ELECTRIC CIRCUITS. The time rate of flow of charge through any cross-section is called current. i.e.,

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1 Lesson- LTI IUITS urrent The time rate of flow of charge through any cross-section is called current. i.e., I avg = q t I = q lim t t 0 dq dt Unit of current is ampere There are two types of current If magnitude and direction of current does not vary with time, it is said to be direct current. If current is periodic (with constant magnitude) with half cycle positive and half negative, it is said to be alternating current. urrent density urrent at any cross section is defined as i = J d S wher J is current density and d S is cross-section area. or J lim S I n S i.e., J d I d S n where n is a unit vector in the direction of flow of current. If cross-section is not perpendicular to the current, the cross-section area normal to current in accordance to given figure will be ds = ds 0 cos. J = lso, J d I d S cos 0 I ds ds 0 J where is conductivity of the material and is electric field across the ends of the conductor. Drift velocity It is defined as average velocity with which charge (free electron for conductor) flows when potential difference is applied across conductor. v d = e m Where is time constant, is electric field across conductor and m is mass of electron.

2 Drift velocity and current are related as I = nev d where n = no. of free electron per unit volume = cross section area of conductor v d = drift velocity of electrons esistivity () It is given by = m ne unit of resistivity is -m for conductor increases with temperature for semiconductor and insulator decreases with temperature Mobility of electron (µ e ) = drift velocity electric field vd, where v d = e/m, = relaxation time. onductivity Inverse of resistivity is defined as conductivity of material. lectric ircuit n electric circuit consists of active and passive elements. n active element like a cell, battery or power supply provides current and electrical energy to an electric circuit. The passive elements like resistor (), capacitor () and inductor (L) consume or store the electrical energy. resistor opposes flow of current. capacitor offers a low resistance to flow of alternating current and does not allow direct current to pass through it at steady state. nergy stored in a capacitor, U = V =. n inductor L opposes the variations of current and it does not oppose steady or direct current. The energy stored in an inductor, U = LI. Ohm s Law Ohm s law states that the ratio of potential difference V maintained across a conductor to the current flowing through the conductor is a constant which is equal to the resistance of the conductor provided the physical conditions remain unchanged. If V is the potential difference in volts and current I is in amperes, the electrical resistance is measured in ohms in the S.I. units. Q i.e., V I esistance esistance of material is given by l

3 Where l is the length and is the area of cross-section of the conducting body of resistance. The resistance of most of the conductors increases with increase in temperature whereas the resistance of few of the materials decreases with temperature. In the case of some of the conductors, the resistivity changes linearly with the increase of temperature. If 0 is the resistivity at lower temperature T 0 and is the resistivity at higher temperature T, then = 0 [ + (T T 0 )] where is called the temperature coefficient of resistivity. For semi-conductors and insulators resistance decreases with increase in tempearture. esistances in series If a number of resistances,, 3,..., n are connected in series to a battery of e.m.f., the same current will pass through all the resistances. The equivalent or effective resistance series for such resistance is equal to the sum of all resistances. series = n v = v = v n esistances in parallel 3 I v v v n I parallel combination of resistances can be made by connecting one end of all resistances to the positive terminal of the battery and the second end of all resistances is connected to the negative terminal of the battery. The potential difference across all the resistances is the same whereas the current I in the main circuit divides itself across the respective resistances,, 3,..., n as I, I, I 3,..., I n so that I = I + I + I I n The equivalent or effective resistance in parallel is given by eq lectromotive Force and Potential Difference 3 If a battery of e.m.f. volts and internal resistance r ohm is connected to an external resistance ohm, the current I flows throughout the circuit. = I + Ir n I I The potential difference across the external resistance is less than the e.m.f. of the cell. This potential difference across when the current is flowing through it is called as the closed circuit voltage. It is less than the e.m.f. of the cell when no current flows through the battery. Terminal potential difference across = I = Ir. Maximum power transfer If a cell of e.m.f. and internal resistance r is connected to a load resistance, current I r I I 3 3, r I

4 lectric Power = P = I = ( r), r If maximum power is transferred, dp d = 0 or 0 d ( ) d ( ) ( ) 3 r r r or = r d p For = r, d < 0 P is maximum i.e., The power transferred is maximum when load resistance is equal to internal resistance of a cell. Grouping of ells Series ombination If n cells are connected in series, as shown in figure, then they can be replaced by a single cell of emf and resistance r, given by I r r r = n and r = r + r + + r n Parallel ombination If m cells are connected in parallel as shown in figure, then their equivalent cell of emf and internal resistance r is given by = r r r r r r m m m r r 3 and r r r r m r 3 Series and Parallel Grouping of ells Suppose there are N identical cells each of emf and internal resistance r. nd, they are grouped together to form an arrangement as shown in figure. In each branch there are n cells in series and there are m such branches in parallel. The group of cells is joined across an external resistor. Since the total number of cells N, therefore n N = mn The equivalent cell of the group has the emf 0 = n and internal resistance r 0 = m n r Thus, the total current flowing through the external resistor is I = r n nr m 0 or I = mn nr m

