BANSAL CLASSES TARGET LIT JEE 2007 XI (PQRS) CALORIMETRY & HEAT TRANSFER CONTENTS KEYCONCEPT EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY

THERMAL Definition of Heat: EXPANSION Heat is a form of energy which is transferred between a system and its surroundi ng as a result of temperature difference only. due to increase in temperature. F or temperature change At change in length Al = l0a At Area AA= A^At volume AV = V yAt 0 Thermal Expansion : Expansion 1. Type of thermal expansion Coefficient of expansion (i) Linear (ii) Superficial (iii) Volume (a) (b) 2. . a = At>0 / 1 A/t Lim A T 0 P = Lim 1 AA AtA0 At y = At>o v1 AV Lim At 0 For isotropic solids otj = a = a = olids p = otj + a and y = a, + a + pansion in X , Y and Z directions. rature volume increases so density 2 3 2 2 3 2 3 0 3 a (let) so P =2a and y = 3a For anisotropic s a Here , a and a are coefficient of linear ex Variation in density : With increase of tempe decreases and vice-versa. H d =(1 + yAt)

Note For solids values of y are generally small so we can write d = d (1-yAt) (u sing bimomial expansion) (0 (ii) y for liquids are in order of 10~ For water den sity increases from 0 to 4C so y is -ve (0 to 4 C) and for 4 C to higher temperatur e y is +ve. At 4 C density is maximum. 3. Thermal Stress: Arod of length 1 is cla mped between two fixed walls with distance 1 . If temperature is changed by amou nt At then F stress A (area assumed to be constant) 0 0 : so, or A/ strain = I F/A F/ Y = A/// AAI F =YAa A t 0 0 F AaAt (!l Bansal Classes Calorimetry & Heat Transfer [3]

4. If a is not constant (i) (a varies with distance) Let a = ax+b Total expansion = Jexpansion of length dx i = |(ax + b)dxAt " x 1 (ii) ( a varies with tempearture) Let a = f (T) T2 0 dx A/ _ j"a/ dT T i Caution: If a is in C then put Tj and T in C. similarly if a is i n K then put Tj and T in K. 2 2 CAL ORIMETR Quantity of heat transfered and specific heat Y The amount ofheat needed to incerase the temperature of 1 gmofwaterfrom 14.5Cto 1 5.5CatSTP is 1 calorie dQ = mcdT Q = m [ C dT (be careful about unit of temperatu re, use units according to the given units of C) T i Heat transfer in phase change 'h Q = rnL L = latent heat of substance in cal/ gm/ C or in Kcal/ kg/ C L = 80 cal/ gm for ic e ice L steam = 5 4 0 C a l / g m (A) (i) (ii) Note: 1.

vibration and collision of medium particles. Steady State : In this state heat a bsorption stops and temperature gradient throughout the rod dT becomes constant i.e. = constant. dx Before steady state : Temp of rod at any point changes If sp ecific heat of any substance is zero, it can be considered always in steady stat e. Let the two ends of rod of length 1 is maintained at temp Tj and T ( Tj > T ) dQ i ~ 2 I Thermal current D 1 = K-XH L T 2 2 T T 1 Conduction : Due to HEATTRANSFER Ohm's law for Thermal Conduction in Steady State : / Where thermal resistance R = K A Th 1 1 2. Differential form of Ohm's Law T-dT dQ dT =KA dT dx dT = temperature gradient dx dx (!lBansal Classes Calorimetry & Heat Transfer [3]

(B) (Q 1. Heat transfer due to movement ofmedium particles. Radiation: Every body radiates electromagnetic radiation of all possible wavelength at all temp>0 K. Stefan's Law: Rate of heat emitted by a body at temp T K from per unit area E = GT J/sec/ m d = P = oAT watt Q Radiation power dl If a body is placed in a surrounding of temperature T dQ Convection: 4 2 4 s valid only for black body heat from general body Emissmty or emmisive power e = ~ Iftemp ofbody falls by dT in time dt dT _ _ j4x (dT/dt=rate of cooling) dt ~ m S h e a t f r o m s ^ =cA(T -T ) 4 s 4 Newton's law of cooling Iftemp difference ofbody with surrounding is small i.e. T = T eA then, dT 4mS -a T ( T - T ) dt dT a ( T - T ) so dt rr3/ 2 s Average form of Newtons law of cooling If a body cools from T j to T in time 51 T - T _ K T, +T, -T (used generally in objective questions) 5t mS s 2 dt 4. mS (for better results use this generally in subjective) At every temperature (>0K) a body radiates energy radiations ofall wavelengths. According to Wein's displacement law if the wavelength corresponding to maximum energy is X then X T = b where b = is a constant (Wein's constant) T=temperature of body m m Wein's black body radiation T3>T2>T, ess (!l Bansal Classes

Calorimetry & Heat Transfer [3]

EXERCISE -1 Q. 1 An aluminium container of mass 100 gm contains 200 gm of ice at - 20C. Heat is added to the system at the rate of 100 cal/s. Find the temperature of the sys tem after 4 minutes (specific heat of ice = 0.5 and L = 80 cal/gm, specific heat of A1 = 0.2 cal/gm/C) Q. 2 A U-tubefilledwith a liquid ofvolumetric coefficient of 10 /C lies in a vertical plane. The height of liquid column in the left vertic al limb is 100 cm. The liquid in the left vertical limb is maintained at a tempe rature = 0C while the liquid in the right limb is maintained at a temperature = 1 00C. Find the difference in levels in the two limbs. _5 Q.3 A thin walled metal tank of surface area 5m is filled with water tank and contai ns an immersion heater dissipating 1 kW. The tank is covered with 4 cm thick lay er of insulation whose thermal conductivity is 0.2 W/m/K. The outer face of the insulation is 25C. Find the temperature of the tank in the steady state 2 Q.4 A glassflaskcontains some mercury at room temperature. It is found that at diffe rent temperatures the volume of air inside the flask remains the same. If the vo lume of mercury in the flask is 300 cm , thenfindvolume of the flask (given that coefficient of volume expansion of mercury and coefficient oflinear expansion o f glass are 1.8 x 10^(C) and9x 10~ (C) respectively) 3 _1 6 _1 Q.5 Q.6 Q.7 A clock pendulum made of invar has a period of 0.5 sec at 20C. If the clock is us ed in a climate where average temperature is 30C, aporoximately. How much fast or slow will the clock run in 10 sec. (a =lxlO /C) 6 ilwar -6 A pan filled with hot food cools from 50.1 C to 49.9 C in 5 sec. How long will it take to cool from 40.1 C to 39.9C if room temperature is 30C? A composite rod made of three rods of equal length and cross-section as shown in the fig. The thermal conductivities of the materials of the rods are K/2, 5K and K respectively. The end A and end B are at constant temperatures. All heat entering the face A goes out of the end B there being no loss of heat from the sides of the bar. Find th e effective thermal conductivity of the bar A I Q.8 Q.9 K/2 I 11 5K 2 6 1 K

1 B An iron bar (Young's modulus = 10 N/m , a = 10" /C) 1 m long and 10~ m in area is heated from 0C to 100C without being allowed to bend or expand. Find the compress ive force developed inside the bar. 3 2 A solid copper cube and sphere, both of same mass & emissivity are heated to sam e initial temperature and kept under identical conditions. What is the ratio of their initial rate of fall of temperature? Q. 10 A cylindrical rod with one end in a stream chamber and other end in ice ca use melting of 0.1 gm of ice/sec. If the rod is replaced with another rod of hal f the length and double the radius of first and thermal conductivity of second r od is 1/4 that of first, find the rate of ice melting in gm/sec (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.ll Three aluminium rods of equal length form an equilateral triangle ABC. Taki ng O (mid point of rod BC) as the origin. Find the increase in Y-coordinate per unit change in temperature ofthe centre ofmass of the system. Assume the length of the each rod is 2m, and a = 4 v3 x10" /C d 6 Q.12 Three conducting rods of same material and cross-section are shown in figur e. Temperature of A, D and C are maintained at 20C, 90C and 0C. Find the ratio of l ength BD and BC if there is no heat flow in AB 20C 90'C 0C Q. 13 If two rods of layer L and 2 L having coefficients of linear expansion a a nd 2a respectively are connected so that total length becomes 3 L, determine the average coefficient of linear expansion of the composite rod. Q.14 A volume of 120 ml of drink (half alcohol + half water by mass) originally at a temperature of 25C is cooled by adding 20 gm ice at 0C. If all the ice melts, find the final t emperature of the drink, (density of drink = 0.833 gm/cc, specific heat of alcoh ol = 0.6 cal/gm/C) Q.15 A solid receives heat by radiation over its surface at th e rate of 4 kW. The heat convection rate from the surface of solid to the surrou nding is 5.2 kW, and heat is generated at a rate of 1.7 kW over the volume of th e solid. The rate of change of the average temperature of the solid is 0.5 Cs . Find the heat capacity of the solid. o -1 Q.16 The figure shows the face and interface temperature of a composite slab con taining offour layers oftwo materials having identical thickness. Under steady s tate condition, find the value of temperature 6. 20C 10C E -5C -10C 2k 2k k = thermal conductivity Q.17 Two identical calorimeter A and B contain equal quantity of water at 20C. A 5 gm piece of metal X of specific heat 0.2 cal g (C) is dropped into A and a 5 gm piece of metal Y into B. The equilibrium temperature in A is 22C and in B 23C. Th e initial temperature of both the metals is 40C. Find the specific heat of metal Y in cal g" (C)~ 4 _1 1 l Q.18 Two spheres of same radius R have their densities in the ration 8 . 1 and t he ratio of their specific heats are 1 : 4. If by radiation their rates of fall of temperature are same, thenfindthe ratio of their rates of losing heat. Q.19 I n the square frame of side I of metallic rods, the corners A and C are maintaine d at Tj and T respectively. The rate of heat flow from A to Cisa. IfA and D are instead maintained Tj & T respectivleyfind,findthe total rate ofheat flow. 2 2 Q.20 A hot liquid contained in a container of negligible heat capacity loses tem perature at rate 3 K/min, just before it begins to solidify. The temperature rem ains constant for 30 min, Find the ratio of specific heat capacity of liquid to

