which do not obey Ohm‟s law are called Non-Ohmic conductor. Example- Vacuum tubes, Transistors, Electrolytes, Semi-conductor devices etc are called as Non-Ohmic conductor. The various graph of a non-ohmic conductor are as shown the figure. Properties of Non-Ohmic conductor:- 𝑽 (1) The 𝑽 − 𝑰 graph is nonlinear i.e. is variable. 𝑰 (2) The 𝑽 − 𝑰 graph may not pass through the origin as in case of non-ohmic conductor. (3) A non-ohmic conductor may conduct poorly or not at all when potential difference is reversed. Electrical resistance (𝑹):- Resistance is the opposition offered to the flow of electric charge in the conductor. “The resistance of a conductor is defined as the ratio between potential differences between the two ends of a conductor to the current flowing through it”. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 9  CHAPTER - 03 CURRENT ELECTRICITY 𝑽 Mathematically, Resistance 𝑹 =. If 𝑰 = 𝟏 unit than, 𝑹 = 𝑽. 𝑰 Thus, “The resistance of a conductor is equal to the difference of potentials across the two ends of the conductor if a unit current flowing through it”. Resistance can measured by an instrument called Ohmmeter. Cause of resistance of a conductor:- When a conductor is connected across a battery or a cell, the free electrons in the conductor flow from negative terminal to the positive terminal of the battery through the conductor. This gives rise to the electric current in the conductor. When free electrons flow from one end to another end of the conductor, they collide with the ions of the conductor. This collision offers opposition or resistance to the flow of electrons through the conductor. Important:- The symbols of resistance are as shown the figure. Units of resistance:- (a) S.I. unit of resistance:- The S.I. unit of resistance is ‘Ohm’. Thus, “The resistance of a conductor is said to be one ohm if a current of one ampere flows through it for a 𝟏 𝑽𝒐𝒍𝒕 𝑽 potential difference of one volt across its ends”. 𝒊. 𝒆. 𝟏 𝑶𝒉𝒎 =. ∵𝑹= 𝟏 𝑨𝒎𝒑𝒆𝒓𝒆 𝑰 (b) C.G.S. unit of resistance:- The C.G.S. unit may be electro-static or electro-magnetic unit. (1) C.G.S. electro-static unit (e.s.u.):- The e.s.u. of resistance is ‘Stat-ohm’. Thus, “The resistance of a conductor is said to be one stat-ohm if a current of one stat-ampere flows through 𝟏 𝑺𝒕𝒂𝒕−𝒗𝒐𝒍𝒕 𝑽 it for a potential difference of one stat-volt across its ends”. 𝒊. 𝒆. 𝟏 𝑺𝒕𝒂𝒕 − 𝒐𝒉𝒎 =. ∵𝑹= 𝟏 𝒔𝒕𝒂𝒕−𝒂𝒎𝒑𝒆𝒓𝒆 𝑰 (2) C.G.S. electro-magnetic unit (e.m.u.):- The e.m.u. of resistance is ‘Ab-ohm’. Thus, “The resistance of a conductor is said to be one ab-ohm if a current of one ab-ampere flows through it 𝟏 𝑨𝒃−𝒗𝒐𝒍𝒕 𝑽 for a potential difference of one ab-volt across its ends”. 𝒊. 𝒆. 𝟏 𝑨𝒃 − 𝒐𝒉𝒎 =. ∵𝑹= 𝟏 𝒂𝒃−𝒂𝒎𝒑𝒆𝒓𝒆 𝑰 𝟏 𝟏 𝑽𝒐𝒍𝒕 𝒔𝒕𝒂𝒕−𝒗𝒐𝒍𝒕 𝟏 𝑺𝒕𝒂𝒕−𝒗𝒐𝒍𝒕 Relation between Ohm and Stat-ohm:- As, 𝟏 𝑶𝒉𝒎 = = 𝟑𝟎𝟎 = 𝟑 × 𝟏𝟎𝟐 ×. 𝟏 𝑨𝒎𝒑𝒆𝒓𝒆 𝟑 × 𝟏𝟎𝟗 𝑺𝒕𝒂𝒕−𝒂𝒎𝒑𝒆𝒓𝒆 𝑺𝒕𝒂𝒕−𝒂𝒎𝒑𝒆𝒓𝒆 𝟑 × 𝟏𝟎𝟗 𝟏 ⟹ 𝟏 𝒐𝒉𝒎 = 𝒔𝒕𝒂𝒕 − 𝒐𝒉𝒎. 