5 ondition for Maximum urrent through the xternal esistance. For maximum current through, we must have nr = m or m n r The maximum current is given by I max = n m r mn r Wheatstone s bridge If a net work of four resistances,, 3 and 4 are connected along arms,, D and D and ends and are connected across the terminals of a battery so that the galvonometer connected across and D does not show any deflection i.e. potential at point and D is equal. Such an arrangement is called a balanced Wheatstone s ridge for which G current I, flows across and I flows across D. I = I 3 or I = I 4 3 or 4 Kirchhoff s Law 3 4 I If an electrical circuit has more than one path of electrical closed circuits, it is called a network. The electric currents in different portions of such electric circuits can be found by the application of the following Kirchhoff s Laws : Kirchhoff s First Law of urrent : The algebraic sum of currents at any junction in a circuit is zero. It implies that the sum of the currents entering at any junction is equal to the sum of currents leaving that junction. t junction O, I + I + I 3 I 4 I 5 I 6 = 0. Kirchhoff s Voltage Law (KVL): round any closed circuit loop the sum of voltage changes (drops or gains) across all the circuit elements while traversing in one direction (clockwise or anticlockwise) must be zero. I 3 D + While moving from higher potential to lower potential while crossing element to apply. Kichhoff s IInd law, P.D. is taken as negative. Potentiometer. Potentiometer consists of a uniform wire of large length of 4 m, 6 m, 8 m, etc. 4

6 ( ). The potential drop across the potentiometer wire is directly proportional to its length (this is because the wire is of uniform area of cross-section). ( ) D V (i) ( ) D D G (ii) K ( ) D D. K G (iii) 3. Through the path of galvanometer, current from the battery of e.m.f. can be varied i.e. if point D is closer to point, less current flows through the path of galvanometer; and if point D is farther from point, then more current flows through the path of galvanometer. 4. For comparing e.m.f.s of two cells, the positive terminals of both the cells and the positive terminal of the battery (of e.m.f. ) are connected to the same end of the potentiometer wire. Now, we locate point D on the potentiometer wire so that no current flows through the path of galvanometer. Then [see figure (ii)].m.f. of cell = P.D. across and D on potentiometer wire...(i) For two cells, D D l l...(ii) 5. For finding internal resistance (r) of a cell : (i) We connect a cell [see figure (iii)] and locate a point D on potentiometer wire so that no current flows through the path of galvanometer, when key K is not used. (ii) Now, we take out some resistance from the resistance-box (..) and use key K also, and locate a point D on the potentiometer wire so that no current flows through the path of galvanometer. l Then, r = l, where l = length D of potentiometer wire l = length D of potentiometer wire

7 onversion of Galvanometer into ammeter and voltmeter In a galvanometer, a moving coil is placed in a uniform magnetic field. When the current to be measured is passed through the coil of a galvanometer, the coil is deflected through a certain angle depending upon the current. The current I can be measured by converting a galvanometer into an ammeter by connecting a small resistance s in parallel to the galvanometer of resistance g through which the current I g is flowing. s I g I I g g I g I G G S The potential difference V between any two points and is measured by connecting a voltmeter across these points. galvanometer is converted into a voltmeter by connecting a high resistance in series with a galvanometer of resistance g. V g g V Vg Heating ffects of urrent If a current I ampere passes through a heating element of resistance for time t when a potential difference V volt is applied across it, the power produced is P. P = VI = I = V lectrical energy or work = VIt J Heat produced = H = Work VIt = cal J J where Joule s constant, J = 4.8 Joule/calorie VIt H = 4. Fuse Wire = 0.4 VIt = 0.4 V t calorie kilowatt hour = = Joule. It is used in series with electrical installations to protect from high electric current, this fuse melts causing breakage in the circuit when high electric current flows. Fuse wire has high resistance and low melting point, it is gernerally prepared from tin lead alloy (63% tin + 37 lead). I G g G V g V V V g

8 SOLVD XMPLS x. Sol.: wire of uniform area of cross-section has a resistance of 0. It is bent into a circle and points and situated at a distance of a quarter of the circumference apart are connected to a battery of e.m.f. 6.0 V and of internal resistance. Find the current in two parts and of the circuit. Total resistance of the wire = 0 0 esistance of wire = r = =.5 ( resistance length) V, 0 3 esistance of wire = r = = Let equivalent resistanceof r & r is I I I r r = I = r 5 8 y current division I = I r r r I = I I = x. Sol.: uniform copper wire of mass.3 gm carries a current of when an e.m.f. of.7 volt is applied across it. Find the length and area of cross-section. If the wire is uniformly stretched to double its length, find the new resistance. Density of copper = kg/m 3 and its resistivity is m. If is area of cross-section and L is length of wire, 3 mass Volume = L = density (i) V.7 esistance of wire, = = I =.7 = L or L.7 = =.7 08 = (ii) L 0 Multiplying (i) and (ii) L 6 = 0 8 = 4 L = 5m = 0 6 = m 0 = 5 4

9 When the wire is stretched to double its length, its volume remains constant. Its new area is given as L = L or = = m New resistance = L 4 = L = 4.7 = 6.8 x.3 alculate the equivalent resistance between points P and Q of the network of resistances shown in figures, and. 7 O P 3 P Q Q P Q Sol.: The network of resistances in these circuits are arranged in various branches such that these can be grouped into series or parallel combinations of resistances. The resistances should be grouped from the sides farthest from points P and Q. esistors 3 and 7 in series have a total resistance of 0 which is parallel to 0 resistance. This resistance of 0 is in parallel to a resistance of 0 along OP which gives an equivalent resistance of = 5. This resistance of 5 along OP is in series with 5 resistance along OQ yielding a total resistance of 0. This resistance of 0 is parallel to 0 along PQ which gives an equivalent resistance of 5 between the points P and Q. esistances, 4 and on extreme right side are in series and gives a total resistance of which in parallel with 8 resistance and gives 8 8 = 4. gain, 4 and in series combine to give a resistance of 8 which is parallel to 8 resistance and it gives total resistance of 4. The resistances of and are in series with this 4 resistance on extreme left combine to yield an equivalent resistance of 8 between points P and Q. onsider the 3 resistance in series with 3 along extreme left side of the network which combines to give 6 which inturn is parallel to 6 and yields a total resistance 3 and so on. t last, there will be a resistance of 3 along PO in series with 3 along OQ which is equal to 6. The resistance of 3 along PQ alongwith 6 in parallel will yield an equivalent of resistance between points P and Q =