specific latent heat of fusion is in Kr (given that rate of losing heat is const ant). 1 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q. 21 A thermostatted chamber at small height h above earth's surface maintained at 30C has a clock fitted in it with an uncompensated pendulum. The clock design er correctly designs it for height h, but for temperature of 20C. Ifthis chamber is taken to earth's surface, the clock in it would click correct time. Find the coefficient oflinear expansion ofmaterial of pendulum, (earth's radius is R) Q.2 2 The coefficient of volume expansion of mercury is 20 times the coefficient of linear expansion of glass. Find the volume of mercury that must be poured into a glass vessel ofvolume V so that the volume above mercury may remain constant at all temperature. Q. 23 Two 50 gm ice cubes are dropped into 250 gm ofwater ion a glass. Ifthe water was initially at a temperature of 25C and the temperature of ice -15C. Find the final temperature of water, (specific heat ofice = 0.5 cal/gm /C and L = 80 cal/gm) Q.24 Water is heated from 10C to 90C in a residential hot wat er heater at a rate of 70 litre per minute. Natural gas with a density of 1.2 kg /m is used in the heater, which has a transfer efficiency of 32%. Find the gas c onsumption rate in cubic meters per hour, (heat combustion for natural gas is 84 00 kcal/kg) 3 Q.25 A metal rod A of 25cm lengths expands by 0.050cm. When its temperature is r aised from 0C to 100C. Another rod B of a different metal of length 40cm expands b y 0.040 cm for the same rise in temperature. A third rod C of 50cm length is mad e up of pieces of rods A and B placed end to end expands by 0.03 cm on heating f rom 0C to 50C. Find the lengths of each portion of the composite rod. Q.26 A subst ance is in the solid form at 0C. The amount of heat added to this substance and i ts temperature are plotted in the following graph. If the relative specific heat capacity of the solid substance is 0.5, find from the graph (i) the mass of the substance; (ii) the specific latent heat of the melting process, and (iii) the specific heat of the substance in the liquid state. Q. 27 One end of copper rod ofuniform cross-section and of length 1.5 meters is in contact with melting ice and the other end with boiling water. At what point along its length should a te mperature of200C be maintained, so that in steady state, the mass ofice melting i s equal to that of steam produced in the same interval oftime? Assume that the w hole system is insulatedfromthe surroundings. Q.28 Two solids spheres are heated to the same temperature and allowed to cool under identical conditions. Compare : (i) initial rates of fall of temperature, and (ii) initial rates of loss of he at. Assume that all the surfaces have the same emissivity and ratios of their ra dii of, specific heats and densities are respectively 1 : a, 1 : p, 1 : y. Q.29 A vessel containing 100 gm water at 0C is suspended in the middle of a room. In 1 5 minutes the temperature of the water rises by 2C. When an equal amount of ice i s placed in the vessel, it melts in 10 hours. Calculate the specific heat offusi on ofice. Q. 3 0 The maximum in the energy distribution spectrum of the sun is a t 4753 A and its temperature is 6050K. What will be the temperature of the star whose energy distribution shows a maximum at 9506 A. (!l Bansal Classes Calorimetry & Heat Transfer [3]

EXERCISE-II Q. 1 A copper calorimeter of mass 100 gm contains 200 gm of a mixture of ice and water. Steam at 100C under normal pressure is passed into the calorimeter and th e temperature of the mixture is allowed to rise to 50C. If the mass of the calori meter and its contents is now 330 gm, what was the ratio of ice and water in the beginning? Neglect heat losses. Given : Specific heat capacity of copper = 0.42 x 10 J kg K" , Specific heat capacity of water = 4.2 x 10 J kg^Kr , Specific he at of fusion of ice = 3.36 x 10 J kg Latent heat of condensation of steam = 22.5 x 1Q Jkg" 3 _1 x 3 1 5 -1 5 1 Q.2 base and two thin rods each of length l and coefficient of linear expansion a fo r the two pieces, ifthe distance between the apex and the midpoint ofthe base re main unchanged as the temperatures /, varied show that 7 2 2 l A n isoscetes triangte is form ed w ith a rod of length l and coefficient of linea r expansion OTJ for the x 2 Q.3 A solid substance of mass 10 gm at - 10C was heated to - 2C (still in the solid st ate). The heat required was 64 calories. Another 880 calories was required to ra ise the temperature ofthe substance (now in the liquid state) to 1C, while 900 ca lories was required to raise the temperature from -2C to 3C. Calculate the specifi c heat capacities of the substances in the solid and liquid state in calories pe r kilogram per kelvin. Show that the latent heat of fusion L is related to the m elting point temperature t by L = 85400 + 200 t . m m Q.4 (a) (b) Q. 5 Q.6 Q. 7 A steel drill making 180 rpm is used to drill a hole in a block of steel. The ma ss of the steel block and the drill is 180 gm. If the entire mechanical work is used up in producing heat and the rate of raise in temperature of the block and the drill is 0.5 C/s. Find the rate of working of the drill in watts, and the tor que required to drive the drill. Specific heat of steel = 0.1 and J = 4.2 J/cal. Use ;P = i o A brass rod of mass m = 4.25 kg and a cross sectional area 5 cm in creases its length by 0.3 mm upon heatingfrom0C. What amount ofheat is spent for heating the rod? The coefficient of linear expansic 1 for brass is 2xl0 /K, its specific heat is 0.39 kJ/kg.K and the density of brass is 8.5 x 10 kg/m . A subm arine made of steel weighing 10 g has to take 10 g of water in order to submerge when the temperature of the sea is 10C. How much less water it will have to take in when the sea is at 15C? (Coefficient of cubic expansion of sea water = 2 x 10 "VC, coefficient of linear expansion of steel = 1.2 x 10- /C) A flow calorimeter i s used to measure the specific heat of a liquid. Heat is added at a known rate t o a stream of the liquid as it passes through the calorimeter at a known rate. T hen a measurement of the resulting temperature difference between the inflow and the outflow points of the liquid stream enables us to compute the specific heat of the liquid. A liquid of density 0.2 g/cm flows through a calorimeter at the rate of 10 cm /s. Heat is added by means of a 250-W electric heating coil, and a temperature difference of 25 C is established in steady-state conditions between

the inflow and the outflow points. Find the specific heat of the liquid. 2 -5 3 3 9 8 5 3 3 (!lBansalClasses Calorimetry & Heat Transfer [3]