𝟗×𝟏𝟎𝟏𝟏 𝟏 𝑽𝒐𝒍𝒕 𝟏𝟎𝟖 𝒂𝒃−𝒗𝒐𝒍𝒕 𝒂𝒃−𝒗𝒐𝒍𝒕 Relation between Ohm and Ab-ohm:- As, 𝟏 𝑶𝒉𝒎 = = 𝟏 = 𝟏𝟎𝟗. 𝟏 𝑨𝒎𝒑𝒆𝒓𝒆 𝒂𝒃−𝒂𝒎𝒑𝒆𝒓𝒆 𝒂𝒃−𝒂𝒎𝒑𝒆𝒓𝒆 𝟏𝟎 ⟹ 𝟏 𝒐𝒉𝒎 = 𝟏𝟎𝟗 𝒂𝒃 − 𝒐𝒉𝒎. Relation between Stat-ohm and Ab-ohm:- 𝟏 As, 𝟏 𝒐𝒉𝒎 = 𝒔𝒕𝒂𝒕 − 𝒐𝒉𝒎 …..…… (1) Again 𝟏 𝒐𝒉𝒎 = 𝟏𝟎𝟗 𝒂𝒃 − 𝒐𝒉𝒎 …..…… (2) 𝟗×𝟏𝟎𝟏𝟏 𝟏 Now, equating (1) and (2) we get, ⟹ 𝒔𝒕𝒂𝒕 − 𝒐𝒉𝒎 = 𝟏𝟎𝟗 𝒂𝒃 − 𝒐𝒉𝒎. 𝟗 × 𝟏𝟎𝟏𝟏 ⟹ 𝟏 𝒔𝒕𝒂𝒕 − 𝒐𝒉𝒎 = 𝟏𝟎𝟗 × 𝟗 × 𝟏𝟎𝟏𝟏 𝒂𝒃 − 𝒐𝒉𝒎, ⟹ 𝟏 𝒔𝒕𝒂𝒕 − 𝒐𝒉𝒎 = 𝟗 × 𝟏𝟎𝟐𝟎 𝒂𝒃 − 𝒐𝒉𝒎. Dimension of resistance:- 𝑽 𝑾𝒐𝒓𝒌 𝑪𝒉𝒂𝒓𝒈𝒆 𝑾𝒐𝒓𝒌 𝑴𝟏 𝑳𝟐 𝑻−𝟐 As, 𝑹 = = = = = 𝑴𝟏 𝑳𝟐 𝑻−𝟑 𝑨−𝟐. 𝑰 𝑪𝒖𝒓𝒓𝒆𝒏𝒕 𝑪𝒉𝒂𝒓𝒈𝒆 × 𝑪𝒖𝒓𝒓𝒆𝒏𝒕 𝑨𝟏 𝑻𝟏 × 𝑨𝟏 Thus, dimensions of resistances are 𝟏, 𝟐, −𝟑 and −𝟐 in mass, length, time and current respectively. Factors effect upon resistance:- (1) The resistance of a conductor increases with increase in temperature i.e. 𝑹 ∝ 𝑻. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 10  CHAPTER - 03 CURRENT ELECTRICITY (2) The resistance of a conductor increases with increase in length of the conductor i.e. 𝑹 ∝ 𝒍. 𝟏 (3) The resistance of a conductor inversely proportional to its area of cross-section i.e 𝑹 ∝. 𝑨 (4) Resistance depends upon nature of material i.e. number of electrons per meter cube of material. The resistance of a nichrome wire is 𝟔𝟎 times that of copper wire of equal length and area of cross-section. (5) The volume of a conductor remains unaffected by stretching the conductor i.e. 𝑨𝟏 𝒍𝟏 = 𝑨𝟐 𝒍𝟐 and resistance changes from „𝑹𝟏 ‟ to „𝑹𝟐 ‟. Where, 𝑨𝟏 and 𝑨𝟐 = Area of cross-section of a conductor before and after stretching and 𝒍𝟏 and 𝒍𝟐 = Length of the conductor before and after stretching. 𝑹𝟏 𝒍𝟏 𝑨𝟐 Now, the ratio of resistances before and after stretching of a conductor is, = ×. 𝑹𝟐 𝒍𝟐 𝑨𝟏 Effect of resistance with temperature:- The resistance of a conductor varies with temperature. Let „𝑹𝟎 ‟ and „𝑹𝒕 ‟ are resistance of conductor at 𝟎𝟎 𝑪 and 𝒕𝟎 𝑪. Thus, 𝑹𝒕 − 𝑹𝟎 is the increase in resistance of the conductor at 𝒕𝟎 𝑪. Now, 𝑹𝒕 − 𝑹𝟎 ∝ 𝑹𝟎 and 𝑹𝒕 − 𝑹𝟎 ∝ 𝒕, ⟹ 𝑹𝒕 − 𝑹𝟎 = 𝜶𝑹𝟎 𝒕 i.e. 𝑹𝒕 = 𝑹𝟎 + 𝑹𝟎 𝜶𝒕. 𝑹𝒕 −𝑹𝟎 Where, 𝜶 = Temperature co-efficient of resistance. As, 𝑹𝟎 𝜶𝒕 = 𝑹𝒕 − 𝑹𝟎 , ⟹ 𝜶 =. 𝑹𝟎 𝒕 Thus, “The temperature coefficient of resistance is defined as change in resistance of the conductor per unit resistance per degree centigrade rise of temperature”. Note:- If the resistance of a conductor is „𝑹𝟐 ‟ at 𝒕𝟐 ℃ and „𝑹𝟏 ‟ at 𝒕𝟏 ℃ 𝒕𝟏 < 𝒕𝟐 then, 𝑹𝟐 = 𝑹𝟏 𝟏 + 𝜶 𝒕𝟐 − 𝒕𝟏. Important:- (1) The unit of „𝜶‟ is 𝑪𝒆𝒏𝒕𝒊𝒈𝒓𝒂𝒅𝒆−𝟏 or 𝑲𝒆𝒍𝒗𝒊𝒏−𝟏 and dimension is 𝑴𝟎 𝑳𝟎 𝑻𝟎 𝑲−𝟏. For all metals and most of alloys the value of „𝜶‟ is positive i.e. their resistance increases with an increase in their temperature. (2) The value of „𝜶‟ is greater for metals and smaller for alloys. Thus metals show more change in resistance as compare to alloy when they are heated. (3) The substances like carbon and semi-conductors the value of „𝜶‟ is negative i.e. their resistance decreases with increase in temperature. For insulators the value of „𝜶‟ is negative. Conductance (𝑮):- The reciprocal of resistance of a conductor is called conductance and is denoted by ‘G’. 