CHAPTER -3 (CURRENT ELECTRICITY ic current Itis the rate of flow of electric charge. ‘The instantaneous current is given by ag dt Steady current is given by 1 1, q~charge, t-time 1 Electric current is a scalar quantity. Sl unit — ampere (A) 1A= 1C/s Other units are mA= 107A, wA=10%A Lightning is an example of transient current. Current in a domestic appliance is of the order of 1A. Current carried by lightning is of the order of 10°A. Current in our nerves is of the order of ipa. By convention direction of motion of positive charges (direction opposite to the ‘motion of electrons) is taken as the direction of current. Current density (J Current flowing through a unit area held normal to the direction of current. Ae Current density is given by L ‘AcosO Where A ~area of cross section, 8 - angle between direction current and area. If the area is normal to the current flow, 60, thus 7 Unit - A/m? and dimensions are [AL”] Current density is a vector quantity. JA Mechanism of current flow in conductors ‘Metals have large number of free electrons nearly 10*electrons /cm?, © Inthe absence of an electric field electrons are in random motion due to thermal energy. ‘* The average thermal velocity of electrons is zero. © Inthe presence of an external electric field, electrons are accelerated and acquire an average velocity. ‘* During the random motion, electrons collide with each other or with positive metal ions. Drift velocity ‘© Average velocity acquired by an electron in the presence of an electric field. Relaxation time ‘© Average time interval between two successive collisions. Path of an electron Relation connecting drift velocity and relaxation Time ‘+ The force experienced by the electron in an electric field is F =e , where €~ electric field ‘* From Newton's second law F = ma, a acceleration, m- mass Thus, ma=—eE Therefore acceleration of electron is E + Ifanelectron accelerates, the velocity attained is given by ¥) =u Faz, ur initial velocity, t- time © Similarly v= tar, vy=u; ary Vy Sly Haty ‘+ Thus the average velocity (drift velocity) is given byO+ar=ar ‘© Therefore the drift velocity is given by eEt m © Where , r= relaxation time Relation connecting drift velocity and current The number of electrons in the length | of the conductor = nAl © Where n- electron density (number of electrons per unit volume), A area of cross section. Thus total charge q=nAle, e— charge of electron © The electron which enter the conductor at the right end will pass through the conductor at left end in time t=, vedate velocity of electrons q_nAle ty aw * Thus the current, = nAve © Thatis 1 =nAve electron density, A area, v- drift velocity, e- electron charge 1 _ nave + Te current density J = 4 = 40 Mobi + Ratio of magnitude of drift velocity to the electric field eke mo a + Stunit of mobility is CmN35* ifference between emf and potential difference emf Potential Difference ‘The difference in The difference in Potential between _| potential between the terminals ofa _| the terminals of a cell cell, when no current | or between any two is drawn from it, points in a circuit when current is drawn from the cell. Exists only between _ | Exists throughout the the terminals of the _| circuit. cell. It is the cause Itis the after effect. ‘Always greater than potential difference ‘Always less than emf Taw, ‘Ohm's law ‘* Atconstant temperature the current flowing through a conductor is directly Proportional to potential difference between the ends of the conductor. © Thus V=IR, V- potential difference, I~ current, R- resistance ce 'y of conductor to oppose electric ea au i © Slunit-ohm (Q) Factors affecting resistance of a conductor ‘© Nature of material '* Proportional to length of the conductor ‘© Inversely proportional to area of cross section. '* Proportional to temperature Relation connecting resistance and resistivity pl A Where p- resistivity, A—area, | length Resistivity (specific resistance) ‘* Resistivity of the material of a conductor is defined as the resistance of the conductor having unit length and unit area of cross section. RA pat‘© Unit-ohm meter (Am) ‘© Resistivity of conductor depends on nature of material and Temperature Conductance (6) ‘© Reciprocal of resistance o-i R © Unit- 0, or mho or siemens (S) Conductivity (0) ‘© Reciprocal of resistivity 1 o e © Unit- m+, or mho m4, or Sm Ohmic conductor ‘* Aconductor which obeys ohm’s law. © Eg: metals \V-1 graph of an ohmic conductor Non ohmic conductors ‘© Conductor which does not obey ohm’s law. © Eg = diode, transistors, electrolytes etc. V-1 graph of a non- ohmic conductor (Diode! Circuit diagram for the experimental study of ‘ohms law $i] wg LS 10° Qm Relation connecting resistivity and