Q.8 Toluene liquid of volume 300 cm at 0C is contained in a beaker an another quantit y of toluene of volume 110 cm at 100C is in another beaker. (The combined volume is 410 cm ). Determine the total volume of the mixture ofthe toluene liquids whe n they are mixed together. Given the coefficient of volume expansion y = 0.001/C and all forms of heat losses can be ignored. Also find the final temperature of the mixture. Q. 9 Ice at -20C isfilledupto height h = 10 cm in a uniform cylindr ical vessel. Water at temperature 9C is filled in another identical vessel upto t he same height h= 10 cm. Now, water from second vessel is poured into first vess el and it is found that level of upper surface falls through Ah = 0. 5 cm when t hermal equilibrium is reached. Neglecting thermal capacity of vessels, change in density of water due to change in temperature and loss of heat due to radiation , calculate initial temperature 0 of water. Given, Density of water, p = 1 gm cm Density of ice, p. =0.9gm/cm Specific heat of water, s = 1 cal/gm C Specific hea t of ice, s = 0.5 cal/gmC Specific latent heat of ice, L = 80 cal/gm Q. 10 A comp osite body consists of two rectangular plates of the same dimensions but differe nt thermal conductivities K and Kg. This body is used to transfer heat between t wo objects maintained at different temperatures. The composite body can be place d such that flow of heat takes place either parallel to the interface or perpend icular to it. Calculate the effective thermal conductivities K. and Kj Of the co mposite body for the parallel and perpendicular orientations. Which orientation will have more thermal conductivity? 3 3 3 w -3 3 w ; A Q. 11 Two identical thermally insulated vessels, each containing n mole of an id eal monatomic gas, are interconnected by a rod of length I and cross-sectional a rea A. Material of the rod has thermal conductivity K and its lateral surface is thermally insulated. If, at initial moment (t = 0), temperature of gas in two v essels is T, and T (< T ), neglecting thermal capacity of the rod, calculate dif ference between temperature of gas in two vessels as a function of time. 2 } Q. 12 A highly conducting solid cylinder of radius a and length I is surrounded by a co-axial layer of a material having thermal conductivity K and negligible h eat capacity. Temperature of surrounding space (out side the layer) is T , which is higher than temperature of the cylinder. If heat capacity per unit volume of cylinder material is s and outer radius of the layer is b, calculate time requi red to increase temperature of the cylinder from T to T Assume end faces to be t hermally insulated. 0 t r Q. 13 A vertical brick duct(tube) is filled with cast iron. The lower end of the duct is maintained at a temperature T, which is greater than the melting point T of cast iron and the upper end at a temperature T which is less than the tempe rature ofthe melting point of cast iron. It is given that the conductivity of li quid cast iron is equal to k times the conductivity of solid cast iron. Determin e the fraction ofthe duct filled with molten metal. Q.14 Water is filled in a no n-conducting cylindrical vessel of uniform cross-sectional area. Height of water column is h and temperature is 0C. Ifthe vessel is exposed to an atmosphere havi ng constant temperature of- 0C (< 0C) at t = 0, calculate total height h ofthe col umn at time t .Assume thermal conductivity ofice to be equal to K.Density ofwate r is p and that of ice is p.. Latent heat offusion ofice isL. m 2 0 ffi (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.15 A lagged stick of cross section area 1 cm and length 1 m is initially at a temperature of 0C. It is then kept between 2 reservoirs of tempeature 100C and 0C. Specific heat capacity is 10 J/kgC and linear mass density is 2 kg/m. Find 100C oc (a) temperature gradient along the rod in steady state. (b) total heat absorbed by the rod to reach steady state. Q.16 A cylindrical block of length 0.4 m an ar ea of cross-section 0.04m is placed coaxially on a thin metal disc ofmass 0.4 kg and ofthe same cross-section. The upper face of the cylinder is maintained at a constant temperature of 400K and the initial temperature of the disc is 300K. I f the thermal conductivity of the material of the cylinder is 10 watt/m-K and th e specific heat of the material of the disc in 600 J/kg-K, how long will it take for the temperature of the disc to increase to 350K? Assume, for purposes of ca lculation, the thermal conductivity of the disc to be very high and the system t o be thermally insulated except for the upper face of the cylinder. 2 2 Q.17 A copper calorimeter of negligible thermal capacity isfilledwith a liquid. The mass of the liquid equals 250 gm. A heating element of negligible thermal ca pacity is immersed in the liquid. It is found that the temperature of the calori meter and its contents risesfrom25C to 30C in 5 minutes when a or rent of 20.5 amp ere is passed through it at potential difference of 5 volts. The liquid is throw n off and the heater is again switched on. It is now found that the temperature ofthe calorimeter alone is constantly maintained at 32C when the current through the heater is 7A at the potential difference 6 volts. Calculate the specific hea t capacity ofthe liquid. The temperature ofthe surroundings is 25C. Q.18 A solid copper sphere cools at the rate of 2.8C per minute, when its temperature is 127C. Find the rate at which another solid copper sphere oftwice the radius lose its t emperature at 327C, ifin both the cases, the room temperature is maintained at 27C . Q.19 A calorimeter contains 100 cm of a liquid of density 0.88 g/cm in which a re immersed a thermometer and a small heating coil. The effective water equivale nt of calorimeter, thermometer and heater may be taken to be 13 gm. Current of 2 A is passed through the coil. The potential difference across the coil is 6.3 V and the ultimate steady state temperature is 55C. The current is increased so th at the temperature rises slightly above 55C, and then it is switched off. The cal orimeter and the content are found to cool at the rate of 3.6C/min. (a) Find the specific heat of the liquid. (b) The room temperature during the experiment was 10C. If the room temperature rises to 26C, find the current required to keep the l iquid at 55C. You may assume that Newton's law is obeyed and the resistance of th e heater remains constant. 3 3 Q.20 End A of a rod AB of length L = 0.5 m and of uniform cross-sectional area i s maintained at some constant temperature. The heat conductivity of the rod is k = 17 J/s-rnK. The other end B of this rod is radiating energy into vacuum and th e wavelength with maximum energy density emitted from this end is XQ = 75000 A. If the emissivity of the end B is e = 1, determine the temperature of the end A. Assuming that except the ends, the rod is thermally insulated. Q.21 A wire of l ength 1.0 m and radius 10" m is carrying a heavy current and is assumed to radia te as a blackbody. At equilibrium temperature of wire is 900 K while that of the surroundings is 300 K. The resistivity of the material of the wire at 300 K is n x 10" O-m and its temperature coefficient of resistance is 7.8 x 10' /C. Find t he current in the wire, [a = 5.68 x 10" w/m K ]. 3 2 8 3 8 2 4 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.22 The temperature distribution of solar radiation is more or less same as tha t of a black body whose maximum emission corresponds to the wavelength 0.483 jam . Find the rate of change of mass due to radiation. [Radius of Sun = 7.0 x 10 m] 8 Q.23 A black plane surface at a constant high temperature T , is parallel to ano ther black plane surface at constant lower temperature T . Between the plates is vacuum. In order to reduce the heatflowdue to radiation, a heat shield consisti ng oftwo thin black plates, thermally isolated from each other, it placed betwee n the warm and the cold surfaces and parallel to these. After some time stationa ry conditions are obtained. By what factor r) is the stationary heatflowreduced due to the presence of the heat shield? Neglect end effects due to thefinitesize of the surfaces. h ; Q.24 The shell of a space station is a blackened sphere in which a temperature T = 500K is maintained due to operation of appliances of the station. Find the te mperature of the shell if the station is enveloped by a thin spherical black scr een of nearly the same radius as the radius of the shell. Blackened envelop Q.25 A liquid takes 5 minutes to coolfrom80C to 50C. How much time will it take to coolfrom60C to 30C ? The temperature of surrounding is 20C. Use exact method. Q .2 6 Find the temperature of equilibrium of a perfectly black disc exposed normally to the Sun's ray on the surface of Earth. Imagine that it has a nonconducting b acking so that it can radiate only to hemisphere of space. Assume temperature of surface of Sun = 6200 K, radius of sun = 6.9 * 10 m, distance between the Sun a nd the Earth = 1.5 x lo m. Stefan's constant = 5.7 x i0~ W/m .K . What will be t he temperature ifboth sides of the disc are radiate? s 11 s 2 4 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q. 1 Q.2 The temperature of 100 gm of water is to be raised from 24 C to 90 C by adding ste am to it. Calculate the mass of the steam required for this purpose. [JEE '96] T wo metal cubes A & B of same size are arranged as shown in figure. The extreme e nds of the combination are maintained at the indicated temperatures. The arrange ment is thermally insulated. The coefficients of thermal conductivity of A & B a re 300 W/mC and 200 W/mC respectively. After steady state is reached the temperatu re T of the interface will be . [JEE' 96] 2 EXERCISE - III o A B Q.3 A double pane window used for insulating a room thermally from outside consists of two glass sheets each of area 1 m and thickness 0.01 m separated by a 0.05m t hick stagnant air space. In the steady state, the room glass interface and the g lass outdoor interface are at constant temperatures of 27C and 0C respectively. Ca lculate the rate of heat flow through the window pane. Also find the temperature s of other interfaces. Given thermal conductivities of glass and air as 0.8 and 0.08 W nr'K- respectively. [JEE'97] 1 Q. 4 The apparatus shown in the figure consists of four glass columns connected by ho rizontal sections. The height of two central columns B & C are 49 cm each. The t wo outer columns A & D are open to the atmosphere. A & C are maintained at a tem perature of 95 C while the columns B & D are maintained at 5 C. The height of the liquid in A & D measured from the base line are 52.8 cm & 51 cm respectively. De termine the coefficient ofthermal expansion ofthe liquid, [JEE '97] A 95 C 95 Q.5 Q.6 Q.7 A spherical black body with a radius of 12 cm radiates 450 W power at 500 K . If the radius were halved and the temperature doubled, the power radiated in watt would be : (A) 225 (B) 450 (C) 900 (D) 1800 Earth receives 1400 W/m of solar pow er . If all the solar energy falling on a lens of area 0.2 m is focussed on to a block of ice of mass 280 grams, the time taken to melt the ice will be minutes. (Latent heat of fusion of ice = 3.3 x 10 J/kg) [JEE '97] 2 2 5 A solid body X of heat capacity C is kept in an atmosphere whose temperature is T = 300K. At time t = 0, the temperature of X is T = 400K. It cools according to Newton's law of cooling. At time tj its temperature is found to be 3 5 OK. At t his time t the body X is connected to a larger body Y at atmospheric temperature T , through a conducting rod of length L, cross-sectional area A and thermal co

