Electric charges in motion constitute an electric current. Any medium having pra ctically free electric charges,freeto migrate is a conductor of electricity. The electric charge flowsfromhigher potential energy state to lower potential energ y state. Positive charge flows from higher to lower potential and negative charg e flowsfromlower to higher. Metals such as gold, silver, copper, aluminium etc. are good conductors. In absence of potential difference across a conductor no ne t current flows through a corss section. When a potential difference is applied across a conductor the charge carriers (electrons in case of metallic conductors ) flow in a definite direction which constitutes a net current in it. These elec trons are not accelerated by electric field in the conductor produced by potenti al difference across the conductor. They move with a constant drift velocity. Th e direction of current is along the flow ofpositive charge (or opposite to flow of negative charge), i = nv eA where V = drift velocity.
d d 2.
ELECTRIC CURRENT I N A CONDUCTOR : 3.
The strength ofthe current i is the rate at which the electric charges are flowi ng. If a charge Q coulomb passes through a given cross section ofthe conductor i n t second the current I through the conductor is , Q gtven by I = — = Coulomb =Q ampere . — t second t dq Ampere is the unit of current. If i is not constant then / = — , where dq is net charge transported at . . . dt a section.in time dt. In a current carrying conductor we can define a vector which gives the direction as c urrent per unit normal, cross sectional area. Thus J = ^ n or I = J • S Where n is the unit vector in the direction of theflowof current.
T
CHARGE A N D CURRENT :
> > For random J or S, we use 1= - J • -ds f
4.
In conductors drift vol. of electrons is proportional to the electric field in s ide the conductor as- v = pE where p is the mobility of electrons current densit y is given as J = — = ne v = ne(pE) = aE
d
RELATION IN J , E AND V D : d
where a = neu is called conductivity of material and we can also write p = — -> re sistivity a of material. Thus E = p J. It is called as differential form of Ohm' s Law.
5.
Dry cells, secondary cells, generator and thermo couple are the devices used for producing potential difference in an electric circuit. The potential difference between the two terminals ofa source when no energy is drawn from it is called
the " Electromotive force" or " EMF " ofthe source. The unit of potential differ ence is volt. 1 volt = 1 Amphere x 1 Ohm.
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SOURCES O F POTENTIAL DIFFERENCE & ELECTROMOTIVE FORCE : Current Electricity
6.
ELECTRICAL RESISTANCE :
The property of a substance which opposes theflowof electric current through it is termed as electrical resistance. Electrical resistance depends on the size, g eometery, temperature and internal structure ofthe conductor.
LAW O F RESISTANCE : 7.
The resistance R offered by a conductor depends on the following factors : R a y (cross section area of the conductor)
R a L (length of the conductor) ;
at a given temperature R= P ~ . Where p is the resistivity ofthe material of the conductor at the given temperature. It is also known as specific resistance of the material.
8. [
The resistance ofmost conductors and all pure metals increases with temperature, but there are a few in which resistance decreases with temperature. If R & Rbe the resistance of a conductor at 0° C and 6° C, then it is found that R = R (1 +aG). c 0
DEPENDENCE O F RESISTANCE O N TEMPERATURE :
Here we assume that the dimensions ofresistance does not change with temperature if expansion coefficient ofmaterial is considerable. Then instead of resistance we use same property for resistivity as p = p (1 + a0) The materials for which resistance decreases with temperature, the temperature coefficient of resistance is negative.
0
Where a is called the temperature co-efficient of resistance. The unit of a is K " of °C reciprocal of resistivity is called conductivity and reciprocal ofresistan ce is called conductance (G). S.I. unit of G is ohm.
1 _1 9.
Ohm's law is the most fundamental of all the laws in electricity. It says that t he current through the cross section or the conductor is proportional to the app lied potential difference under the given physical condition. V = R I . Ohm's la w is applicable to only metalic conductors. I - Law (Junction law or Nodal Analy sis) :This law is based on law of conservation of charge. It states that" The al gebric sum of the currents meeting at a point is zero" or total currents enterin g a junction equals total current leaving the junction. I I = I I . It is also k nown as KCL (Kirchhoffs current law).
in out
O H M ' S LAW : 10.