10 x.4 alculate the equivalent resistances between points P and Q of the network of resistances shown in figures, and. P Q P 3 8 D D 4 Q D P Q Sol.: ll these network of resistances represent balanced Wheatstone s ridge in which no current flows through the side D. The resistance 0 through D is ineffective in balanced Wheatstone s ridge. The resistance across P and Q is equivalent to 4 and 6 in series which is parallel to 8 and in series. quivalent resistance eq is eq eq = In balanced Wheatstone s ridge D, 3 across D is ineffective. rm D and have respective resistances and. eq across PQ is sum of resistances and in seires which is parallel to resistance, on side D in series with 4 along D. eq eq = The circuit D is equivalent to balanced Wheatstone s ridge with each arm having equal resistance and resistance along D is ineffective. The equivalent resistance eq is given by: eq or eq = = x.5 battery of 6.0 V is connected to an infinite network of resistances of connected in series and in parallel as shown in the circuit. Find effective resistance between and the current through resistance nearest to the battery V Sol.: Let be equivalent resistance between and. ssume that one more set of resistances and is connected between them. s there are infinite resistances having a total resistance between and, the addition of resistances and will not affect the resistance. Let

11 be total resistance between and. s addition of one resistance does not make any difference for infinite resistances. = If is resistance between and. or = esistance between = = 3 = + 3 = 0 or = 0 ( + ) ( ) = 0 or = +, = is not possible = ohm. esistance between and = = esistance across = + = Let a current I flows through and current I flows from to. urrent I = 6 = 3 ampere P.D. across = I = 3 = 3V current across = 3 =.5 ampere. 6 V I I I I x.6: Twelve wires, each of resistance 6 are connected to form a network of skelton cube. current Sol.: enters at one corner and leaves from the diagonally opposite corner. Find the effective resistance between the opposite corners. Let i be the current flowing towards junction which will divide itself across different wires as shown. This is due to symmetry., and D are symmetrically placed so that current equal to i/3 flows through each of them, similarly i/3 current flows through each of the resistance G, FG & HG such that the current i flows out of junction G. Let be equivalent resistance between and G. onsider path G, V G = V + V + V G i eq i i i 5i r r r r eq 6 r 6 = 5 ohm. 6 x.7. Find the potential difference between points and in the circuit shown in the figure where the battery of.5 V has an internal resistance of and the battery of V has an internal resistance of. i i/3 D i/3 i/3 i/6 H i/6 i/3 i/6 i/3 i/6 i/6 i/3.5 V 8 F i/3 G i D 3 3 V F

12 Sol.: Figure between show current distribution according to Kirchhoff s Ist Law pplying Kirchhoff s second law for loops D and F..5 + i + 3i + 8i + i = 0 or 7i + 8i =.5...(i) and 8i + 3(i i ) + (i i ) + + (i i ) = 0 or 3i 7i =...(ii) Solving (i) and (ii) 3 3 i = and i = Potential difference between and, i i i V i.5 V 8 D 3 3 F 3 9 V V = 8 = V x.8 In the circuit shown, find : potential difference between and D potential difference across cell of V between and and cell of 3V between and D V D V 3 3 V V Sol.: Figure shows distribution of current according to Kischoff s first law pplying Kirchhoff s second law to loop D and D i i i + = 0 or 3i + i =...(i) and (i i ) + 3 3(i i ) + i = 0 or 4i 6i = = 0...(ii) Solving i and i i = 5, i 3 = 3 D V i 3 3 V i i i V P.D across and = V V D = 3 V i i = V = + = V V D = 3 3 = V 3 3

13 x.9: Two capacitors = 0 F and = 0 F are connected Sol.: in the circuit diagram showing three cells of.m.f. V, 6V and 3V connected alongwith various resistors. In the steady state, find : the currents I, I and I 3 the energy stored in capacitors and In steady state, capacitor acts as an open circuit. I = 0. pplying Kirchhoff s First Law at junction D, I I I 3 = 0. or I 3 = I...(i) pplying Kirchhoff s Second Law to mesh GHDFG, I =. 3 5I 3 + 5I 0I 3 6 = 0 0 V I 3 V 6 V 5 0 or 5I 3 5( I 3 ) = 9 0 V I 9 9 I 3 =, I = I = 0, I =, I3 =. 0 0 It shows that the direction of current I 3 is clockwise and is opposite to what is shown in the figure. 0 F G F I 3 3 V 6 V I 3 H 5 5 pplying Kirchhoff s Second Law to mesh HG, where the potential difference across condenser is V, I I I D I 3 I 0 F 0 F 5I V = 0 V = 5I or V = 3 = V (Negative sign shows that polarity of charge on capacitor plates 0 4 are opposite to what shown in figure) I 3 nergy stored in capacitor = V = = J pplying Kirchhoff s Second Law to mesh DH, where the potential difference across is V, + V I 5 = 0 V = 5I 9 or V = V = = V i.e. is at a higher potential than D. 4 4 nergy stored in capacitor = V = = J.