nductivity K. The heat capacity of Y is so large that any variation in its tempe rature may be neglected. The cross-sectional area A of the connecting rod is sma ll compared to the surface area of X. Find the temperature of X at time t = 3t [ JEE' 98] A 0 p A r Q.8 A black body is at a temperature of2880 K. The energy ofradiation emitted by thi s obj ect with wavelength between 499 nm and 500 nm is U between 999 nm and 1000 nm is U and between 1499 nm and 1500nmisU . TheWienconstantb = 2.88 x 10 nmK. T hen [JEE' 98] (A) Uj = 0 (B)U = 0 (C) Uj > U (D)U >U p 2 3 6 3 2 2 1 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.9 A bimetallic strip is formed out oftwo identical strips one ofcopper and the oth er ofbrass. The coefficient of linear expansion ofthe two metals are a and ctg. On heating, the temperature ofthe strip goes up by AT and the strip bends to for m an arc of radius of curvature R. Then R is: (A) proportional at AT (B) inverse ly proportional to AT [JEE' 99] (C) proportional to lOg - a | (D) inversely prop ortional to |a - a | c c B c Q.10 A block of ice at - 10C is slowiy heated and converted to steam at 100C. Whic h of the following curves represents the phenomenon qualitatively? [JEE (Scr) 20 00] (A) Heat supplied (B) Heat supplied \ (C) Heat supplied (D) Heat supplied Q. 11 The plots of intensity versus wavelength for three black bodies at tempera ture T, , T and T, respectively are as shown. Thentemperatures are such that [JE E (Scr) 2000] (A)T >T >T (B) T j > T > T (C) T > T > T (C) T. > T > T 2 1 2 3 3 2 2 3 1 2 t Q.12 Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and rig ht ends are kept at 0C and 90C respectively. The temperature of the junction of th e three rods will be [JEE(Scr)2001 ] oc(A) 45C (B) 60C (C) 30C (D)20C ,S0C "90C Q. 13 An ideal black body at room temperature is thrown into a furnace. It is ob served that (A) initially it is the darkest body and at later times the brightes t. (B) it the darkest body at all times (C) it cannot be distinguished at all ti mes. (D) initially it is the darkest body and at later times it cannot be distin guished. [JEE(Scr)2002] Q. 14 An ice cube of mass 0.1 kg at 0C is placed in an is olated container which is at 227C. The specific heat S of the container varies wi th temperature T according the empirical relations = A + BT, where A= 100 cal/kg -K and B = 2 x 10~ cal/kg-K . If the final temperature of the container is 27C, d etermine the mass of the container. (Latent heat of fusion for water = 8 x \ o c al/kg. Specific heat of water = 103 cal/kg-K) [JEE' 2001] 2 2 4 Q.15 Two rods one of aluminium of length /, having coefficient of linear expansi on a , and other steel of length l having coefficient of linear expansion a are joined end to end. The expansion in both the a 2 s

[JEE (Scr) 2003] rods is same on variation of temperature. Then the value of , h is . n +/2 ac a0 (D) None of these (A) a + a (B) a s (C) Otc r a s a - a (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.16 2 kg ice at - 20C is mixed with 5 kg water at 20C. Thenfinalamount ofwater in the mixture would be; Given specific heat of ice = 0.5cal/gC, specific heat ofwa ter = 1 cal/gC, Latent heat of fusion of ice = 80 cal/g. [JEE (Scr) 2003] (A) 6 k g (B) 5 kg (C) 4 kg (D) 2 kg Q.17 If emissivity of bodies X and Y are e and e an d absorptive power are A and Ay then [JEF (Scr) 2003] (A) e > e ; Ay > A (B) e < e ; A < A (C)e >e ;A Kj_, K| = K 1 B 3 T A R V A B ; x EXERCISE-II Q.6 Q.9 Q.ll t m m m 9.02 x 10 gm 45C 5 \n i (T, ~T )e "3 R J 2 2 ( 4KAt N | Q.12 a s. ^log 2 (-) l0geV. 0 ~ 2 J T T Q 1 3 k(T - T ) I k(T -T ) + (T -T ) 1 Q.14 h + 0

Q.17 21000 Jkg^Kr Q.20 T = 423 K a 1 - JBL V / \ 1 \ Pi f L 12k;6t Q.15 (a) 100 C/m, (b) 1000 J Q.18 9.72C/min Q.21 36A 0 x Q.16 166.3 sec 9 1 Q.19 (a)0.42 cal/gmC, (b) 1.6A Q.23 r| = 3 Q.25 10 minutes Q.I Q.4 Q.7 12 gm e Q.22 ~dt = 5.06 x 10 kg/s Q.24 T" = 500 = 600 K Q.26 T = 420 K, T = 353.6 K Q.2 60 C EXERCISE-III Q.3 Q.6 0 41.53 Watt; 26.48 C;0.55C 5.5 min Q.14 0.5 kg Q.19 B Q.24 A 2 x 10^ C Q.5 D log 2 ; T = 300 + 50 exp. k= Q.9 B, D Q.10 Q.16 A Q.17 Q.21 C Q. 26 A Q.22 Q.27 Q.8 D Q.15 A Q.20 D Q.25 C (!l Bansal Classes [LC tj A Q.ll B Q.12 B Q.13 D A Q.18 (a) 595 watt/m , ( b ) T * 4 2 0 K K y,= 2a s Q.23 4eaLTf+K A Q.28 B 2 0 Calorimetry & Heat Transfer [3]

BA TARGET IIT JEE 2007 XII (ALL) COHTENTS KEYCONCEPTS EXERCISE-1 EXERCISE-II EXERCISE-III ANSWER KEY

KEY 1. CAPACITANCE O F A N 0 ( CONCEPTS C = 471 e e R in a medium ISOLATED SPHERICAL CONDUCTOR : C = 47C G R in air This sphere is at infinite distance from all the conductors. The Capacitance C = 4T E R exists between the surface of the sphere & earth . 7 Q It consists of tw o concentric spherical shells as shown infigure.Here capacitance of region betwe en the two shells is C and that outside the shell is C . We have 471 e ab C = an d C = 471 e b b-a Depending on connection, it may have different combinations of C, and -C . t 2 n 2 Q 2 SPHERICAL CAPACITOR : 3. PARALLEL PLATE CAPACITOR : If two parallel plates each of area A & separated by a distance d are charged wi th equal & opposite charge Q, then the system is called a parallel plate capacit or & its capacitance is given by, ^ S)6 A C = ; .in a medium C= with air as medi um r (i) UNIFORM DI-ELECTRIC M E D I U M : This result is only valid when the electricfieldbetween plates of capacitor is c onstant, (ii) M E D I U M PARTLY A I R : C = U d-lt-i r So A When a di-electric slab of thickness t & relative permittivity e is l l l l intr oduced between the plates of an air capacitor, then the distance between P3 the plates is effectively reduced by irrespective ofthe position of BSSSSii V ^rJ the di-electric slab . (iii) COMPOSITE M E D I U M : c= GA I I -rl r2 0 r3

4. CYLINDRICAL CAPACITOR : It consist oftwo co-axial cylinders ofradii a& b, the outer conductor is earthed . The di-electric constant ofthe mediumfilledin the space between the cylinder i s Farad e . The capacitance per unit length is C = 2ne-ne m in r y r (fe^Bansal Classes CAPACITANCE 121

CONCEPT o r VARIATION OF PARAMETERS: 6. e kA , ifeither ofk, A or d varies in the region between As capacitance ofa para llel plate capacitor isC = the plates, we choose a small dc in between the plate s and for total capacitance of system. dx -, If all dC's are in parallel C = } d C If all dC's are in series 1 e k(x)A(x) 0 T J 0 COMBINATION (i) OF CAPACITORS SERIES : : In this arrangement all the capacitors when uncharged get the same charge Q but the potential difference across each will differ (if the capacitance are unequal ). 1 +1 1 1 + + + 1 (ii) CAPACITORS I N rIMHh v, v, v, Q Q Q C| C2 C3 C 3 When one plate of each capacitor is connected to the positive terminal of the ba ttery & the other plate of each capacitor is connected to the negative terminals of the battery, then the capacitors are said to be in parallel connection. The capacitors have the same potential difference, V but the charge on each one is d ifferent (if the capacitors are unequal). eq. C CAPACITORS I N PARALLEL : I + C 2 + C 3 + +c

s 1 jC3,y 1 Q + v % 1Cj.V c,,v % ENERGY Capacitance C, charge Q & potential difference V; then energy stored is 1 U = -1 CV = QV = 1 Q . This energy is stored in the electrostatic field set up in the di-electric - medium between the conducting plates of the capacitor . 2 2 STORED IN A CHARGED CAPACITOR : HEAT PRODUCED IN SWITCHING IN CAPACITIVE CIRCUIT Due to charge flow always some amount of heat is produced when a switch is close d in a circuit which can be obtained by energy conservation as Heat = Work done by battery - Energy absorbed by capacitor. 9. 10 When two charged conductors of capacitance C & C at potential V & V respectively are connected by a conducting wire, the charge flows from higher potential cond uctor to lower potential conductor, until the potential of the two condensers be comes equal. The common potential (V) after sharing of charges; C,V C V q + V =n etnet charge _ C,j + q capacitance C C+C charges after sharing qj = C,'V & q = C V. In this process energy is lost in the connecting wire C C (V,-V ) as heat. T his loss of energy is U - U = ^ r ^ g s 2 } 2 2 1+ 2 2 SHARING O F CHARGES : 2 2 t 2 2 2 2 2 initial eal