EL - Law(Loop analysis) :The algebric sum ofall the voltages in closed - v, circ uit is zero. I I I R + 2 EMF = 0 in a closed loop. The closed loop can be traver sed in any direction . While traversing a loop if higher potential point is > en tered, put a + ve sign in expression or if lower potential point is i + 4 entere d put a negative sign. -Vj -V +V -V = 0. Boxes may contain resistor or batteiy o r any other element (linear or non-linear). It is also known as KVL (Kirchhoffs voltage law). + e V 2 3 4 il.Bansal Classes Current Electricity [5]
11.
A number of resistances can be connected and all the v. V, V„ complecated combinat ions can be reduced to two different types, namely series and parallel. V (i) RE SISTANCE IN SERIES : When the resistances are connected end toend then they are said to be in series, The current through each resistor is same. The effective r esistance appearing across the batter}', R = RJ + R J + R + + R and
r/WV\—fyWv—A-WV3 N COMBINATION O F RESISTANCES : ••-VWV-H + Rn V = VJ + V 2 + V 3 +
The voltage across a resistor is proportional to the resistance R„ i V V;V = R,+R„+. .+R R,+R-+. +R_
R 2 +V„. (ii)
Aparallel circuit of resistors is one in which the same voltage is applied acros s all the components in a parallel grouping of resistors R R,, R3, , R,,.
1;
RESISTANCE IN PARALLEL : CONCLUSIONS :
(a) (c)
Potential difference across each resistor is same. I = Ij + I + I + I 1 Effectiv e resistance (R) then ±-J_ ^ Current in different resistors is inversally proporti onal to the resistance. ,,.111; I,:l : R_ Rj R , R
2 3 2 3 (b) (d) 1 R.n A -WW-iR -WW 12.
I, etc, I,l G,+G~+. + _ G G . + G2 + . . . + G _ 1 n I where G - — = C onductance ofa resistor. R Ij =
1 2 0 13.
If a cell of emf E an d internal resistance r be connected with a resistance R t he total resistance of the circuit is (R+r). £,r E,RE,R E,? upton I = — AB ^ 7 R+r ; E = Terminal voltage of the batten .If r 0, cell is Ideal & V -> E. AVvV
V = WHERE
E M F O F A CELL & ITS INTERNAL RESISTANCE : 7
GROUPING O F CELLS : (i) If n r « R t h e n I
Let there be n cells each ofemf E, arranged in series,Let r be the internal resi stance of each cell, nE The total emf = n E. Current in the circuit I R+nr
nE R
CELLS IN SERIES : If nr » K then I E
» Series combination should be used. Series combination should not be used Current Electricity
il.Bansal Classes [5]
(ii)
C E L L S I N PARALLEL :
If m ceils each of emf E & internal resistance r be connected in parallel and if this combination be connected to an external resistance then the emf ofthe circ
uit=E. Internal resistance ofthe circuit = m -^1—wU— mE 1= R+— mR+r m R •m— mE Parallel co mbination should be used. If m R « r ; 1 = If m R » r : 1 = R -» Parallel combination should not be used.
upto (iii)
mn=number ofidentical cells. n=number of rows m=number of cells in each rows. Th e combination ofcells is equivalent to single cell of: mr (a) emf = mE & (b) int ernal resistance = n For maximum current N = mr or R Current I = mE
R+mr n
C E L L S LN M U L T I P L E A R C : 12 3 m HHH>m R
HHH»
mr R= — = internal resistance of battery. T _ nE_mE ~ 2r~2R ' m a x
W H E A T STONE N E T W O R K :
When current through the galvanometer is zero (null point or balance point) — = — . When PS > QR; V < V & PS <QR; V > V or Q S PS = QR => products of opposite arms are equal. Potential difference between C & D at null point is zero. The null po int is not affected by resistance of G & E. It is not affected even ifthe positi ons of G & E are inter changed. I a (QR-PS).
c D c D C D 14.
A potentiometer is a linear conductor ofuniform cross-section with a steady curr ent set up in it. This maintains a uniform potential gradient along the length o fthe wire. Any potential difference which is less then the potential difference maintained across the potentiometer wire can be measured using this. The • • i Ii po tentiometer equation is — =— . 2 I2 E L E POTENTIOMETER : il.Bansal Classes Current Electricity [5]
15.
AMMETER :
It is a modified form of suspended coil galvanometer it is used to measure curre nt . A shunt (small resistance) is connected in parallel with I-Rgalvanometer to convert into ammeter. S = ; An ideal ammeter has zero resistance. where I = Max imum current that canflowthrough the galvanometer. I = Maximum current that can be measured using the given ammeter.
g i
J« -vwv g
16.