14 x.0. n electric tea kettle has two heating coils. When one of the coils is switched on, the water in the kettle begins to boil in 6 minutes. When only the other coil is switched on, the boiling starts in 8 minutes. Find the time it will take so that boiling begins when both the coils are switched on simultaneously if these are connected in series in parallel. Sol.: If and are the respective resistances of both coils and a potential difference V is applied for time t, let H be heat required so that boiling begins, V t H = For constant H, t t t 4 = For series connection, = + = + Let ' t be time taken for boiling, Then ' ' t t 7 3 For parallel connection, or 3 7, ' 4 4 ' Let t be time taken, Then or ' ' t t ' t = 6 = 3.43 min. 7 4 = 3 7 ' t = 6 = 4 min ' 7 3 x.: cell of steady emf..0 V is put across a potentiometer wire. For finding internal resistance of a cell of emf.5 V, it is connected as shown under. The balance point for this cell in open circuit is 76.3 cm. When a resistance of 9.5 is put across this cell, the balance point shifts to 64.8 cm. Find the internal resistance of the cell. V.5 V G 9.5 Sol.: = 9.5 l = 76.3 l = 64.8 r =? l 76.3 r = 9.5 l 64.8 = 9.5 [.775 ] = =.686

15 OJTIV QUSTIONS. alculate the resistance between P and Q in the following network. ach element has resistance The resistance between P and Q in the following combination of resistances is The effective resistance between points P and Q is P Q Six resistances, each equal to 4 are connected as shown in the figure. The effective resistance between any two vertices is The equivalent resistance between the points and of the following circuit is

16 6. Seven resistors, each of, are connected as shown in the figure. The effective resistance between P and Q is P Q 7. The resistance across points D and of the circuit is D / /3 /4 8. Five cells each of internal resistance 0. and e.m.f. V are connected in series with a resistance of 4. The current through the external resistance is ssume that the internal resistance of battery is zero and the key is closed in the following circuit, the reading of the ammeter is In the following circuit, the reading of the voltmeter is 5 V 0 V 5 V 6.5 V. cell of e.m.f. is connected across a resistance. The potential difference between the terminals of a cell is measured equal to V. The internal resistance of the cell is ( V) ( V) V ( V) ( V) V. In the circuit shown in the figure, the heat produced in the 5 resistor, due to the current flowing through it, is 0 calories per second. The heat generated in the 4 ohm resistor is cal/s cal/s 3 cal/s 4 cal/s 4 6 5

17 3. In the circuit shown in the figure, the current through 3 9 V resistor is resistor is resistor is resistor is The current through the resistor in the given circuit is V, 3 V, 5. The resistance of votlmeter in the given circuit is 800. The voltmeter will read 40 V V V 6 V 3 V 40 V 6. The temperature coefficient of resistance of a wire is perº. t 300 K, its resistance is. The resistance of the wire will be at 00 K 7 K 54 K 400 K 7. Two electric lamps and having power 00 watt and 00 watt respectively are rated on the same voltage. The ratio of resistance of lamp to that of is : : : 4 4 : 8. In the circuit shown in the figure, the final voltage drop across the capacitor in steady state is V r r 3 r V r r r V r r r V( r r r ) V( r r r r ) r 3 9. Three capacitors of capacitance 3 F each are connected in series. The energy stored in the capacitances, when a potential of kilo volt is applied to these for charging, is 5 J 0.5 J J 50 J

18 0. battery of e.m.f. 0 V is connected to resistances as shown in the figure. The potential difference between and is V + V 5 V 5 V. torch bulb rated as 4.5 W,.5 V is connected as shown in the figure. The e.m.f. of the cell of internal resistance.67, needed to make the bulb glow at full intensity is 4.5 V 6.75 V 3.5 V 7 V + 0 V 4.5 W.5 V, r =.67. Four equal resistors, when connected in series, dissipate a power of 5 watts. If these resistors are connected in parallel, the power dissipated will be 0 watts 40 watts 50 watts 80 watts 3. If a wire is stretched so that its length increases by 0. %, the percentage change in its resistance will be 0. % 0. % 0.3 % 0.4 % 4. uniform wire of resistance 0 having resistance /m is bent in the form of a circle as shown in the figure. If the equivalent resistance between and is.8, the length of the shorter section is.8 m.0 m.5 m 4 m 5. The current in the circuit in the figure is + V amp. amp. amp. amp In the given circuit currents I and I are 5, 4 6, I 6, 4 I

19 7. In the circuit shown, = 0 ohm, = 0 ohm and 3 = 30 ohm. The potentials of points, and are 0V, 6V and 5V respectively. The current through resistance is D In the given circuit, the potential difference between the points V, I and is.5 V 3.0 V.67 V.33 V 3 V, In the following circuit, emf across and is 3.0 V 3. V.5 V V, 3V,.8 V 4V, 30. simple potentiometer circuit is shown in the figure. The internal resistance of the 4V battery is negligible. is a uniform wire of length 00 cm and resistance. For what length, the galvanometer shows no deflection? 4 V cm 8.5 cm 84.5 cm 80.5 cm.5 V G MO THN ON OT HOI 3. The charge flowing in a conductor varies with line as Q = at bt. Then, the current decreases linearly with time fall to zero after time t = a/b reaches a maximum and then decreases changes at a rate b 3. Two batteries and and three resistors are connected. Internal resistance of both batteries is each as shown. MF of battery is 5V, the potential difference between P and Q is zero. Which of the following is/are O, P 5V, Q true the current through 5 is the current through the battery is 8 the emf of the source is 47 the p.d. between O and P is 8V