250 ps, I = - 0.1 i -4000(t-250)xi (r 00t e 0.04 0.015 -o.n 6 a m p ; t(xIO^s) Q.18 400 ^ - P EXERCISE # III Q.l Q.2 Q.5 (i) 0.2 x 10" 9 8 F, 1.2 x lO" J ; (ii) 4.84 x 10" J ; (iii) 1.1 x 10" 5 5 5 J 4.425 x 10~ Ampere QA = 90 Q.3 B q.4 F C K ^ /n K, (Ka-KO K, = 18 MJ pC, Q B = 150 pC, Q C = 210

pC, UJ = 4 7 . 4 MJ, U D Q= 0 CVR, Q ' 6 2^9A V48 C & Q.7 Q.10 Q.8 Ri+R2 ^ 2 e0 2 s0 Q.9 R1+R2 anda= Q.ll

XII (ALL) quesjjommm. R ifE,=E (B)C R.C, (D) f Q. 15 Four capacitors and a batteiy are connected as shown. The potential drop a cross the 7 pF capacitor is 6 V. Then the : J (A) potential difference across th e 3 pF capacitor is 10 V (B) charge on the 3 pF capacitor is 42 pC (C) e.m.f. of the battery is 3 0 V (D) potential difference across the 12 pF capacitor is 10 V. A H 3.9(.IF Jn 7F "puF 2 Q. 16 A circuit shown in the figure consists of a battery of emf 10 V and two ca pacitance C, and C of capacitances 1.0 pF and 2.0 pF respectively. The potential difference V - V is 5 V (A) charge on capacitor Cj is equal to charge on capaci tor C Ao| || | | | o B (B) Voltage across capacitor Cj is 5V. c' e q, (C) Voltage across capacitor C is 10 V (D) Energy stored in capacitor C. is two times the en ergy stored in capacitor C . B 2 2 2 Q.17 A capacitor C is charged to a potential difference V and batteiy is disconn ected. Now if the capacitor plates are brought close slowly by some di stance: ( A) some +ve work is done by external agent (B) energy of capacitor will decrease (C) energy of capacitor will increase (D) none of the above (fe Bansal Classes Question Bank on Capacitance [13]

Q.18 The capacitance of a parallel plate capacitor is C when the region between the plate has air. This region is nowfilledwith a dielectric slab of dielectric constant k. The capacitor is connected to a cell of emf E, and the slab is taken out (A) charge CE(k - 1 ) flows through the cell (B) energy E C(k - 1) is absor bed by the cell. (C) the energy stored in the capacitor is reduced by E C(k - 1 ) (D) the external agent has to do ^E C(k -1) amount ofwork to take the slab out . 2 2 2 Q.19 Two capacitors of capacitances 1 pF and 3 pF are charged to the same voltag es 5 V. They are connected in parallel with oppositely charged plates connected together. Then: (A) Final common voltage will be 5 V (B) Final common voltage wi ll be 2.5 V (C) Heat produced in the circuit will be zero. (D) Heat produced in the circuit will be 37.5 pJ Q. 20 The two plates X and Y of a parallel plate cap acitor of capacitance C are given a charge of amount Q each. X is now joined to the positive terminal and Yto the negative terminal of a cell of emfE = Q/C. (A) Charge of amount Q willflowfromthe negative terminal to the positive terminal o fthe cell inside it (B) The total charge on the plate X will be 2Q. (C) The tota l charge on the plate Y will be zero. (D) The cell will supply CE amount of ener gy. 2 Q.21 A dielectric slab is inserted between the plates of an isolated charged cap acitor. Which of the following quantities will remain the same? (A) the electric fieldin the capacitor (B) the charge on the capacitor (C) the potential differen ce between the plates (D) the stored energy in the capacitor. Q.22 The separatio n between the plates of a isolated charged parallel plate capacitor is increased . Which of the following quantities will change? (A) charge on the capacitor (B) potential difference across the capacitor (C) energy of the capacitor (D) energ y density between the plates. Q.23 Each plate ofa parallel plate capacitor has a charge q on it. The capacitor is now connected to a battery. Now, (A) the facin g surfaces of the capacitor have equal and opposite charges. (B) the two plates of the capacitor have equal and opposite charges. (C) the battery supplies equal and opposite charges to the two plates. (D) the outer surfaces ofthe plates hav e equal charges. Q. 24 Following operations can be performed on a capacitor: X connect the capacitor to a battery of emf E. Y - disconnect the battery Z - rec onnect the battery with polarity reversed. W - insert a dielectric slab in the c apacitor (A) In XYZ (perform X, then Y, then Z) the stored electric energy remai ns unchanged and no thermal energy is developed. (B) The charge appearing on the capacitor is greater after the action XWY than after the action XYW. (C) The el ectric energy stored in the capacitor is greater after the action WXY than after the action XYW. (D) The electricfieldin the capacitor after the action XW is th e same as that after WX. Q.25 A parallel plate capacitor is charged and then dis connectedfromthe source of potential difference. Ifthe plates of the condenser a re then moved farther apart by the use of insulated handle, which one of the fol lowing is true? (A) the charge on the capacitor increases (B) the charge on the capacitor decreases (C) the capacitance of the capacitor increases (D) the poten tial difference across the plate increases (fe Bansal Classes Question Bank on Capacitance [13]

Q.26 Aparallel plate capacitor is charged and then disconnected from the source steady E.M.F. The plates are then drawn apart farther. Again it is connected to the same source. Then: (A) the potential difference across the plate increases, while the plates are being drawn apart. (B) the charge from the capacitorflowsin to the source, when the capacitor is reconnected. (C) more charge is drawn to th e capacitor from the source, during the reconnection. (D) the electric intensity between the plates remains constant during the drawing apart of plates. Q.27 Wh en a parallel plates capacitor is connected to a source of constant potential di fference, (A) all the charge drawnfromthe source is stored in the capacitor. (B) all the energy drawnfromthe source is stored in the capacitor. (C) the potentia l difference across the capacitor grows very rapidly initially and this rate dec reases to zero eventually. (D) the capacity of the capacitor increases with the increase of the charge in the capacitor. Q.28 When two identical capacitors are charged individually to different potentials and connected parallel to each othe r, after disconnecting themfromthe source: (A) net charge on connected plates is less than the sum of initial individual charges. (B) net charge on connected pl ates equals the sum of initial charges. (C) the net potential difference across them is differentfromthe sum ofthe individual initial potential differences. (D) the net energy stored in the two capacitors is less than the sum ofthe initial individual energies. Q. 29 Aparallel plate capacitor of plate area A and plate s eperation d is charged to potential difference V and then the battery is disconn ected. A slab of dielectric constant K is then inserted between the plates ofthe capacitor so as tofillthe space between the plates. If Q, E and W denote respec tively, the magnitude of charge on each plate, the electricfieldbetween the plat es (after the slab is inserted) and the work done on the system, in question, in the process of inserting the slab, then e AV s KAV V AV 1 - 1 K Q. 3 0 A parall el plate capacitor is connected to a battery. The quantities charge, voltage, el ectricfieldand energy associated with the capacitor are given by Q , V , E and U respectively. A dielectric slab is introduced between plates of capacitor but b attery is still in connection. The corresponding quantities now given by Q, V, E and U related to previous ones are (A)Q>Q (B) V > V (C) E > E (D)U 6X0 ~ + 10Cr + 171^0 3 X ^ 3 + 4Cr 0 " + 26H -> 6X0 - + 8Cr + 13H 0 2 2 7 4 4 2 7 2 + 4 3+ 2 7 2 + 4 3+ 2 Q.19 Near Mount Kailash is the sacred lake, Mansorvar. In the crystal clear wate r of the lake, things at the bottom of the lake are also clearly visible. On a h ot sunny day, when the temperature at the surface is 27C an algae at the bottom o fthe lake produces a 25 ml bubble ofpure oxygen. As the bubble rises to the top, it gets saturated with the water vapours and has a volume of 100 ml of the surf ace. The pressure at the surface is 720 mm Hg. Ifthe depth ofthe lake is 27.2 m, findthe temperature at the bottom of the lake. Vapour pressure of water at 27C is 20 mm Hg. dj^ci = 1 gm/ml, d = 13.6 g/ml. Hg Q.20 A beam of light Ijas three X, 4144 A, 4972 A and 6216 A with a total intens ity of 3.6 x 10~ Wnr equally distributed amongst the three X. The beam falls nor mally on an area 1.0 cm of a clean metallic surface ofwork function 2.3 eV Assum e that there is no loss oflight by reflection etc. Calculate the no. of photoele ctrons emitted in 2 sec. 3 2 2 J E E Humour. A Physics teacher, a Maths teacher and a Chemistry teacher were wa lking on a sea shore. Fascinated by sea waves the physics teacher said, "I want to study the wave nature of sea waves" and went into the sea and never returned back. The maths teacher said, "I want to measure the volume of sea water" and we nt into the Sea and never returned back. The chemistry teacher concluded "Both p hysics and maths teacher are soluble in sea water under condition of 1 atm and 2 98 K. ^Bansal Classes RAkslia Bandhan Holidays Assignment [3]