A high resistance is put in series with galvanometer. It is used to measure pote ntial difference. V I R I = — ^ g —WW— R„+R " * + v R-»oo , Ideal voltmeter. 8 s 8 0
VOLTMETER : 17.
While solving an electric circuit it is convinient to chose a reference point an d assigning its voltage as zero. Then all other potential are measured with resp ect to this point, This point is also called the common point. The energy libera ted per second in a device is called its power. The electrical power P delivered by an electrical device is given by P = VI , where V=potential difference acros s device & I = current. Ifthe current enters the higher potential point ofthe de vice then power is consumed by it (i.e. acts as load). If the current enters the lower potential point then the device supplies power (i.e. acts as source). V P ower consumed by a resistor P = I R = VI = — .
2 2
RELATIVE POTENTIAL : 18.
ELECTRICAL POWER : 19.
When a current is passed through a resistor energy is wested in over coming the resistances ofthe wire . This energy is converted into heat. V W = Vlt Joule; = I Rt Joule ;= — t Joule. R
2 2
HEATING EFFECT O F ELECTRIC CURRENT : 20.
The heat generated (in joules) when a current ofI ampere flows through a resista nce of R ohm for T second is given by: I H = I RT Joules ; = —RT Calories. 4.2 If current is variable passing through the conductor then we use for heat produced in resistance in time t 0 tot is: =jl Rdt
JOULES LAW O F ELECTRICAL HEATING : 21.
UNIT O F ELECTRICAL ENERGY CONSUMPTION :
1 unit of electrical energy = Kilowatt hour = 1 KWh = 3.6 x 10 Joules. 6
il.Bansal Classes Current Electricity [5]
EXERCISE # I
Q. 1 Anetwork ofnine conductors connects six points A B, C, D, E and F as shown infigure.Thefiguredenotes resistances in ohms. Find the equivalent resistance be tween A and D. Q.2
1
In the circuit shown infigurepotential difference between point A and B is 16 V. Find the current passing through 2Q resistance. " a 2n Find the current I & vol tage V in the circuit shown.
AO 4fi 9V i n 3V 4n VW-r-4 I I—I I W OB so "1 T 20V la ,60V Q. 3 Q. 4 Q.5 Q.6 Q. 7 0.443 - 4Q3 2Q<
Find the equivalent resistance of the circuit between points A and B shown in fi gure is: (each branch is ofresistance = 10) ^ |10V |SV J 2 0 V J30V Find the cur rent through 25V cell & power supplied by T ~r 20V cell in thefigureshown. 9s If Ss
t 25V
Ifa cell of constant E.M.F. produces the same amount ofthe heat during the same time in two independent resistors R and R^,, when they are separately connected across the terminals of the cell, one after the another,findthe internal resista nce ofthe cell. Find the effective resistance ofthe network (seefigure)between t he points A and B. Where R is the resistance of each part.
R
Q.8 Q. 9
In the circuit shown infigure,all wires have equal resistance r. Find the equiva lent resistance between A and B. Find the resistor in which maximum heat will be produced.
Q. 10 For what value of Rin circuit, current through 4f2 resistance is zero. Q.l l In the circuit shown infigurethe reading of ammeter is the same with both swit ches open as with both closed. Thenfindthe resistance R. (ammeter is ideal) 4y loon _ —wwh—®—f, . — w wJWt w 1 ( ison [5] il.Bansal Classes
Current Electricity W^tlv
Q.12 Ifthe switches S , S and S in thefigureare arranged such that current throu gh the battery is minimum,findthe voltage across points A and B.
t 2 3 >J 6D -r-Vv 24V 6n - 9fJ w h 3£!
Q.13 Thefigureshows a network ofresistor each heaving value 12H. Find the equiva lent resistance between points Aand B.