relaxation time Substituting vin = pate ml + Thatis Y=© From ohm’s law R + Therefore R=—™ nAe’r *+ Comparing withthe equation x = A orn ee aes i used as fr making connecting wie Copper has low resistivity. Nichrome is used as heating element of electrical devices «Nichrome has High sitivity High meting pine. Why material ike constantan and manga are used to make standard resistances? Resistance doesnot change with temperature Material has high resistivity Resistors © The resistor is a passive electrical ‘component to create resistance in the, flow of electric current. Symbol Constant resisstance Variable resistance wn whe Commercial resistors Wire bound resistors ‘* Made by winding the wires of an alloy, like, manganin , constantan, nichrome or similar ones, around a ceramic, plastic, or fiberglass core. * They are relatively insensitive to temperature. «Large length is required to make high resistance. Carbon resistors ‘¢ Made from a mixture of carbon black, clay and resin binder. ‘+ Are enclosed in a ceramic or plastic jacket. © Carbon resistors are small in size, and inexpensive. Colour code of resistors Colour | Digit | Multiplier | Tolerance Black 0 10° Brown 1 10 Red 2 10 Orange 3 10° Yellow 4 10" Green 5 10° Blue 6 10° Violet 7 107 ‘Grey 8 10" ‘White 9 10° Gold 10" 35% Silver 107 210% No color 220% eizay 6 B B ROY of Great Britain has a Very Good Wife 4- Band code Resistance = (47 x 10) £ 5% 5-Band code ‘olor [Digit [Maitipir Tolerance (] Bax [0 [17 Bom | 1] 10 [7 [Rea [2 [107 2 range] 3 | 10" Yetow | 4_| 104 ween [5 | 10] 05 ‘ue [6 | ot | 0s wot [7 | 107 [08 Gey |e | 10h [wie [9 [10 (Gold io | 5 ‘Siver 107 | 10 ‘none) 2 © The colors brown, red, green, blue, and violet are used as tolerance codes on S- band resistors only.© All S-band resistors use a colored tolerance band. ‘ati Dit mer Tern a ‘Temperature dependence of resistivity ‘* When temperature is increased, average speed of the electrons increases and hence number of collision increases. ‘Thus the average time of collisions t, decreases with temperature. Metals (Conductors) ‘+ Inametal number of free electrons per unit volume does not depend on temperature. ‘© When temperature is increased, relaxation time decreases and hence resistivity increases. ‘© The temperature dependence of resistivity of a metallic conductor is given by br =p[l+ac-T,)] © Where pr— resistivity at a temperature T, po resistivity at a lower temperature To, a -temperature coefficient of resistivity. © We have ‘© Thus for conductors resistivity increases with temperature, ‘Temperature coefficient of resistivity (a) (er=Po) PT -T) © Unit of ais °C". ‘For metals ais positive. ‘Temperature ~ resistivity graph (copper Temperature 70 + Alloys Page 5 of 9 ‘© Resistivity of alloys like, Nichrome, Manganin, Constantan is almost independent of temperature. ‘+ Thus alloys are used to make wire bound resistors. Temperature — resistivity graph (Nichrome] $1001. 300 400 600 800 “Temperature TUK) —* ‘Semiconductors ‘© The electron density (n) increases with temperature. ‘© Thus resistivity decreases with temperature. © ais negative. ‘Temperature resistivity graph o| I Insulators ‘© The electron density (n) increases with temperature. ‘© Resistivity decreases with temperature. ‘Combination of resistors Resistors in Series wonton tw ang q a Rs |, 7 1 s v ‘© Inseries connection same current pass through all resistors. © The potential drop is different for each resistor. ‘© The applied potential is given byPage 6 of 9 V=V,+V,4¥, Where Vi, Voand V3 are the potential drop across resistors Ri, Ro and Rerespectively. {fall the resistors are replaced with a single effective resistance Rs, we get V=IR, Thus IR, = IR, +IR, + IR, ‘Therefore the effective resistance is R=R +R AR, For n resistors Re ER AR ARF oR, ‘Thus effective resistance increases in series combination, Resistors in parallel In parallel connection current is different through each resistors. The potential drop is same for all resistors. The total current =hthth fall resistors are replaced with an effective resistor of resistance Rr, we get Thus Therefore the effective resistance in parallel combination is ipa 1 ee R, Ry Rs Ry For n resistors in parallel roijigd Sto tot Ry Ry RRs For two resistors __RR, rR ‘Thus effective resistance decreases in parallel combination. Internal resistance of a cell (r) Resistance offered by the electrolytes and electrodes of a cell Factors affecting internal resistance Nature of electrolytes Directly proportional to the distance between electrodes Directly proportional to the concentration of electrolytes. Inversely proportional to the area of the electrodes. Inversely proportional to the temperature of electrolyte. Relation