ANSWER KEY soxbj^61 VZ, z,l'b SITTING-I e/oos orb pneero 6ib o1ho 9ib %z,-0i x 8'i srb I r(i-u )+ii z p= i Yn A 1a A3 E J 0 . M' b Z 8 (p) V 8 I'D d ,[X)(l-u)+l] = 0 erb 0 ld a z/b g b xua Z6 6 8 = A ' = %'l = Tu nt) - r /96 E A 0 = 3X 'A3 9'I = 'A9 9'l~ = H lib 9 d ' raro wrzz orb 6b ss'o'ao 8 b a 9b a sb a Kb q (p) vc>) 'a(q)'a00 rb o rb SITTIN GII 9'862 x n orb V sASgi-oixzrt? z,rb 6 8I"b 6 S/Da01x8> 7 v srb r V H'6 rat>-0l x t l'b > ? X 0 1 '0i7 '0 '0 '09 'SL ' 0 '081' % SZ 0lbJxyu^v 0 0 T-x D V 9"b m ). 2 Ik A spring of mass m is pulled such that a given instant,, velocity of both of its end is v in the opposite direction. Find the kinetic energy ofthe spring. A par ticle of mass 3 kg is rotating in a circle ofradius 1 m such that the angle rota ted by its radius is given by 0 = 3 (t + sint). Find the net force acting on the particle when t = n/2. For a particle rotating in a vertical circle with unifor m speed, the maximum and minimum tension in the string are in the ratio 5 :3. If the radius of vertical circle is 2m, then find the speed of revolving body. Q.10 Two strings of length /=0.5 m each are connected to a block of mass m=2 at one end and their ends are attached to the point A and B 0.5 m apart on a tical pole which rotates with a constant angular velocity co=7 rad/sec. Find ratio T, oftension in the upper string (T,) and the lower string (T ). [Use 9.8 m/s ] 2 2 0.5 Q.ll A force F = -k(x i + y j) [where k is a positive constant] acts on a partic le moving in the x-y plane. Startingfromorigin, the particle is taken to (a, a) and then to (a/V2,o). Find the total work done by the force F on the particle. u , it does not slide on the hemisphere (i.e. leaves the surface at the top itself). O (a) For u = 2u , it lands at point P o n ground Find OP. (b) For u = u /3, Find the height from the ground at which it leaves the hemisphere. (c) Find its net acceleration at the instant it leaves th e hemisphere. Q.8 The track in Fig is straight in the horizontal section AB and is a semicircle of radius R in the vertical part BCD. A particle of mass m is gi ven a velocity of /(22gR)/5 to the left along the track. The particle moves up t he vertical section JZL and ultimately loses contact with it. How far from point B will the mass land. Q.9 A small particle of mass 1 kg slides without friction from height H=45 cm shown in figure and then loops the vertical loop of radius R from where a section of angle 6 = 60 has been removed. Find R such that after l osing contact at A and flying through the air, the particle will reach at the po int B. Also find the normal reaction between particle and path at A. Q.10 A ring of mass m slides on a smooth vertical rod. A light string is attached to the ri ng and is passing over a smooth peg distant a from the rod, and at the other end ofthe string is a mass M (> m). The ring is held on a level with the peg and re leased: Show that it first comes to rest after falling a distance: =0 2mMa M Q 0 0 A M Q.5 Q.ll Ablock ofmass m is held at rest ona smooth horizontal floor. Alight fiictio nless, small pulley isfixedat aheight of 6 mfromthe floor. Alight inextensible s tring of length 16 m, connected with Apasses over the pulley and another identic al block B is hungfromthe string. Initial height of B is 5mfromthe floor as 6m s hown in Fig. When the system is releasedfromrest, B starts to move vertically do wnwards and A slides on the floor towards right. (i) Ifat an instant string make s an angle 0 with horizontal, calculate relation between velocity u ofA and v of B Calculate v when B strikes the floor. M 2 -m2 l i l t 7 777 0, the functional form of the potential energy U (x) of the particle is [JEE (Scr.)'2002] 2 U(x) U(x) U(x)f X U(x) X (A) (B) (C) (D) X Q.13 An ideal spring with spring-constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring in itially unstretched. Then the maximum extension in the spring is [JEE (Scr.)'200 2] (A) 4 Mg/k (B) 2 Mg/k (C)Mg/k (D)Mg/2k Q.14 A spherical ball of mass m is kep t at the highest point in the space between two fixed, concentric spheres Aand B (see figure). The smaller sphere A has a radius R and the space between the two spheres has a width d. The ball has a diameter veiy Sphere B slightly less than d. All surfaces are frictionless. The ball is given a gentle push (towards the right in the figure). The angle made by the radius vector ofthe ball with Sphere A the upward vertical is denoted by 9 (shown in the figure). [JEE' 2002] (a) Ex press the total normal reaction force exerted by the spheres on the ball as a fu nction of angle 9. (b) Let N and N denote the magnitudes of the normal reaction force on the ball exerted by the spheres A and B, respectively. Sketch the varia tions of N and N as functions of cos0 in the range 0 < 9 < T by T drawing two se parate graphs in your answer book, taking cos9 on the horizontal axes. Q.15 In a region of only gravitational field of mass 'M' a particle is shifted from Ato B via three different paths in the figure. The work done in different paths are W ,, W , W respectively then [JEE (Scr.)'2003] (A) W, = W = W (B) W, = w > w (C) W j > W~ > w (D) Wi < W < W Q.16 A particle ofmass m, moving in a circular path of radius R with a constant V2 V L speed v is located at point (2R, 0) at time t =

0 and a man starts moving with a velocity v, along the +ve y-axisfromorigin at t ime t=0. (0,0) Calculate the linear momentum ofthe particle w.r.t. the man as a function oftime. [JEE 2003] Q.17 A particle is placed at the origin and a force F = kx is acting on it (where k is a positive constant). If U(0)=0, the graph of U(x) versus x will be (where U is the potential energy function) a B A B 2 3 2 3 2 3 3 2 3 2 U(x) U(x) U(x) U(x) (A) (B) (C) (D) [JEE' 2004(Scr)] c = M Momentum : The total momentum of a system of particles is p = Mv Kinetic Energy: The kinetic energy of a system ofparticles c onsisits of two parts. K = K + K' 1 2 where K - Mv , kinetic energy due to motio n of c.m. relative to the fixed origin O, c a c c c c 5. 2s,(c) ^28565 ~ 169,256 m/s (d) 44rad Q.24 0.1875 Q.25 P 2 2 2 T^. -+-> 2 4 2V2V 71R 4^5 m/s 2m g k Q.10 9 Q.14 ^2g rad/s Q.18 m i IS Q.12 (l-V3/2)mg Q.16 6mg Q.20 2 sec 1 Q.19 - j T rad/s EXERCISE-II Q.l QF = -3ax + b, x 2 , KE 2 2b b 3V3 Q.2 2 m/s Q.3

v = v , 57ia/v 0 0 N=^^ /ra) -l g Q.5 Q.6 (i) ^,(ii)2V^g, 2a 500N/m : Q.7 40 (a) V2 r, (b) h = 2 19r , (c) g Q.8 N 1.19R Q.9 V 2 R=0.2m, ION R(vt-R)v.1/2 (2Rt-vt ) 2 3/2 Q.ll u = vsec9, v A/41 m/s Q.13 up, 10cm N Q.l Q.3 Q.8 mg max R Q.12 a = (2Rt-vt ) ' EXERCISEIII Q.14 9 =7r/2, T=mg(3sin9+3cos9-2) Q.15 4, -J^fis 25 Q.16 24 > Vo> , a=5V3 g/8, N=3mg/8 if C Q.2 Q.5 (i)36N,(ii) 11.66rad/sec,(iii) 0.1m, 0.2m D Q.6 V u = - J g | 3 2 3 y Q.7 Q.13 B Q.12 D L + 2 F=-8mgi-mgj, h=3R Q.4 A Q.9 A A Q.10 5.79 m/'s Q.ll C is 20 ohm. The ammeter reading is 0.10 Amp and voltmeter reading is 12 volt. Q Then R is equal to (A) 1 22 O (B) 140 O (C) 116 O (D)1000 Q 52 By error, a student places moving-coil vol tmeter V (nearly ideal) in series with the resistance in a circuit in order to r ead the current, as shown. The voltmeter reading will be (A) 0 (B)4V (C)6V (D) 1 2V E = 12V, R = 2Q 4FI Q.53 In a balanced wheat stone bridge, current in the galvanometer is zero. It r emains zero when; [1] battery emf is increased [2] all resistances are increased by 10 ohms \ [3 ] all resistances are madefivetimes [4] the battery and the gal vanometer are interchanged (A) only [ 1 ] is correct (B) [ 1 ], [2] and [3 ] are correct (C) [ 1 ], [3] and [4] are correct (D) [1] and [3] are correct 6Q current flows through the branch CF, then answer the f 1A following questi ons H G F E Q.21 The current through (A) branch DE is 1A (B) branch BC is 2A (C) branch BG is 4A (D) branch HG is 6 A Q.22 The emfE ofthe batteiy is (C) 18V (D) 6V (A) 24 V (B) 12 V If a zero resistance -wire is connected in parallel to bran ch CF Q.23 The current through (B) branch BC is zero (A) branch DE is zero (D) b ranch AB is 1.5 A (C)branchBGis0.5A Q.24 The emfE of the battery is (D) 10.5V (E ) 12V (C) 5.25 V (A) 9V (B) 6.6V Question No. 21 to 24 (4 questions) T 11 Inside a super conducting ring six identical resistors each of resistance R are connected as shown in figure. Q.25 The equivalent resistance(s) (A) between 1 & 3 is zero. (B) between 1 & 3 is R/2 (C) between 1 & 2, 2 & 3, 3 & 1 are all equa l. (D) between 1 & 3 is two times that between 1 & 2. Q.26 The equivalent resist ance(s) (A) between 0 & 1 is R. (B) between 0 & 1 is R/3 (C) between 0 & 1 is ze ro. (D) between 0 & 1, 0 & 2 and 0 & 3 are all equal. Q.27 Imagine a battery of emf E between the point 0 and 1, with its positive terminal connected with O. (A ) The current entering at O is equally divided into three resistances. (B) the c urrent in the other three resistances R , R , R^ is zero. (C) The resistances R^ and R^ have equal magnitudes of current while the resistance Rq, have different current. (D) Potential V = V >V,. 12 13 2 3 Question No. 25 to 27 (3 questions) The figure shows a tetrahedron, each side ofwhich has a resistance r Q.28 Choose the correct statements) related to the resistance between any two points. ( ) AB ( ) AB A R B R = = R R Question No. 28 to 30 (3 questions) (C) R is the least c d BD AC = = R R BC AD BC = = R R CD BD R = = R R CA AD BC * ^ D