𝟏 𝟏 𝑨 Mathematically, 𝑮 = = 𝒍 =. 𝑹 𝝆× 𝝆𝒍 𝑨 𝟏 𝟏 Unit of conductance:- As, 𝑮 = = = 𝑶𝒉𝒎−𝟏 𝑶𝒓 𝒎𝒉𝒐 𝑶𝒓 𝑺𝒊𝒆𝒎𝒆𝒏. 𝑹 𝑶𝒉𝒎 𝟏 𝟏 Dimension of conductance:- As, Conductance, 𝑮 = = = 𝑴−𝟏 𝑳−𝟐 𝑻𝟑 𝑨𝟐. 𝑹 𝑴𝟏 𝑳𝟐 𝑻−𝟑 𝑨−𝟐 Thus, dimensions of conductance are −𝟏, −𝟐, 𝟑 and 𝟐 in mass, length, time and current respectively. Resistivity (or) Specific resistance (𝝆):- Resistivity means the capacity of a conductor to resist flow of current or flow of the electron. The reciprocal of conductivity is called as resistivity. 𝟏 𝟏 𝑹𝑨 𝑽 𝑽 𝑽 𝒎𝒍 Mathematically, 𝝆 = = 𝒍 =. From Ohm‟s law, 𝑹 = = = 𝒆𝑽𝝉 =. 𝝇 𝑹𝑨 𝒍 𝑰 𝒏𝑨𝑽𝒅 𝒆 𝒏𝑨 𝒆 𝒏𝑨𝒆𝟐 𝝉 𝒎𝒍 From this equation it is clear that, Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 11  CHAPTER - 03 CURRENT ELECTRICITY (i) Resistance is directly proportional to the length of the conductor i.e. 𝑹 ∝ 𝒍 ……… (1) 𝟏 (ii) Resistance is inversely proportional to the area of cross-section of the conductor i.e. 𝑹 ∝ ……… (2) 𝑨 𝒍 𝒍 𝑹𝑨 Now, combining equation (1) and (2) we get, ⟹ 𝑹 ∝ , ⟹ 𝑹 = 𝝆 × , ⟹𝝆=. 𝑨 𝑨 𝒍 Where, „𝝆‟ is the proportional constant and is called resistivity or specific resistance of material. If 𝒍 = 𝟏 unit and 𝑨 = 𝟏 unit than, ⟹ 𝑹 = 𝝆. Thus, “The resistivity of material is defined as resistance of a conductor made up of material of unit length and unit area of the cross-section”. 𝑹𝑨 𝑶𝒉𝒎 × 𝑴𝒆𝒕𝒆𝒓 𝟐 Unit of resistivity:- As, 𝝆 = =. 𝒍 𝑴𝒆𝒕𝒆𝒓 In M.K.S. ⟶ 𝑶𝒉𝒎 × 𝒎𝒆𝒕𝒆𝒓. In C.G.S. ⟶ 𝑨𝒃 − 𝒐𝒉𝒎 × 𝒄𝒎 (Or) 𝑺𝒕𝒂𝒕 − 𝒐𝒉𝒎 × 𝒄𝒎. 𝑹𝑨 𝑴𝟏 𝑳𝟐 𝑻−𝟑 𝑨−𝟐 × 𝑳𝟐 Dimension of resistivity:- As, 𝝆 = = = 𝑴𝟏 𝑳𝟑 𝑻−𝟑 𝑨−𝟐. 𝒍 𝑳𝟏 Thus, dimensions of resistivity are 𝟏, 𝟑, −𝟑 and −𝟐 in mass, length, time and current respectively. Important:- 𝒍 (1) As we know, 𝑹 = 𝝆 ×. If 𝒍 = 𝟏𝒎 and 𝑨 = 𝟏𝒎𝟐 than 𝑹 = 𝝆. 𝑨 Thus, “Resistivity of a material is the resistance offered by 𝟏𝒎 length of wire of the material having area of cross-section of 𝟏𝒎𝟐 ”. For example, resistivity of copper is 𝟏. 𝟕 × 𝟏𝟎−𝟖 𝛀𝒎. It means that if you take a copper wire of 𝟏𝒎 long and having an area of cross-section 𝟏𝒎𝟐 , then resistance of this copper wire will be 𝟏. 𝟕 × 𝟏𝟎−𝟖 𝛀. 𝒍 (2) As we know, 𝑹 = 𝝆 ×. If 𝒍 = 𝟏𝒎 and 𝑨 = 𝟏𝒎𝟐 than 𝝆 = 𝑹. 𝑨 Thus, “Resistivity of a material is numerically equal to the resistance offered by a unit cube of that material when current flows perpendicular to the opposite faces of cube as shown in figure (a). 𝒍 (3) As we know, 𝑹 = 𝝆 ×. If 𝒍 = 𝟏𝒎 and 𝑨 = 𝟏𝒎𝟐 than 𝝆 = 𝑹. 𝑨 Thus, “Resistivity of a material is numerically equal to the resistance offered by a cylindrical conductor of that material of unit cross-sectional area and unit length when current flows perpendicular to the opposite ends of the cylinder as shown in figure (b). (4) If we take a cube of the material of each side 𝟏𝒎, then area of cross-section of each face is 𝟏𝒎𝟐 and length between opposite faces is 𝟏𝒎. Thus, resistivity may be defined as the resistance between the opposite faces of a meter cube of the material. 