20 33. In a potentiometer arrangement, is the cell establishing current in primary circuit, is the cell to be measured. is the potentiometer wire and G is a galvanometer. Which of the following are the essential condition for balance to be obtained. The emf of must be greater than the emf of. ither the positive terminals of both and or the negative terminals of both and must be joined to one end of potentiometer wire. The positive terminals of and must be joined to one end of potentiometer wire. The resistance of G must be less than the resistance of 34. Two cells of unequal emfs and and internal resistances r and r are joined as shown in figure. V p and V Q are the potential at P and Q respectively. The potential difference across both the cells will be equal One of the cell, will supply energy to the other cell. The potential difference across one of the cells will be greater than its emf. r r VP VQ r r P,r,r Q 35. conductor is made of an isotropic material and has shape of a truncated cone. battery of constant emf is connected across it and its left end is earthed as shown in figure. If at a section distance x from left end, electric field intensity, potential and the rate of generation of heat per unit length are, V and H respectively, which of the following graphs is/are correct? 36. In the circuit shown in figure power supplied by the battery is 00 watt current flowing in the circuit is 5 4 potential difference across 4 resistance is equal to the potential difference across 6 resistance 6 current in wire is zero 0V

21 37. In the given circuit. urrent measured by ammeter is 0.4 amp Potential of point and F are equal Potential of point and are equal If 5 ohm resistances are replaced by 0 ohm resistance each, then current in all the 0 ohm resistance will be same. 38. Two heaters designed for the same voltage V have different power ratings. When connected individually across a source of voltage V, they produce H amount of heat each in times t and t respectively. When used together across the same source, they produce H amount of heat in time t if they are in series, t = t + t if they are in series, t = (t + t ) if they are in parallel, tt t (t t ) if they are in parallel, tt t (t t ) 39. In the circuit shown, F, G and H are cells of e.m.f. V, V, 3V and V respectively and their internal resistance are,, 3 and respectively. VD V V 3 VD V V 3 F D G H V G 3 V = Potential difference across G V H 9 V Potential difference across H Which of the following statements is/are correct? Potential difference between terminals of a non-ideal battery can never be greater than its emf. If a non-ideal battery is short circuited by a wire, heat generated in the wire is less than electric energy developed in the battery MF of an ideal battery is, first, measured by a potentiometer and than by a voltmeter. oth the measurements are equally correct. The terminal potential difference of a battery can never be zero in any closed circuit containing other batteries.

22 MISLLNOUS SSIGNMNT omprehension- Two persons are pulling a square of side a along one of the diameters horizontally to make it rhombus. Plane of rhombus is always vertical and uniform magnetic field exist perpendicular to plane. They start pulling at t = 0 and with constant velocity v.. The induced emf in the frame when angle at corner being pulled is 60 av av. If the resistance of the frame is the current induced is a v a v a v a v 3. Finally square frame reduces to straight wire. The total charge flown is a v 4 a v 4 a a a 4a omprehension- In case of analysis of circuits, containing cells, resistences and inductances two things are very important one is conservation of charge which leads to the fact that at any junction of circuit incoming current is equal to out going current. The other thing is that sum of voltage drop in a closed loop is equal to zero. Inductors have a unique property by which they oppose the change in magnetic flux linked to them. The voltage drop across resistor is di V = I and across inductor is L. In the steady state dt current through inductor becomes constant which leads to zero voltage drop across inductor. ie. it behaves like short circuit. efer to circuit given in the figure. = 0 V, =, = 3, 3 = 6 and L = 5H. Now answer the following questions 4. The current I just after pressing the switch S is S I I I 3 3 L The current I long after pressing the switch S is The current I long after pressing the switch S is

23 7. The current through just after releasing the switch S is In the circuit, the battery is ideal. voltmeter of resistance 600 is connected in turn across and, giving readings V and V respectively. olumn I olumn II. V (p) 30 volt. V (q) 60 volt. rror in V reading (r) 0 volt D. rror in V reading (s) 0 volt 9. For the circuit shown in figure, match the columns olumn I olumn II. The current in db (p) /3 amp. The PD across db (q) /3 volt. The PD across the cell () (r) /3 volt D. The PD across the cell () (s) 9/3 volt INTG TYPS OF QUSTIONS 0. In a metre bridge, (Figure) the balance point is found to be at 39.5 cm from the end, when the resistor Y is of.5. The resistance of X is 4.n. Find the value of n. X Y D G. In the circuit shown the potential at D is 3n volt. Find the value of n V 6 V 0 V D. battery of emf 0V and internal resistance 3 is connected to a resistor. If the current in the circuit is 0.5? The terminal voltage of the battery (when the circuit is closed) is.7n volt. Find the value of n

24 3. fuse made of lead wire has an area of cross-section 0. mm. On short circuiting, the current in the fuse wire reaches 30 amp. fter n second, short circuiting the fuse begins to melt. Find the value of n. Specific heat capacity of lead = 34.3 J/kg-K. Melting point of lead = 37. Density of lead = 340 kg/m 3. esistivity of lead = 0 8 ohm-h. Initial temperature of the wire = 0. Neglect heat loss. 4. Find the current in the three resistors shown in figure. V V V V V V 5. Flow of charge through a surface is given as : Q = 4t + t (for 0 to 0 sec.) the average current for (0 0 sec) is 7n ampere. Find the value of n 6. Figure shows a conductor of length l having a circular crosssection. The radius of cross-section varies linearly from a to b. The resistivity of the material is. ssuming that (b a) << l, the resistance of the conductor is ln /ab. Find the value of n. 7. In the circuit shown, Find current i in ampere. 8. ight resistances each of resistance 5 are connected in the circuit shown in figure. The equivalent resistance between and is n/3. Find the value of n. 9. The equivalent resistance between points and is (3/n). Find the value of n