Q.14 A battery of emfs = 10 Vis connected across a i m long uniform wire having resistance 1 OQ/m. Two cells ofemfgj = 2V and e = 4V having internal resistances 1Q and 5Q respectively are connected as shown in thefigure.If a galvanometer sh ows no deflection at the point P,findthe distance ofpoint P from the point a. 0 2
Q.15 A potentiometer wire AB is 100 cm long and has a total resistance of lOohm. If the galvanometer shows zero deflection at the position C, thenfindthe value ofunknown resistance R. Q.16 In thefigureshown for gives values ofRj and fL the balance point for Jockey is at 40 cmfromA When R, is shunted by a resistance of 10 O, balance shifts to 50 cm.findR, and R,. (AB = lm):
-w R 3 -W2 R
Q.17 A part of a circuit is shown in figure. Here reading of ammeter is 5 R -A/W WWV ampere and voltmeter is 96V & voltmeter resistance is 480 ohm. Then find the resistance R Q.18 An accumulator of emf 2 Volt and negligible internal resistan ce is connected across a uniform wire of length 10m and resistance 30Q. The appr opriate terminals ofa cell of emf 1.5 Volt and internal resistance 10 is connect ed to one end ofthe wire, and the other terminal ofthe cell is connected through a sensitive galvanometer to a slider on the wire. What length ofthe wire will b e required to produce zero deflection of the galvanometer ? How will the balanci ng change (a) when a coil ofresistance 5fi is placed in series with the accumula tor, (b) the cell of 1.5 volt is shunted with 5Q resistor ? Q.19 The resistance ofthe galvanometer G in the circuit is 25f2. The meter deflects Ri R-, full scal e for a current of 10 mA. The meter behaves as an ammeter of -v-AVrvWv- 'vVvVthr ee different ranges. The range is 0-10 A ifthe terminals O and P are taken; rang e is 0 - 1 A between O and Q; range is 0 - 0.1A between O 10A 1A 0.1 A R and R. Calculate the resistance Rj, R2 and R . List of recommended questions from I.E. Irodov, 3,147, 3.149, 3.150,3.154,3.155,3.169, 3.175, 3.176, 3.179,3.186, 3.189, 3.190, 3.194,3.196, 3.207
3
Current Electricity [5]
Q. 1 Atriangle is constructed using the wires AB, BC & CAof same material and of resistance a, 2a & 3a respectively. Another wire of resistance a/3 from A can m ake a sliding contact with wire BC. Find the maximum resistance ofthe network be tween points A and the point of sliding wire with BC. Q.2(a) The current density across a cylindrical conductor of radius R varies according to the equation , w here r is the distancefromthe axis. Thus the current density is a maximum J at t he axis r = 0 and decreases linearly to zero at the surface r = R. Calculate the current in terms of J and the conductor's cross sectional areaisA=7iR Suppose t hat instead the current density is a maximum J at the surface and decreases line arly to zero at the axis so that J = J —. Calculate the current.
0 0 2 0 0 EXERCISE # II (b) Q.3 Q4
What will be the change in the resistance of a circuit consisting of five identi cal conductors iftwo similar conductors are added as shown by the dashed line in figure. The current I through a rod of a certain metallic oxide is given by 1 = 0.2 V , where V is the potential difference across it. The rod is connected in series with a resistance to a 6V battery ofnegligible internal resistance. What value should the series resistance have so that: the current in the circuit is 0 .44 the power dissipated in the rod is twice that dissipated in the resistance. 5/2
© 00 Q.5 Q.6
(I) 01) Q.7 Q. 8
Apiece ofresistive wire is made up into two squares with a common side of length 10 cm. A currant enters the rectangular system at one ofthe corners and leaves at the diagonally opposite corners. Show that the current in the common side is l/5th of the entering current. What length of wire connected between input and o utput terminals wouid have an equivalent effect. A network of resistance is cons tructed with R, & R^ as shown inthefigure.The potential at the points 1,2,3,.., N are Vj, V , V ,.., V respectively each having a potential k tune smaller than previous one Find: Rj R p and p in terms of k current that passes through the re sistance R2 nearest to the V in terms V , k &R .
2 3 R 2 0 0
A hemisphere network ofradius a is made by using a conducting wire of resistance per unit length r. Find the equivalent resistance across OP. Three equal resist ance each of R ohm are connected as shown infigure.A battery of2 volts of intern al resistance 0.1 ohm is connected across the circuit. Calculate Rfor which the heat generated in the circuit is maximum.
c 3 r XL. R / 2V
il.Bansal Classes Current Electricity [5]
Q.9
A person decides to use his bath tub water to generate electric power to run a 4 0 watt bulb. The bath tube is located at a height of 10 m from the ground & it h olds 200 litres ofwater. If we install a water driven wheel generator on the gro und, at what rate should the water drain from the bath tube to light bulb? How l ong can we keep the bulb on, ifthe bath tub was full initially. The efficiency o f generator is 90%. (g= lOm/s" )
2 |36V Q.10 C O m
en: In the circuit shown infigure,calculate the following: Potential difference between points a and b when switch S is open. Current through S in the circuit w hen S is closed. 3Q-"
•6Q
Q.ll The circuit shown infigureis made of a homogeneous wire ofuniform cross-sec tion. ABCD is a square. Find the ratio ofthe amounts of heat liberated per unit time in wire A-B and C-D.