𝒍 𝒍 𝒍 𝒍 𝟒𝝆𝒍 𝟏 (5) As we know, 𝑹 = 𝝆 × = 𝝆 × =𝝆 × 𝑫 𝟐 =𝝆 × 𝝅𝑫𝟐.⟹𝑹=. As „𝝆‟ and „𝒍‟ are constant thus, 𝑹 ∝. 𝑨 𝝅𝒓𝟐 𝝅 𝝅𝑫𝟐 𝑫𝟐 𝟐 𝟒 Thus, resistance of a conductor is inversely proportional to the square of its diameter. Factors effect upon resistivity:- (1) Resistivity is always constant for the material. Two wires having different lengths and thickness but made up of same material will have same resistivity. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 12  CHAPTER - 03 CURRENT ELECTRICITY (2) Resistivity of a conductor is much small while that of insulator is large. (3) Resistivity of a conductor increases with an increase in temperature. (4) Resistivity of an insulator increases with a decrease in temperature. (5) The resistivity is independent of shape and size of the conductor and it depends upon nature of material. Important:- 𝟏 (1) The resistivity of a conductor is inversely proportional to average relaxation time i.e. 𝝆 ∝. 𝝉 𝟏 𝒏𝑨𝒆𝟐 𝝉 𝒎𝒍 𝒎 𝒍 𝒍 𝒎 𝟏 ∵ = ,⟹ 𝑹 = = … … ….. 𝒊 𝒂𝒏𝒅 𝑹 = 𝝆 … … ….. 𝒊𝒊 On comparing (i) & (ii) 𝝆 = i.e. 𝝆 ∝. 𝑹 𝒎𝒍 𝒏𝑨𝒆𝟐 𝝉 𝒏𝒆𝟐 𝝉 𝑨 𝑨 𝒏𝒆𝟐 𝝉 𝝉 Hence, resistivity of a conductor depends upon (i) Number of electrons per unit volume (𝒏) and (ii) relaxation time 𝝉. 𝑽 𝑬 𝑹𝑨 ×𝑨 𝑽𝑨 𝑽 𝑨 𝟏 𝑬 𝑰 𝑰 (2) Resistivity and current density vector are related as, 𝝆 =. 𝝆 = = = = × =𝑬× = and 𝑱 = 𝑱 𝒍 𝒍 𝑰𝒍 𝒍 𝑰 𝑱 𝑱 𝑨 (3) The resistivity increases with impurity and mechanical stress. 𝑽 𝑽 ×𝑨 𝒏𝑨𝝁𝑽𝒆 ×𝑨 𝟏 𝑹𝑨 𝑰 𝒍 𝟏 (4) Resistivity and mobility of electron are related as, 𝝆 =. 𝝆= = = = 𝒏𝒆𝝁 𝒍 𝒍 𝒍 𝒏𝒆𝝁 𝑬 𝑽 𝟏 𝟏 𝑬 𝑽 𝑽 𝒍 (5) Resistivity and drift velocity are related as, 𝝆 = =. 𝝆= = 𝒗 = = = 𝒏𝒆𝒗𝒅 𝒏𝒆𝒍𝒗𝒅 𝒏𝒆𝝁 𝒏𝒆× 𝒅 𝒏𝒆𝒗𝒅 𝒏𝒆𝒗𝒅 𝒏𝒆𝒍𝒗𝒅 𝑬 𝟏 𝟏 𝟏 (6) Conductivity and mobility are related as, 𝝇 = 𝒏𝒆𝝁. 𝝆= , ⟹ = , ⟹ 𝝇 = 𝒏𝒆𝝁 𝒏𝒆𝝁 𝝇 𝒏𝒆𝝁 𝒏𝒆𝒗𝒅 𝒗𝒅 𝒏𝒆𝒗𝒅 (7) Conductivity and drift velocity are related as, 𝝇 =. 𝑨𝒔, 𝝇 = 𝒏𝒆𝝁 & 𝝁 = ,⟹ 𝝇 = 𝑬 𝑬 𝑬 Note:- Resistance is a property of an object while resistivity is a property of material of the object. So, resistance of objects of different shapes and sizes but made up of same materials is different. On the other hand, resistivity of the objects of different shapes and sizes but of same material is same. Conductivity (𝝇):- The reciprocal of resistivity is called as conductivity of material and is denoted by „𝝇‟. 𝟏 𝟏 𝒍 Mathematically, 𝝇 = = 𝑹𝑨 =. 𝝆 𝒍 𝑹𝑨 𝟏 𝟏 Unit of conductivity:- As, 𝝇 = = = 𝑶𝒉𝒎−𝟏 × 𝒎𝒆𝒕𝒆𝒓−𝟏. 𝝆 𝑶𝒉𝒎 × 𝑴𝒆𝒕𝒆𝒓 𝟏 𝟏 Dimension of conductivity:- As, 𝝇 = = = 𝑴−𝟏 𝑳−𝟑 𝑻𝟑 𝑨𝟐. 𝝆 𝑴𝟏 𝑳𝟑 𝑻−𝟑𝑨−𝟐 Thus, dimensions of conductivity are −𝟏, −𝟑, 𝟑 and 𝟐 in mass, length, time and current respectively. 𝒎 𝟏 𝟏 𝒏𝒆𝟐 𝝉 Important:- As we know, resistivity, 𝝆 =. Now conductivity, 𝝇 = = =. 