25 omprehension PVIOUS Y QUSTIONS IIT-J/J-DVN QUSTIONS The capacitor of capacitance can be charged (with the help of a resistance ) by a voltage source V, by closing switch S while keeping switch S open. The capacitor can be connected in series with an inductor L by closing switch S L and opening S.. Initially, the capacitor was uncharged. Now, switch S is closed and S is kept open. If time constant of this circuit is, then after time interval, charge on the capacitor is V/ after time interval, charge on the capacitor is V( e ) the work done by the voltage source will be half of the heat dissipated when the capacitor is fully charged after time interval, charge on the capacitor is V( e ). fter the capacitor gets fully charged, S is opened and S is closed so that the inductor is connected in series with the capacitor. Then, at t = 0, energy stored in the circuit is purely in the form of magnetic energy at any time t > 0, current in the circuit is in the same direction at t > 0, there is no exchange of energy between the inductor and capacitor at any time t > 0, instantaneous current in the circuit may 3. If S is opened and S is closed at t = 0 and total charge stored in the L circuit is Q 0, then for t 0 the charge on the capacitor is Q = Q 0 cos t L the charge on the capacitor is Q = Q 0 cos t L Q the charge on the capacitor is Q = L d dt d Q the charge on the capacitor is Q = L dt 4. Two wires PQ and Q of radius r and r respectively having equal length l/ and equal resistivity are joined as shown. urrent I is flowing through the wires. Power loss in PQ is four times power loss in Q urrent density will be equal in both wire P.D. across wire PQ is four time compare to wire Q lectricfield is equal in both wires V V L S S P l/ Q l/ r r

26 F 5. 4 F F F 4 F 4 F Time constant (in micro seconds) of circuits in order are 8, 4, 8/9 8, 8/9, 4 4, 8, 8/9 4, 8/ circuit is connected as shown in the figure with the switch S open. When the switch is closed, the total amount of charge that flows from Y to X is resistance of is connected across one gap of a metre-bridge (the length of the wire is 00 cm) and an unknown resistance, greater than, is connected across the other gap. When these resistances are interchanged, the balance point shifts by 0 cm. Neglecting any corrections, the unknown resistance is For the circuit shown in the figure the potential difference across L is 8V ratio of powers dissipated in and is 3 if and are interchanged, magnitude of the power dissipated in L will decrease by a factor of 9 9. Two metallic rings and, identical in shape and size but having different resistivities and, are kept on top of two identical solenoids as shown in the figure. When current I is switched on in both the solenoids in identical manner, the rings and jump to heights h and h, respectively, with h > h. The possible relation (s) between their resistivities and their masses m and m is (are) () () and m = m () and m > m (D) and m = m and m < m 0. Match the following

27 Dielectric ring uniformly charged (p) Time independent electrostatic field out of system Dielectric ring uniformly charged (q) Magnetic field rotating with angular velocity onstant current in ring i 0 (r) Induced electric field D i = i 0 cos t (s) Magnetic moment. Six point charges, each of the same magnitude q, are arranged in different manners as shown in column-ii. In each case, a point M and a line PQ passing through M are shown. Let be the electric field and V be the electric potential at M (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line PQ. Let b be the magnetic field at M and be the magnetic moment of the system in this condition. ssume each rotating charge to be equivalent to a steady current. olumn -I olumn-ii. = 0 (p) harges are at the corners of a regular hexagon. M is at the centre of the hexagon. PQ is perpendicular to the plane of the hexagon.. V 0 (q) harges are on a line perpendicular two PQ at equal intervals. M is the mid-point between the two innermost charges.. = 0 (r) harges are placed on two coplanar insulating rings at equal intervals M is the common centre of the rings. PQ is perpendicular to the plane of the rings. D. µ 0 (s) harges are placed at the corners of a rest angle of sides a and a and at the mid points of the longer sides. M is at the centre of the rectangle. PQ is parallel to the longer sides. (t) harges are placed on two coplanar, identical insulating rings at equal

28 intervals. M is the mid point between the centres of the rings. PQ is perpendicular to the line joining the centres and coplanar to the rings.. To verify Ohm s law, a student is provided with a test resistor T, a high resistance, a small resistance, two identical galvanometers G and G, and a variable voltage source V. The correct circuit to carry out the experiment is G G G G T T V V G G G G T T V V 3. Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 00 W, 60 W and 40 W bulbs have filament resistances 00, 60 and 40, respectively, the relation between these resistances is = > 60 > onsider a thin square sheet of side L and thickness t, made of a material of resistivity. The resistance between two opposite faces, shown by the shaded areas in the figure is directly proportional to L t directly proportional to t independent of L independent of t L 5. When two identical batteries of internal resistance each are connected in series across a resistor

29 , the rate of heat produced in is J. When the same batteries are connected in parallel across, the rate is J. If J =.5 J then the value of in is 6. meter bridge is set-up as shown, to determine an unknown resistance X using a standard 0 ohm resistor. The galvanometer shows null point when tapping key is at 5 cm mark. The end-corrections are cm and cm respectively for the ends and. The determined value of X is X 0. ohm 0.6 ohm 0.8 ohm. ohm 7. Two batteries of different emfs and different internal resistances are connected as shown. The voltage across in volts is 6V 8. For the resistance network shown in the figure, choose the correct option (s) 3V The current through PQ is zero I = 3 The potential as S is less than the at Q I = 9. Two ideal batteries of emf V and V and three resistances, and 3 are connected as shown in the figure. The current in resistance would be zero if V = V and = = 3 V V = V and = = 3 V = V and = = 3 3 V = V and = = 3 0. Heater of an electric kettle is made of a wire of length L and diameter d. It takes 4 minutes to raise the