= R R ( ) AB D R = R AC = R A N D CD = AD = R BD ->R Q.30 If a battery is connected between any two points ofthe tetrahedron, t hen identify the correct statement(s). (A) The potentials of the other two point s are always equal. (B) There always exists a branch through which no current fl ows. (C) The current coming out ofthe battery in each case is same. (D) None oft hese 40/\4fi Question No, 31 to 33 (3 questions) A ^ tAC The givenfigureshows a network of resistances and a battery. Q.31 Identify the correct statements) E=!2 V (A) The circuit satisfies the condition of a balanced Wheatstone bridge. (B) V - V - 0 (C) V - V = 8 (D) no currentflowsin the branch BD B D b d Q.32 Which ofthe two batteries is getting charged? (A) 8V battery (B) 12 ry (C) none Q.33 Choose the correct statement(s). (A) The current coming he 8V battery is 2A (B) The current coming out of the 12V battery is 3 A currentflowingin all the 4 0 branches is same. (D) The currentflowingin gonally opposite branches is same (D) can't be said > 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. il.Bansal Classes SOURCES O F POTENTIAL DIFFERENCE & ELECTROMOTIVE FORCE : Current Electricity [5]

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. KRICHHOFF'S LAW'S :

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\fyWvA-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 -^1wU 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 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 2 2 H 2

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 II 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 _ wwhf, . 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 il.Bansal Classes

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 52 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

[5]

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'VWVI 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) 11 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 rence across the resistance of400 ohm, as will be measured by the voltmeter V of resistance 400 ohm, either by applying Kirchhoffs rules or otherwise. [JEE'96, 6] 4000 -VvVv100Q 100Q 2001 rwv-WAVivwv-h -Wr 100Q MV O Q.2(i) A steady current flows in a metallic conductor ofnonuniform cross-section . The quantity /quantities constant along the length ofthe conductor is / are: [ JEE' 97,1 +2+5] (A) current, electricfieldand drift speed (B) drift speed only ( C) current and drift speed (D) current only (ii) The dimension of electricity co nductivity is . (iii) Find the emf (E) & internal resistance (r) ofa single batt ery which is equivalent to a parallel combination oftwo batteries ofemfs V, &V & internal resistances r. & r respectively with their similar polarity connected to each other ^Wr-Wr-rW, Q.3 In the circuit shown in thefigure,the current throu gh: (A) the 3fi resistor is 0.50 A (B) the 3Q resistor is 0,25 A ^yL sq| 40(C) 4 Q resistor is 0.50 A (D) the 4Q resistor is 0.25 A 20 21 2SI M/W^wM-VM [JEE'98,2 ] 2 2 Q.4 In the circuit shown, P # R, the reading ofthe galvanometer is same with switch S open or closed. Then L-VWV L-(g> (A)I = I (B) I = I (C)I = I (D)I = I [JEE'99,2 ] r 0 p G Q G Q r r-Wv p Q Wr K I; Q. 5 The effective resistance between the points P and Q of the electrical circuit sh own in thefigureis (A)2Rr/ (R+r) (B) 8R(R+r)/(3R+r) (C)2r + 4R (D) 5 R/2 + 2r [J EE 2002 (Scr), 3] 2 3 2R WW-^-WA:2R t2R 2R -Wv -VWvf-AMA 2R VM2R

Q.6 A100 W bulb Bj, and two 60 W bulbs B and B , are connected to a 250 V source, as shown in the figure. Now W W and W are the output powers ofthe bulbs B,,B and B respectively. Then (A) W > W = W (B) W, > W > W (C)Wj < w = w (D) Wj P >P >P (B)P >P >P >Pj (C)P >P >P >Pj (D)P >P >P >P Q.9 In the given circu it, no current is passing through the galvanometer. If the cross-sectional diame ter of AB is doubled then for null point of galvanometer the value ofAC would [J EE' 2003 (Scr)] (A)x (B)x/2 (C)2x (D) None Q.10 How a battery is to be connected so that shown rheostat will behave like a potential divider? Also indicate the points about which output can betaken. [JEE '2003] Q.ll Six equal resistances are connected between points P, Q and R as sho wn in thefigure.Then the net resistance will be maximum between (A) P and Q (B) Q and R (C) P and R (D) any two points [JEE' 2004 (Scr)] AWv*B r c Q Q.12 In an RC circuit while charging, the graph of In I versus time is as shown by the dotted line in the adjoining diagram where I is the current. When the val ue ofthe resistance is doubled, which of the solid curves best represents the va riation of In I versus time? [JEE' 2004 (Scr)] (A)P (B)Q (C)R (D)S " M 1 -R -s "Q p il.BansalClasses Current Electricity [5]

Q.13 For the post office box arrangement to determine the value ofunknown resist ance, the unknown resistance should be connected between [JEE' 2004 (Scr)] (A) B and C (B)CandD (C) A and D (D)B andC 1 1 ooo*oogTi 'jaTotoo o o fESuSsjEEOQi 3 Q. 14 Draw the circuit for experimental verification of Ohm's law using a source ofvariable D.C. voltage, a main resistance of 100 O, two galvanometers and two resistances ofvalues 10 Q and 10* O respectively. Clearly show the positions oft he voltmeter and the ammeter. [JEE' 2004] 6 Q.15 In thefigureshown the current through 2Q resistor is (A) 2 A (B) OA (C) 4 A (D) 6 A , 10V f50 VWv 10Q 2fJ Wr 20V [JEE' 2005 (Scr)] Q.16 An uncharged capacitor of capacitance 4pF, a battery of emf 12 volt and a r esistor of 2.5 MO are connected in series. The time after which v = 3v is (take /n2 = 0.693) (A) 6.93 sec. (B) 13.86 sec. (C) 20.52 sec, (D) none of these [JEE' 2005 (Scr)] c R Q.17 A galvanometer has resistance 100Q and it requires current lOOpAforfull sca le deflection. Aresistor 0. ID is connected to make it an ammeter. The smallest current required in the circuit to produce the full scale deflection is (A) 1000 .1mA (B) 1.1mA (C) 10.1mA (D) 100.1mA [JEE' 2005 (Scr)] Q.18 An unknown resistan ce X is to be determined using resistances R,, R or R,. Their corresponding null points are A, B and C. Find which of the above will give the most accurate read ing and why? [JEE 2005] 2 1| VWv -sC or R 3 A B R=R, or R 2 Q.19 Consider a cylindrical element as shown in thefigure.Current , flowing the through element is I and resistivity ofmaterial ofthe 4 cylinder is p. Choose th e correct option out * the following. (A) Power loss in second half is four time s the power loss infirsthalf. (B) Voltage drop infirsthalfis twice ofvoltage dro p in second half. (C) Current density in both halves are equal. (D) Electricfiel din both halves is equal.

r B $2r 1/2 1/2 [JEE 2006] il.Bansal Classes Current Electricity [5]