𝒏𝒆𝟐 𝝉 𝝆 𝒎 𝒏𝒆𝟐 𝝉 𝒎 From the above equation we have, 𝝇 ∝ 𝒏 and 𝝇 ∝ 𝝉. Thus, conductivity of a conductor depends upon – (i) Nature of the material of a conductor: Different materials have different number density of electrons (𝒏), so conductivity of different material is different. (ii) Temperature of the conductor: When temperature of a conductor increases relaxation time 𝝉 decreases. Hence the conductivity of conductor decreases. The variation of conductivity of a conductor with temperature is as shown above the figure. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 13  CHAPTER - 03 CURRENT ELECTRICITY Classification of materials on the basis of electrical conductivity:- On the basis of electrical conductivity, the materials are classified as (i) Insulators (ii) Conductors and (iii) Semi-conductors. (i) Insulators:- Those materials whose electrical conductivity is negligible are called as insulators. Example- Mica, Glass, Wood, Rubber etc. When a small potential difference is applied across an insulator practically no current flows through it. There are practically no free electrons in an insulator. For this reason they are poor conductors of electricity as well as heat. (ii) Conductors:- Those materials whose electrical conductivity is very high are called as conductors. Example- Copper, Silver, Aluminium etc. When a small potential difference is applied across a conductor a large current flows through it. There are large free electrons in a conductor. For this reason they are good conductors of electricity as well as heat. (iii) Semi-conductors:- Those material whose electrical conductivity lies in between insulator and conductor are called as semi-conductors. Example- Germanium and Silicon. When a small potential difference is applied across a semi- conductor a very weak current flows through it. The conductivity of a semi-conductor can be increased by adding controlled amount of suitable impurities. Semi-conductors are being widely used in the manufacture of a variety of electronic devices. Difference between Resistance and Conductance:- Resistance Conductance 1) It is the ratio of the P.D. across two ends of 1) The reciprocal of resistance is called as conductor to the current flowing through it. conductance. 2) Mathematically, 𝑹 = 𝑽 𝑰. 2) Mathematically, 𝑮 = 𝑨 𝝆𝒍. 𝟏 𝟐 −𝟑 −𝟐 3) Its unit is 𝑶𝒉𝒎 and dimension is 𝑴 𝑳 𝑻 𝑨. 3) Its unit is 𝑶𝒉𝒎−𝟏 and dimension is 𝑴−𝟏 𝑳−𝟐 𝑻𝟑 𝑨𝟐. Difference between Resistivity and Conductivity:- Resistivity Conductivity 1) The reciprocal of conductivity is called resistivity. 1) The reciprocal of resistivity is called conductivity. 2) Mathematically, 𝝆 = 𝑹𝑨 𝒍. 2) Mathematically, 𝝇 = 𝒍 𝑹𝑨. 3) Its unit is 𝛀𝒎 and dimension is 𝑴𝟏 𝑳𝟑 𝑻−𝟑 𝑨−𝟐. 3) Its unit is 𝛀−𝟏 𝒎−𝟏 and dimension is 𝑴−𝟏 𝑳−𝟑 𝑻𝟑 𝑨𝟐. Difference between Resistance and Resistivity:- Resistance Resistivity 1) It is the ratio of the P.D. across two ends of 1) It is the capacity of a conductor to resist flow of conductor to the current flowing through it. current or flow of electron. 