30 temperature of 0.5 kg water by 40 K. This heater is replaced by a new heater having two wires of the same material, each of length L and diameter d. The way these wires are connected is given in the options. How much time in minutes will it take to rise the temperature of the same amount of water by 40 K? 4 if wires are in parallel if wires are in series if wires are in series 0.5 if wires are in parallel. galvanometer gives full scale deflection with current. y connecting it to a 4990 resistance, it can be converted into a voltmeter of range 0 30V. If connected to a an ammeter of range 0.5. The value of n is n 49 resistance, it becomes. During an experiments with a metre bridge, the galvanometer shows a null point when the jockey is pressed at 40.0 cm using a standard resistance of 90, as shown in the figure. The least count of the scale used in the metre bridge is mm. The unknown resistance is 40.0 m () 60 ± 0.5 () 35 ± 0.56 () 60 ± 0.5 (D) 35 ± 0.3 D QUSTIONS. potentiometer is used to determine the emf of a cell. No current flows through the galvanometer. If the cell is shunted by a resistance, the balancing length becomes half. What is the internal resistance of the cell? 4 none of these. If the colours are blue, blue, red, silver, then the resistance is ± 0% 0 6 ± 0% ± 5% 66 0 ± 0% 3. uniform wire of resistance 36 is bent in form of a circle. The effective resistance across the points M and N is [D-00] What is the current in D? 30 M N I

31 30 30 G 30 D 30 V none of these 5. Two wires having resistances and 3 are connected in parallel, the ratio of heat generated in 3 and is : 3 : : 4 4 : 6. Null point with V cell comes out to be 55 cm and with = 0, it is 50 cm. What is the internal resistance of the cell? cm 7. In the given circuit I is =0 30 I V I V I In the balanced Wheatstone bridge circuit as shown in the figure, when the key is pressed, what will be the change in the reading of the galvanometer remains same increased decreased zero V G 9. What is? none of these i = 0.5 amp V n electric bulb illuminates a plane surface. The intensity of illumination on the surface at a point m

32 away from the bulb is phot (lumens/cm ). The line joining the bulb to the point makes an angle of 60 with the normal to the surface. The intensity of the bulb in candela (candle power) is esistance of rod is. It is bent in form of square. What is resistance across adjoint corner 3 3 (3/6) 4. ulb is 00 W 50 V and bulb is 00 W 00 V are connected across 50 V. What is potential drop across? b b 00 V 50 V 98 V 48 V MINS QUSTIONS. Shown in the figure below is a meter-bridge set up with null deflection in the galvanometer. The value of the unknown resistor is V battery with internal resistance and V battery with internal resistance and is connected to a 0 resistor as shown in the figure 0.03 P to P 0.03 P to P 0.7 P to P 0.7 P to P Passage Directions : Question No. 3 and 4 are based on the following paragraph. onsider a block of conducting material of resistivity shown in the figure. urrent I enters at and leaves from D. We apply superposition principle to find voltage V developed between and. The calculation is done in the following steps : (i) Take current I entering from and assume it to spread

33 (ii) (iii) over a hemispherical surface in the block. alculate field (r) at distance r from by using Ohm s law = J, where J is the current per unit area at r. From the r dependence of (r), obtain the potential V(r) at r (iv) epeat (i), (ii) and (iii) for current I leaving D and superpose results for and D. 3. For current entering at, the electric field at a distance r from is I I I r r 4r 4. V measured between and is I I I I I a ( a b) a ( a b) ( a b) I 8r I I a ( a b) 5. Statement-: The temperature dependence of resistance is usually given as = 0 ( + t). The resistance of a wire change from 00 to 50 when its temperature is increased from 7 to 7. This implies that = /. Statement-: = 0 ( + t) is valid only when the change in the temperature T is small and = ( 0 ) << 0. Statement- is true, Statement- is true, Statement- is the correct explanation of Statement- Statement- is true, Statement- is true, Statement- is not the correct explanation of Statement- Statement- is false, Statement- is true Statement- is true, Statement- is false 6. n electric current is passed through a circuit containing two wires of the same material, connected in parallel. If the lengths and radii of the wires are in the ratio of (4/3) and (/3), then the ratio of the currents passing through the wires will be 8/9 / In a metre bridge experiment null point is obtained at 0 cm. from one end of the wire when resistance X is balanced against another resistance Y. If X < Y, then where will be the new position of the null point from the same end, if one decides to balance a resistance of 4 X against Y? 40 cm 80 cm 50 cm 70 cm 8. The thermistors are usually made of metal oxides with high temperature coefficient of resistivity metals with high temperature coefficient of resistivity metals with low temperature coefficient of resistivity semiconducting materials having low temperature coefficient of resistivity

34 9. Time taken by a 836 W heater to heat one litre of water from 0 to 40 is 50 s 00 s 50 s 00 s 0. heater coil is cut into two equal parts and only one part is now used in the heater. The heat generated will now be one fourth halved doubled four times. In the circuit, the galvanometer G shows zero deflection. If the batteries and have negligible internal resistance, the value of the resistor will be G V V. Two sources of equal emf are connected to an external resistance. The internal resistances of the two sources are and ( > ). If the potential difference across the source having internal resistance is zero, then = ( = ( ( ) ) ) = ( = 3. In a potentiometer experiment the balancing with a cell is at length 40 cm. On shunting the cell with a resistance of, the balancing length becomes 0 cm. The internal resistance of the cell is ) The resistance of hot tungsten filament is about 0 times the cold resistance. What will be the resistance of 00 W and 00 V lamp when not in use? n energy source will supply a constant current into the load if its internal resistance is non-zero but less than the resistance of the load equal to the resistance of the load very large as compared to the load resistance 6. The Kirchhoff s firs law ( i = 0) and second law ( i = ), where the symbols have their usual meanings, are respectively based on conservation of charge, conservation of momentum conservation of energy, conservation of charge conservation of momentum, conservation of charge conservation of charge, conservation of energy 7. The current I drawn from the 5 volt source will be