connecting emf and internal resistance R Rer € Effective resistance Thus the current is / Rer Where s-emf, R-external resistance, r- internal resistance. Thatis (R+N=e>IR+h From ohm’s law, V=IR, therefore e-V 1 The potential is given by v Combination of cells Cells in series In series connection current is same; the potential difference across the cells is different. The potential difference across the first cellis Vag = 6) - Mh Similarly Vgc =€,— Ir, Thus total potential across AC is Vac =Van +Vac Thatis Vic =6,- I +e,— IPage 7 of 9 Vgc = (6 +6 )- 10H, +) ‘Ifthe two cells are replaced with a single cell of emf ¢,, and internal resistance rea, wehave, Vac = ~My ‘* Comparing the equations we get eq = (6 + &) rg = (+n) + Forncellsin series B= 8464+ fag = HAH tet hy ® If the negatives of the cells are connected together y= h (>) Cells in parallel 4 Bore 4 ‘¢ In parallel connection current is different and potential is same. © Forthe first cell, V=6,— 1h av 7 ‘© Similarly for the second cell, V -v Thus 4 ‘© The total current is given by ath (e)-4) ‘© Comparing this with the equation ‘If the negative terminal of the second is connected to positive terminal of the first, the equations are valid with (¢ >—#;) Joule’s law of heating ‘© The heat energy dissipated in a current flowing conductor is given by H=PR © [-current, R-resistance, t-time Electric power © Itis the energy dissipated per unit time. © Power, © Also P= © Slunitis watt (W) © Lkilo watt (1kW) = 10000 © -Imega watt (MW) = 10°W © Another unit horse power (hp) © Lhp=746W Electrical energy ‘Electrical energy = electrical power X time * Slunit joule UW) © Commercial unit - kilowatt hour (kWh) © 1kWh=3.6% 10°). Efficiency «The efficiency of an electrical device is ‘output power input power Kirchhoff’s rulePage 8 of 9 First rule (junction rule or current rule] ‘© Algebraic sum of the current meeting at junction is zero. ‘+ Thus,, Current entering a junction = current leaving the junction Lthth-L-b=0 Sign convention ‘© Current entering the junction — positive Current leaving the junction - negative Second rule (loop rule or voltage rule! ‘© Algebraic sum of the products of the current and resistance in a closed circuit is equal to the net emf in it. © This rule isa statement of law of conservation of energy. Sign convention ‘© Current in the direction of loop - positive ‘© Current opposite to loop - negative eee — —_ J —ercisve LR,- LR, Loop CDEFC + th) R = b Wheatstone’ s bridge ® vis Viet | I]t If galvanometer current is zero, P_® os Derivation of balancing condition ‘© Applying voltage rule to the loop ABDA IP+1,G-LR=0 ‘+ For the loop BCDB (1)-1,)0-(t.+1,)S-1,6= When the bridge is balanced ly Thus ,P—1,R=0 and 1Q-1,8=0 © Or, LP=1R and 10=1,8 © Thus PLR os ‘© This is the balancing condition of a Wheatstone bridge. ‘* Works on Wheatstone’s principle. '* Used to find resistance of a wire. Circuit diagram e555 ‘© Where k~ key, X— unknown resistance, R- known resistance, HR- high resistance, G-Galvanometer, J - Jockey Equation to find unknown resistance ‘+ From wheatstone’s principle P_R Qs ‘© Here P—unknown resistance , Q- known resistance, R- resistance of the wire ofPage 9 of 9 length | , S- resistance of wire of length (200-1) ‘The length I for which galvanometer shows zero deflection — balancing length. © Thus Xo R (00-Dr + Where r~ resistance per unit length of the meterbridge wire. # Therefore the unknown resistance is given by RI 00-1) ‘+ The resistivity of the resistance wire can be calculated using the formula Where r—radius of the wire, |-length of the wire, meter © Adevice used to measure an unknown emf or potential difference accurately. Principle ‘* When a steady current (\) flows through a wire of uniform area of cross section, the potential difference between any two points of the wire is directly proportional to the length of the wire between the two points, From ohm's law , V =/R Ip © thatis, Y= © Therefore, Val or V =Ad Thus £ = k, where k— constant — potential gradient. Uses of potentiometer ‘* To compare the emf of two cells, ‘© To find the internal resistance of a cell Comparison of emfs Circuit diagram : Tee SS Shia ee i weaee es *hebalancing length with cell E ‘© le-balancing length with cell E2 ‘+ Toget the balancing length E:>E To find internal resistance ‘© when the key Ks is open ex, * when the key kz is closed Vol, © But we have v 1r— internal resistance c Therefore & UR+r) _(R+r) IR R © Thus (R+r) L «The interna esitance even by Rh) 1 ‘+ Where hi balancing length, key Kiopen, ly- balancing length, key Ki closed. Why potentiometer is preferred over voltmeter for measuring emf of a cell? ‘+ Inpotentiometer null method is used, so no energy loss in measurement.