2) Mathematically, 𝑹 = 𝑽 𝑰. 2) Mathematically, 𝝆 = 𝑹𝑨 𝒍. 3) It depends upon geometry of the conductor. 3) It is independent of geometry of the conductor. Electrical energy:- It is defined as, The amount of work done in moving a charge between any two points of varying potential in an electric field. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 14  CHAPTER - 03 CURRENT ELECTRICITY Let us consider an electrical device or a circuit element (an electric lamp, heater etc.) of resistance „𝑹‟ through which current „𝑰‟ flows from end „𝑨‟ to end „𝑩‟ for time „𝒕‟ as shown the figure. Let a charge „𝒒‟ flowing from „𝑨‟ to „𝑩‟ in time „𝒕‟. Let „𝑽‟ is the potential difference between „𝑨‟ and „𝑩‟. 𝑾 Now, the work done to carry the charge from „𝑩‟ to „𝑨‟ is, 𝑾 = 𝑽 × 𝒒............ (1) ∵𝑽= 𝒒 𝒒 But as current, 𝑰 = , ⟹ 𝒒 = 𝑰𝒕. Now, equation (1) becomes, ⟹ 𝑾 = 𝑽𝑰𝒕............ (2) 𝒕 𝑽 But current, 𝑰 = , ⟹ 𝑽 = 𝑰𝑹. Now equation (2) becomes, ⟹ 𝑾 = 𝑰𝑹 × 𝑰𝒕 = 𝑰𝟐 𝑹𝒕............ (3) 𝑹 𝑽 𝑽 𝑽𝟐 𝒕 Since, 𝑰 =. Now, equation (2) becomes, ⟹ 𝑾 = 𝑽𝑰𝒕 = 𝑽 × × 𝒕 = …...…… (4) 𝑹 𝑹 𝑹 𝑽𝟐 𝒕 Now, from equation (1), (2), (3) and (4) we have, ⟹ 𝑾 = 𝑽𝒒 = 𝑽𝑰𝒕 = 𝑰𝟐 𝑹𝒕 =. 𝑹 𝑽𝟐 𝒕 Since work done is equal to energy, thus electrical energy, 𝑾 = 𝑽𝒒 = 𝑽𝑰𝒕 = 𝑰𝟐 𝑹𝒕 =. 𝑹 This above equation gives the different mathematical form of electrical energy. Unit of electrical energy:- If „𝑽‟ is in volt, „𝑰‟ is in ampere and „𝒕‟ is in second than electrical energy „𝑾‟ is taken in Joule. ⟹ 𝟏 𝒋𝒐𝒖𝒍𝒆 = 𝟏 𝒗𝒐𝒍𝒕 × 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆 × 𝟏 𝒔𝒆𝒄 = 𝟏 𝒘𝒂𝒕𝒕 × 𝟏 𝒔𝒆𝒄. ⟹ 𝟏 𝒋𝒐𝒖𝒍𝒆 = 𝟏 𝒘𝒂𝒕𝒕 × 𝒔𝒆𝒄𝒐𝒏𝒅. So, 1 watt second or joule is the energy consumed if the power maintained is 1 watt for a second in a device. For example bulb, heater etc. Dimension of electric energy:- As, 𝑾 = 𝑽 × 𝒒 = 𝑴𝟏 𝑳𝟐 𝑻−𝟑 𝑨−𝟏 × 𝑨𝟏 𝑻𝟏 = 𝑴𝟏 𝑳𝟐 𝑻−𝟐. Thus, the dimension of electric energy are 𝟏, 𝟐 and −𝟐 in mass, length and time respectively. Important:- (1) For calculation purpose of electrical energy, watt second is considered as a too small unit. The bigger unit of electric energy is ‘watt hour’. ⟹ 𝟏 𝒘𝒂𝒕𝒕 𝒉𝒐𝒖𝒓 = 𝟏 𝒘𝒂𝒕𝒕 × 𝟏 𝒉𝒐𝒖𝒓 = 𝟏 𝒘𝒂𝒕𝒕 × 𝟑𝟔𝟎𝟎 𝒔𝒆𝒄𝒐𝒏𝒅 = 𝟑𝟔𝟎𝟎 𝒘𝒂𝒕𝒕 𝒔𝒆𝒄. ⟹ 𝟏 𝒘𝒂𝒕𝒕 𝒉𝒐𝒖𝒓 = 𝟑𝟔𝟎𝟎 𝒋𝒐𝒖𝒍𝒆 = 𝟑𝟔𝟎𝟎 × 𝟏𝟎𝟕 𝒆𝒓𝒈. ⟹ 𝟏 𝒘𝒂𝒕𝒕 𝒉𝒐𝒖𝒓 = 𝟑𝟔𝟎𝟎 × 𝟏𝟎𝟕 𝒆𝒓𝒈 = 𝟑𝟔 × 𝟏𝟎𝟐 𝒋𝒐𝒖𝒍𝒆. (2) 𝟏 𝒌𝒊𝒍𝒐 𝒘𝒂𝒕𝒕 𝒉𝒐𝒖𝒓 = 𝟏𝟎𝟎𝟎 = 𝟏𝟎𝟑 𝒘𝒂𝒕𝒕 𝒉𝒐𝒖𝒓. 𝟏 𝒎𝒆𝒈𝒂 𝒘𝒂𝒕𝒕 𝑴𝑾 = 𝟏𝟎𝟔 𝒘𝒂𝒕𝒕 = 𝟑. 𝟔 × 𝟏𝟎𝟔 𝑱𝒐𝒖𝒍𝒆. This unit is called as 𝑩. 𝑶. 𝑻. 𝑼. (Board Of Trade Unit) or unit of electricity. It is defined as the amount of work done when a power of one kilowatt is consumed for one hour. 𝑽 𝒊𝒏 𝒗𝒐𝒍𝒕 × 𝑰 𝒊𝒏 𝒂𝒎𝒑𝒆𝒓𝒆 × 𝒕 (𝒊𝒏 𝒔𝒆𝒄𝒐𝒏𝒅) (3) Electrical energy in 𝑲𝒘𝒉 is, 𝑲𝒘𝒉 = 𝑷𝒐𝒘𝒆𝒓 𝒊𝒏 𝑲𝑾 × 𝑻𝒊𝒎𝒆 𝒊𝒏 𝒉𝒐𝒖𝒓𝒔 =. 𝟏𝟎𝟎𝟎 Electric power:- 𝑾𝒐𝒓𝒌 The time rate of spending electrical energy is called as electric power. As, Power, 𝑷 =. 