35 I In a Wheatstone s bridge, three resistances P, Q and are connected in the three arms and the fourth arm is formed by two resistances S and S connected in parallel. The condition for the bridge to be balanced will be P Q S S P Q (S S) S S P Q (S S) S S P Q S S 9. The circuit has two oppositely connected ideal diodes in parallel. What is the current flowing in the circuit? V D D 3 0. n electric bulb is rated 0 volt 00 watt. The power consumed by it when operated on 0 volt will be 75 watt 40 watt 5 watt 50 watt. Two electric bulbs marked 5 W-0 V and 00 W - 0 V are connected in series to a 440 V supply. Which of the bulbs will fuse? oth 00 W 5 W Neither. The supply voltage to a room is 0 V. The resistance of the lead wires if 6. 60W bulb is already switched on. What is the decrease of voltage across the bulb, when a 40 W heater is switched on in parallel to the bulb? 3.3 Volt 0.04 Volt zero Volt.9 Volt 3. The current voltage relation of diode is given by I = (e 000V/T ) m, where the applied voltage V is in volts and the temperature T is in degree Kelvin. If a student makes an error mesuring ±0.0 V while measuring the current of 5 m at 300 K, what will be the error in the value of current in m? 0.5 m 0.05 m 0. m 0.0 m

36 SI POLMS. Two cells and, having e.m.f. 4 V and 8 V and internal resistance 0.5 and respectively are connected as shown. Find the current in each resistor and potential difference across each cell. 4V 8V I M 4.5 I 3 I 6. galvanometer having 30 divisions has a current sensitivity of 0 per division and a resistance of 5 ohm. Find the shunt required to convert it into an ammeter of ampere range. Find the resistance which should be connected in series to convert this ammeter into a voltmeter of range V. 3. heater is designed to operate with a power of 000 watts in a 00-volts line. It is connected, in combination with a resistance of 0 ohms and a resistance, to a 00-volts mains as shown in the figure. What should be the value of so that the heater operates with a power of 6.5 watts? 4. In the circuit shown a voltmeter reads 30 volts when it is connected across the 400 ohms resistance. alculate what the same voltmeter will read when it is connected across the 300 ohm resistance? V 5. galvanometer of unknown resistance in series is connected across two identical cells each of.5 volt. When the cells are connected in series, the galvanometer records a current of and when the cells are in parallel, the current is 0.6. Find the internal resistance of the cell. 6. Determine the charge and energy stored in the capacitor in the figure for the circuit in which internal resistance of battery, r = 4. lso, find the value of 3 P r for which the charge on the capacitor is zero. 6V, r S 7 3µF Q r

37 7. battery of e.m.f. 0 V is connected to a network of resistors as shown in the figure. The potential difference across a resistor of 400 is measured by a voltmeter of resistance 400. Find the reading of the volt meter. V D 00 0 V 8. battery of internal resistance 4 is connected to the one single network of resistances as shown in the figure. Find the value of so that maximum power is delivered to the network. Find the maximum power In the circuit shown in the figure, the e.m.f. of the cell is.8 V and its internal resistance r is 3. Find the current in 3 resistance and power consumed by the circuit from the battery when K is closed r K 0. Two resistors, 400 ohms and 800 ohms, are connected in series with a 6-volt battery. It is desired to measure current in the circuit. n ammeter of 0 ohms resistance is used for this purpose. What will be the reading in the ammeter? Similarly, if a voltmeter of 0,000 ohms resistance is used to measure the potential difference across the 400-ohm resistor, what will be the reading in the voltmeter?. carbon and an aluminium wire are connected in series. If the combination has resistance of 30 ohm at 0, what is the resistance of each wire at 0 so that the resistance of the combination does not change with temperature? [ = ( ) and = ( ) ]

38

39 DVND POLMS. Find the equivalent resistance between the terminal points and in the network shown in the figure Find the potential difference between the plates of the capacitance in the circuit shown in the figure. The internal resistances of batteries are negligible. 3. n electric circuit is shown in figure where the cells have got negligible internal resistance. Find : the current in 3 resistance and in the cell of 8 volt. the charge on the capacitor. 4. material of resistivity is formed in the shape of a truncated cone of altitude h as shown in figure. The top end has a radius a while bottom b. ssuming a uniform current density through any circular crosssection of the cone, show that the resistance between the two ends is: = a V h ab. h I b V

40 5. battery of 9.0 V is connected to three resistances of k, 5 k and 5 k and a condenser of 0 F through a switch S as shown in the circuit diagram. If the switch is closed for a sufficiently long time so that the capacitor becomes fully charged, find : the steady state current I, I and I 3. the charge on the capacitor. the time it takes for the charge on the capacitor to fall to half its value if the switch S is opened at t = 0. the variation of the current through the resistance of 5 k as a function of time after opening the switch. 6. In the circuit shown in the figure, = 3 volts, = volts, 3 = volt, and = r = r = r 3 = ohm. k 5k 0F 5k r r r 3 3 Find the potential difference between the points and, and the currents through each branch. If r is short-circuited and the point is connected to point, find the currents through,, 3 and the resistor. 7. In the circuit shown in the figure, the battery is an ideal one, with e.m.f. V. The capacitor is initially uncharged. The switch S is closed at time t = 0. Find the charge Q on the capacitor at time t. Find the current in at time t. What is its limiting value as t? 8. fuse wire of lead has an area of cross-section 0. mm. The current in the fuse wire reaches 30 when there is a short circuit. Find the time after the short circuit when the fuse will begin to melt. The specific heat and melting point for lead is 0.03 cal g (º) and 37º respectively. The density and resistivity of fuse wire is.34 g.cm 3 and. 0 5 ohm-cm. Neglect heat losses. The initial temperature of wire of 0º.