ANSWER KEY Q I in 12A-20W 4Q 9n 20 ohm Q.2 Q.5 Q.9 Q.13 Q.17 Q.l Q.4 22 I = 2.5 A V = 3.5 Volts Q.4 ^n 3r Q.7 8/7R Q.8 Q.6 V i 2 Q.10 lQ Q.ll 600n Q. 12 I V 10 Q.16 y n , 5 n Q.14 46.67 cm Q.15 4 ohm Q.18 7.5 m, 8.75m, 6.25m Q.19 Rj = 0.0278 n , R2 = 0.25 n , R = 2.5 n 3.5 A R EXERCISE # I Q.3 R R _3 Q.3 R! 5 (3/1 l)a Q.2 (a)J A/3;(b)2J A/3 Q.5 7/5 times the length of any si de of the square (i)10.52n;(u)0.3125n (2 + 7i)ar Q.6 (i) (k - l ) ' ( kk- l )(ii ) ((k-l)/k )v Q7 R 8 Q.8 0.3n Q.9 4/9 kg/sec., 450 sec Q.10 (i) V = - 12 V, (ii) 3 amp from b to a Q.ll II + 6V2 < ^ V Af - ^ Q.12 R PoL . - I ; i = ;v = V" (e" 1 - e -1 ) A Po e - l , Q.13 r + R ( f - f ) ' m a f = 0 , l ; I f = l / 2 Q.14 7.2m Q.15 (a) 5/7R, (b) 9R/14 22 Q eff ^ Q.19 1A Q.20 233.3n; 144V Q.21 (a) 6m, (b) i n Q.17 - V Q.22 (a) 1.01 W, (b) 0-5A 0-10V, (c) 0.05 A 2 0 0 2 2 0 k w 3 ab n 0 e EXERCISE # II 3 L v 0 2 f o r I X m in 1 8 a = a EXERCISE # III Vir +V r! l 2 2 2 r + r

Q. 1 20/3 V Q-4 Q.2 (i) D; (ii) M L~ T A ; (iii) Q.5 Q.6 D _1 3 3 2 ( Y ) rr t + r 2 r l 2 Q.3 D Q.7 (a) No, (b) A J 0-y) ^ B VWV 12 O D (c)8n Q.8 A Q9 Q.10 Battery should be connected across Aand B. Out put can be taken across the terminals Aand C or B and C Q.ll A Q.12meterB Q.13 C Volt 10' n\ r t - ^ Q.14 t 2 Q.15 B Q.16 B Q.17 D Q.19 A [5] Q.18 This is true for r = r ; So R, given most accurate value il.BansalClasses Current Electricity

XII (ALL) ELECTROMAGNETIC INDUCTION ALTERNATING CURRENT CONTENTS & KEY CONCEPTS EXERCISE-I EXERCISE-II EXERCJSE-llI ANSWER KEY

When a conductor is moved across a magnetic field, an electromotive force (emf) is produced in the conductor. If the conductors forms part of a closed circuit t hen the emf produced caused an electric current to flow round the circuit. Hence an emf (and thus a current) is induced in the conductor as a result of its move ment across the magnetic field. This is known as "ELECTROMAGNETIC INDUCTION." 1. MAGNETIC FLUX : KEY CONCEPTS (]) = B . A ^BA cos 9 weber for uniform B . (j) = j B . d A for non uniform B . 2. (i) (ii) An induced emf is setup whenever the magnetic flux linking that circuit changes. The magnitude of the induced emf in any circuit is proportional to the rate of change of the magnetic flux linking the circuit, s a . dt The direction of an in duced emf is always such as to oppose the cause producing it. LAW O F EMI: LENZ'S LAWS : FARADAY'S LAWS O F ELECTROMAGNETIC INDUCTION : 3. 4. e = - . The neaative sign indicated that the induced emf opposes the change of t he flux. dt 5. E = BLV sin 0 voltwhere B = flux densi ty in wb/m ; L = length of the conductor (m); V=velocity of the conductor (m/s); 9 = angle between direction of motion of conductor & B . 2 E M F INDUCED IN A STRAIGHT CONDUCTOR IN UNIFORM MAGNETIC FIELD : 6. COIL ROTATION IN MAGNETIC FIELD SUCH THAT A X I S O F ROTATION I s PERPENDICULAR T O Instantaneous induced emf. N = number of turns in the coil ; B = magnetic induct ion ; E = maximum induced emf. 0 THE MAGNETIC FIELD : A = area of one turn; = uniform angular velocity ofthe coil; E = NABco sin cot = EQ sin cot, where 7. When a current flowing through a coil is changed the flux linking with its own w

inding changes & due to the change in linking flux with the coil an emf is induc ed which is known as self induced emf & this phenomenon is known as self inducti on. This induced emf opposes the causes ofInduction. The property ofthe coil or the circuit due to which it opposes any change ofthe current coil or the circuit is known as SELF - INDUCTANCE . It's unit is Henry. Coefficient of Self inducta nce L = or 4> = Li s SELF INDUCTION & SELF INDUCTANCE : fe Bansal Classes Electromagnetic Induction [10]

L depends only on; (i) (ii) shape of the loop & medium i = current in the circui t. II axis of solenoid J 0 vj 1H Q)re 10. R = 0 ; E = 0. Therefore (j) = constant. Thus in a superconducting loop flux nev er changes, (or it opposes 100%) total SUPER CONDUCTION LOOP IN MAGNETIC FIELD : 11.

(i) ENERGY STORED IN A N INDUCTOR : (ii) W = -2 LI . Energy of interation of two loops U = l,(j) = I ^ , = M I j I , wher e M is mutual inductance . 2 2 2 fe Bansal Classes Electromagnetic Induction [10]

12. GROWTH O F A Rt/L CURRENT IN A N L - R CIRCUIT : I = (1 - e~ ) . [ If initial current = 0 ] R L R = time constant of the circuit. E (i) (ii) 13. L behaves as open circuit at t = 0 [If /' = 0 ] L behaves as short circuit at t = oo always. L Curve (1) > Large R L Curve (2) Small R 0 DECAY O F CURRENT : Initial current through the inductor = I ; Current at any instant i = I e~ M 0 Rt/L ^Bansal Classes Electromagnetic Induction [4]

EXJER CISEI Q.l Q.2 The horizontal component ofthe earth's magneticfieldat a place is 3 x 10^T and t he dip is tan '(4/3). A metal rod of length 0.25 m placed in the north-south pos ition is moved at a constant speed of lOcm/s towards the east. Find the e.rnf. i nduced in the rod. A wire forming one cycle of sine curve is moved in x-y plane with velocity V = V i + V j. There exist a magneticfieldB = - B k Find the motio nal emf develop across the ends PQ of wire. x y 0 Q.3 A conducting circular loop is placed in a uniform magneticfieldof0.02 T, with it s plane perpendicular to thefield.If the radius ofthe loop starts shrinking at a constant rate of 1.0 mm/s, thenfindthe emfinduced in the loop, at the instant w hen the radius is 4 cm. Find the dimension of the quantity 7-7-7 , where symbols have usual meaimng. RCV A rectangular loop with a sliding connector of length I = 1.0 m is situated in a uniform magnetic field B = 2T perpendicular to the pla ne of loop. Resistance of connector is r = 2 f l Two resistances of 6 0 and 3Q a re connected as shown infigure.Find the external force required to keep the conn ector moving with a constant velocity v = 2m/s. B 3Q:? Q. 4 Q.5 >6Cl Q. 6 Two concentric and coplanar circular coils have radii a and b(a)as shown in figur e. Resistance of the inner coil is R. Current in the outer coil is increased fro m 0 to i, thenfindthe total charge circulating the inner coil. A horizontal wire isfreeto slide on the vertical rails of a conductingframeas shown infigure.The wire has a mass m and length I and the resistance ofthe circuit is R. If a unifo rm magneticfieldB is directed perpendicular to the frame, thenfindthe terminal s peed ofthe wire as it falls under the force ofgravity. Q, 7 *B x -yww xR X Q.8 Q.9 A metal rod of resistance 200 isfixedalong a diameter of a conducting ring of ra dius 0.1 m and lies on x-y plane. There is a magnetic field B = (50T)k- The ring rotates with an angular velocity 0 = 20 rad/ sec about its axis. An external re sistance of 10Q is connected across the centre of the ring andrim.Find the curre nt through external resistance. 6Q r-VW\A 2 mH In the given current,findthe rati o of i, to i where i, is the initial (at t = 0) current and i i s steady state ( at t = 0 ) current through the battery. 0

2 2 10 Q 10 In the circuit shown, initially the switch is in position 1 for a long time . Then the switch is shifted to position 2 for a long time. Find the total heat produced in R,. R. -WVVfe Bansal Classes H HVWVR; Electromagnetic Induction [10]

Q.ll Two resistors of 1OQ and 20Q and an ideal inductor of 1 OH are connected to a 2V battery as shown. The key K is shorted at time t = 0. Find the initial (t = 0) and final (t oo) currents through battery. L = 10H IW-j I V V W R= ion H>J 2on Q.12 There exists a uniform cylindrically symmetric magneticfielddirected along the axis ofa cylinder but varying with time as B = kt. Ifan electron is released fromrest in thisfieldat a distance of' r'fromthe axis of cylinder, its accelerat ion, just after it is released would be (e and m are the electronic charge and m ass respectively) Q.13 An emf of 15 volt is applied in a circuit containing 5 H inductance and 10 Q resistance. Find the ratio of the currents at time t = oo an d t = 1 second. Q. 14 A uniform magnetic field of 0.08 T is directed into the pl ane of the page and perpendicular to it as shown in thefigure.A wire loop in the plane of the page has constant area 0.010 m . The magnitude ofmagneticfielddecr ease at a constant ra