𝑻𝒊𝒎𝒆 When a current „𝑰‟ flows through a conductor for a time „𝒕‟ than work done „𝑾‟ is, 𝑾 = 𝑽𝑰𝒕. 𝑽𝑰𝒕 ∴ Electric power, 𝑷 = = 𝑽 × 𝑰................. (1) 𝒕 Thus, “Electric power is defined as product of current and P.D. due to which the current flows”. Since, 𝑽 = 𝑰𝑹. Now, equation (1) becomes, 𝑷 = 𝑰𝑹 × 𝑰 = 𝑰𝟐 𝑹. 𝑽 𝑽 𝑽𝟐 Since, 𝑰 =. Now, equation (1) becomes, 𝑷 = 𝑽 × =. 𝑹 𝑹 𝑹 Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 15  CHAPTER - 03 CURRENT ELECTRICITY 𝑽𝟐 Thus, electric power may be written as, ⟹ 𝑷 = 𝑽𝑰 = 𝑰𝟐 𝑹 =. 𝑹 This above equation gives the different mathematical form of electric power. Unit of electric power:- Power is measured in ‘watt’, As,𝟏 𝒘𝒂𝒕𝒕 = 𝟏 𝒗𝒐𝒍𝒕 × 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆. Thus, “The power of an electric circuit is said to be 1 watt if a current of 1 ampere flows through it when there is a potential difference of 1 volt across its ends”. Dimension of electric power:- As, electric power, 𝑷 = 𝑽 × 𝑰. ⟹ 𝑷 = 𝑴𝟏 𝑳𝟐 𝑻−𝟑 𝑨−𝟏 × 𝑨 = 𝑴𝟏 𝑳𝟐 𝑻−𝟑. Thus, the dimension of electric power are 𝟏, 𝟐 and −𝟑 in mass, length and time respectively. Important:- 𝑱𝒐𝒖𝒍𝒆 𝑪𝒐𝒖𝒍𝒐𝒎𝒃 (1) As, 𝟏 𝒘𝒂𝒕𝒕 = 𝟏 𝒗𝒐𝒍𝒕 × 𝟏 𝒂𝒎𝒑𝒆𝒓𝒆 = 𝟏 ×𝟏 = 𝟏 𝑱/𝒔, i.e. 𝟏 𝑾𝒂𝒕𝒕 = 𝟏 𝑱𝒐𝒖𝒍𝒆/𝑺𝒆𝒄𝒐𝒏𝒅. 𝑪𝒐𝒖𝒍𝒐𝒎𝒃 𝑺𝒆𝒄𝒐𝒏𝒅 Since watt is a unit of power, its higher units like kilowatt (𝑲𝑾), megawatt (𝑴𝑾), horse power (𝑯𝑷) etc. are used as its practical units. (2) 𝟏 𝒌𝒊𝒍𝒐 𝒘𝒂𝒕𝒕 = 𝟏𝟎𝟎𝟎 𝒘𝒂𝒕𝒕 = 𝟏𝟎𝟑 𝒘𝒂𝒕𝒕. (3) 𝟏 𝒎𝒆𝒈𝒂 𝒘𝒂𝒕𝒕 = 𝟏𝟎𝟔 𝒘𝒂𝒕𝒕 and 𝟏 𝑯𝑷 = 𝟕𝟒𝟔 𝒘𝒂𝒕𝒕. (4) 𝟏 𝒌𝒊𝒍𝒐 𝒘𝒂𝒕𝒕 – 𝒉𝒐𝒖𝒓 = 𝟏 𝒌𝒊𝒍𝒐 𝒘𝒂𝒕𝒕 × 𝟏 𝒉𝒐𝒖𝒓 = 𝟏𝟎𝟎𝟎 𝒘𝒂𝒕𝒕 × 𝟑𝟔𝟎𝟎 𝒔𝒆𝒄𝒐𝒏𝒅. = 𝟑𝟔 × 𝟏𝟎𝟓 𝒘𝒂𝒕𝒕 𝒔𝒆𝒄𝒐𝒏𝒅 = 𝟑𝟔 × 𝟏𝟎𝟓 𝒋𝒐𝒖𝒍𝒆𝒔 = 𝟑𝟔 × 𝟏𝟎𝟓 × 𝟏𝟎𝟕 𝒆𝒓𝒈𝒔. ⟹ 𝟏 𝑲𝒘𝒉 = 𝟑𝟔 × 𝟏𝟎𝟏𝟐 𝒆𝒓𝒈𝒔. 𝑽 𝒊𝒏 𝒗𝒐𝒍𝒕 × 𝑰 𝒊𝒏 𝒂𝒎𝒑𝒆𝒓𝒆 (5) Electrical power in 𝑲𝒘𝒉 is given by, 𝑷 𝒊𝒏 𝑲𝒘𝒉 =. 𝟏𝟎𝟎𝟎 Relation between electric energy and electric power:- Electric energy, 𝑾 = 𝑽𝑰𝒕.................. (1) Again electric power, 𝑷 = 𝑽𝑰.................. (2) 𝑾 𝑬𝒍𝒆𝒄𝒕𝒓𝒊𝒄 𝒆𝒏𝒆𝒓𝒈𝒚 On comparing (1) and (2) we get, 𝑾 = 𝑷𝒕, ⟹ 𝑷 = i.e. 𝑬𝒍𝒆𝒄𝒕𝒓𝒊𝒄 𝒑𝒐𝒘𝒆𝒓 =. 𝒕 𝑻𝒊𝒎𝒆 Maximum power theorem:- It states that, “The output power delivered by a source of e.m.f. is maximum when external resistance is equal to internal resistance of the source”. Proof:- Let us consider a battery of e.m.f. „𝑬‟ having internal resistance „𝒓‟ and it is delivered power to external resistance „𝑹‟ as shown figure. 𝑵𝒆𝒕 𝒆.𝒎.𝒇. 𝑬 Now, net resistance of the circuit = 𝑹 + 𝒓. Now, current in the circuit, 𝑰 = =. 𝑵𝒆𝒕 𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝑹+𝒓 𝑬 𝟐 𝑬𝟐 𝑹 Now, power delivered to the external resistance, 𝑷 = 𝑰𝟐 𝑹 = ×𝑹=. 𝑹+𝒓 𝑹+𝒓 𝟐 𝒅𝑷 𝒅 𝑬𝟐 𝑹 𝒅 −𝟐 But power delivered will be maximum if, = 𝟎, ⟹ = 𝟎, ⟹ 𝑹 𝑹+𝒓 =𝟎 ∵ 𝑬𝟐 ≠ 𝟎. 𝒅𝑹 𝒅𝑹 𝑹+𝒓 𝟐 𝒅𝑹 𝒅 𝑹+𝒓 −𝟐 −𝟐 𝒅𝑹 𝒅 𝒅𝒗 𝒅𝒖 ⟹ 𝑹. + 𝑹+𝒓. = 𝟎. ∵ 𝒖. 𝒗 = 𝒖. + 𝒗. 𝒅𝑹 𝒅𝑹 𝒅𝒙 𝒅𝒙 𝒅𝒙 𝒅 𝑹+𝒓 −𝟐 𝒅 𝑹+𝒓 −𝟐 −𝟐−𝟏 𝒅𝑹 𝒅𝒓 −𝟐 ⟹ 𝑹. × + 𝑹+𝒓. 𝟏 = 𝟎, ⟹ 𝑹 (−𝟐) × 𝑹 + 𝒓 × + + 𝑹+𝒓 = 𝟎. 