T
Q.12 Arod oflength L and cross-section area Alies along the x-axis between x = 0 and x = L. The material obeys Ohm's law and its resistivity varies along the ro d according to p (x) = p e . The end ofthe rod at x = 0 is at a potential V and it is zero at x = L. (a) Find the total resistance of the rod and the current in the wire. (b) Find the electric potential in the rod as a function ofx.
0 _xL 0
Q.13 In the figure. PQ is a wire of uniform cross-section and of resistance Rq. Ais an ideal ammeter and the cells are ofnegligible resistance. Thejockey J canf reelyslide over the wire PQ making contact on it at S. If the length ofthe wire PS is f= l/n* ofPQ, find the reading on the ammeter. Find the value of'f for max imum and minimum reading on the ammeter. Q.14 An ideal cell having a steady emfo f2 volt is connected across the potentiometer wire oflength 10 m. The potentiome ter wire is ofmagnesium and having resistance of 11.5 Q/m. An another cell gives a null point at 6.9 m. Ifa resistance of 5£2 is put in series with potentiometer wire,findthe new position ofthe null point. Q.15 Find the equivalent resistance of the following group of resistances between A and B. Each resistance of the ci rcuit is R (a) -w-*A v Vr—, v»—— -oB -Vyx 2
Q.16 An enquiring physics student connects a cell to a circuit and measures the current drawn from the cell to Ij. When he joins a second identical cell is seri es with the first, the current becomes I . When the cells are connected are in p arallel, the current through the circuit is I,. Show that relation between the c urrent is 31 1 = 2 I (I +1 ) iv iv iv iv
3 2 t 2 3 n
Q.17 Find the potential difference V - V for the circuit shown in the figure. A B
il.Bansal Classes Current Electricity
Q.18 A resistance R of thermal coefficient of resistivity = a is connected in pa rallel with a resistance = 3R, having thermal coefficient of resistivity = 2a. F ind the value of a . 40 -AV-2Q. w 2/3 f2 -W- 4nw Q.19 Find the current through — O resistance in thefigureshown. 2Q
eff I
Q.20 A galvanometer having 50 divisions provided with a variable shunt s is used to measure the current when connected in series with a resistance of 90 Q and a battery of internal resistance 10 Q. It is observed that when the shunt resista nce are 10Q, 500, respectively the deflection are respectively 9 & 30 divisions. What is the resistance ofthe galvanometer? Further ifthe full scale deflection ofthe galvanometer movement is 300 mA find the emf ofthe cell. Q.21 In the prima ry circuit of potentiometer the rheostat can be variedfrom0 to 100. Initially it is at minimum iov ion resistance (zero). ^-HpvWv vwv (a) Find the length AP oft he wire such that the galvanometer shows zero 9n deflection. 12m (b) Now the rhe ostat is put at maximum resistance (100) and the switch S is closed. New balanci ng length is found to 8m. Find the internal resistance r 4.5V ofthe 4.5 V cell. 2n Q.22 A galvanometer (coil resistance 99 D) is converted into a ammeter using a shunt of 1Q and connected as shown in thefigure(i). The ammeter reads 3 A The same galvanometer is converted into a voltmeter by connecting a resistance of 10 1 O in series. This voltmeter is connected as shown infigure(ii).Its reading is
found to be 4/5 of the full scale reading. Find 12V r 12 V r |H' VWv—| H'—VWV—I intern al resistance r ofthe cell (a) 2n (b) range ofthe ammeter and voltmeter -AAAA —W/v I full scale deflection current ofthe galvanometer 2n (c) (ii) G)
1£1 10 V V. il.Bansal Classes Current Electricity [5]
EXERCISE # III
Q. 1 An electrical circuit is shown in the figure. Calculate the potential diffe