𝒅 𝑹+𝒓 𝒅𝑹 𝒅𝑹 𝒅𝑹 −𝟑 −𝟐 −𝟑 −𝟐 ⟹ −𝟐𝑹 × 𝑹 + 𝒓 ×𝟏+𝟎+ 𝑹+𝒓 = 𝟎, ⟹ −𝟐𝑹 × 𝑹 + 𝒓 + 𝑹+𝒓 = 𝟎. −𝟑 −𝟑 ⟹ 𝑹+𝒓 −𝟐𝑹 + 𝑹 + 𝒓 = 𝟎, ⟹ 𝑹 + 𝒓 𝒓 − 𝑹 = 𝟎. −𝟑 As, 𝑹 + 𝒓 ≠ 𝟎, so, 𝒓 − 𝑹 = 𝟎, ⟹ 𝒓 = 𝑹. This proves the maximum power theorem i.e. for maximum power transfer to external resistance the external resistance must be equal to internal resistance of source. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 16  CHAPTER - 03 CURRENT ELECTRICITY 𝑬 Thus, maximum power transferred, 𝑷𝒎𝒂𝒙. = 𝑰𝟐 𝑹. Since, current, 𝑰 =. 𝑹+𝒓 𝑬 𝟐 𝑬𝟐 𝑬𝟐 𝑬𝟐 𝑬𝟐 𝑬𝟐 ⟹ 𝑷𝒎𝒂𝒙. = ×𝑹= ×𝑹 = ×𝑹= ×𝑹= , i.e. ⟹ 𝑷𝒎𝒂𝒙. =. 𝑹+𝒓 𝑹+𝒓 𝟐 𝑹+𝑹 𝟐 𝟒𝑹𝟐 𝟒𝑹 𝟒𝑹 Important:- When the battery is shorted, „𝑹‟ becomes zero and hence output power is zero. In this case, entire power of the battery is dissipated as heat inside the battery due to its internal resistance. 𝑬 𝟐 𝑬𝟐 Now power dissipated inside the battery = 𝑰𝟐 𝒓 = × 𝒓, ⟹ 𝑷 =. 𝒓 𝒓 Relation between Current (𝑰), potential difference (𝑽), Resistance (𝑹) and Power (𝑷):- 𝑽 𝑷 𝑷 (a) Electric Current: 𝑰 = = =. 𝑹 𝑽 𝑹 𝑷 (b) Potential Difference: 𝑽 = 𝑰𝑹 = = 𝑷𝑹. 𝑰 𝑽 𝑽𝟐 𝑷 (c) Electrical Resistance: 𝑹 = = =. 𝑰 𝑷 𝑰𝟐 𝑽𝟐 (d) Electric Power: 𝑷 = 𝑽𝑰 = 𝑰𝟐 𝑹 =. 𝑹 The above relations can be summarized by a chart is called as “Electrician chart”. Electric power in series and parallel combinations of bulbs:- Let us consider three electric bulbs capable of consuming powers 𝑷𝟏 , 𝑷𝟐 and 𝑷𝟑 respectively when 𝑽𝟐 connected across a source of e.m.f. „𝑽‟. As, electric power, 𝑷 =. 𝑹 𝑽𝟐 𝑽𝟐 𝑽𝟐 Now, power of bulbs are, 𝑷𝟏 = , 𝑷𝟐 = and 𝑷𝟑 =. 𝑹𝟏 𝑹𝟐 𝑹𝟑 Where, 𝑹𝟏 , 𝑹𝟐 and 𝑹𝟑 are resistance of bulbs. Case-(1), When all bulbs are connected in series:- In series connection the three resistances are connected in the following manner. Now, from diagram, the net resistance is, 𝑹𝑺 = 𝑹𝟏 + 𝑹𝟐 + 𝑹𝟑. 𝑽𝟐 𝑽𝟐 Now, net power consumed by circuit is, ⟹ 𝑷𝑺 = =. 𝑹𝑺 𝑹𝟏 +𝑹𝟐 +𝑹𝟑 𝟏 𝑹𝟏 +𝑹𝟐 +𝑹𝟑 𝑹𝟏 𝑹𝟐 𝑹𝟑 𝟏 𝟏 𝟏 𝟏 𝑽𝟐 𝟏 𝑹 ⟹ = = + +.⟹ = + + ∵𝑷= ,⟹ = 𝑷𝑺 𝑽𝟐 𝑽𝟐 𝑽𝟐 𝑽𝟐 𝑷𝑺 𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑹 𝑷 𝑽𝟐 Thus, “In a series combination of resistances, the reciprocal of the net power consumed is equal to the sum of the reciprocals of the individual powers consumed by them”. 𝟏 𝟏 𝟏 𝟏 Important:- If „𝒏‟ bulbs of same power „𝑷‟ are in series than, = + + … …. 𝒏 𝒕𝒊𝒎𝒆𝒔. 𝑷𝑺 𝑷 𝑷 𝑷 𝟏 𝟏 𝒏 𝑷 ⟹ = 𝒏 = , ⟹ 𝑷𝑺 =. As bulbs are connected in series, current „𝒊‟ through each bulb is same. 𝑷𝑺 𝑷 𝑷 𝒏 𝑽 Now the current in the circuit is, 𝒊 =. Where „𝑹𝟏 ‟, „𝑹𝟐 ‟ and „𝑹𝟑 ‟ are resistances of three bulbs. 𝑹𝟏 +𝑹𝟐 +𝑹𝟑 𝟏 𝑽𝟐 Now the power of three bulbs will be: 𝑷𝟏 = 𝒊𝟐 𝑹𝟏 , 𝑷𝟐 = 𝒊𝟐 𝑹𝟐 and 𝑷𝟑 = 𝒊𝟐 𝑹𝟑. As, 𝑹 ∝. ∵𝑹= 𝑷 𝑷 Thus, the bulb of lowest wattage (power) will have maximum resistance and it will glow with maximum brightness. When current in the circuit exceeds the safety limit, bulb of low watt will be fuse first. Prepared By: Sugyan Kumar Sahu (7751814888 / 9438725999) 17  CHAPTER - 03 CURRENT ELECTRICITY Case-(2), When all bulbs are connected in parallel:- In parallel connection the three resistances are connected in the following manner.