BANSAL CLASSES TARGET LIT JEE 2007 XI (PQRS)
CALORIMETRY & HEAT TRANSFER
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 At—A0 At y = At—>o v1 AV Lim At 0
For isotropic solids otj = a = a = a (let) so P =2a and y = 3a For anisotropic s olids p = otj + a and y = a, + a + a Here , a and a are coefficient of linear ex pansion in X , Y and Z directions. Variation in density : With increase of tempe rature volume increases so density decreases and vice-versa. H d =(1 + yAt) 2 3 2 2 3 2 3 0 3
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 4°C 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
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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 14.5°Cto 1 5.5°CatSTP 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 HEAT-TRANSFER
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
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(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
Calorimetry & Heat Transfer [3]
EXERCISE -1
Q. 1 An aluminium container of mass 100 gm contains 200 gm of ice at - 20°C. 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 = 0°C while the liquid in the right limb is maintained at a temperature = 1 00°C. 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 25°C. 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 20°C. If the clock is us ed in a climate where average temperature is 30°C, 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 39.9°C if room temperature is 30°C? 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 0°C to 100°C 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
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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 20°C, 90°C and 0°C. Find the ratio of l ength BD and BC if there is no heat flow in AB
20°C 90'C 0°C
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 25°C is cooled by adding 20 gm ice at 0°C. 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.
20°C 10°C E -5°C -10°C 2k 2k k = thermal conductivity
Q.17 Two identical calorimeter A and B contain equal quantity of water at 20°C. 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 22°C and in B 23°C. Th e initial temperature of both the metals is 40°C. 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
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Q. 21 A thermostatted chamber at small height h above earth's surface maintained at 30°C has a clock fitted in it with an uncompensated pendulum. The clock design er correctly designs it for height h, but for temperature of 20°C. 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 25°C and the temperature of ice -15°C. 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 10°C to 90°C 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 0°C to 100°C. 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 0°C to 50°C. Find the lengths of each portion of the composite rod. Q.26 A subst ance is in the solid form at 0°C. 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 of200°C 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 0°C is suspended in the middle of a room. In 1 5 minutes the temperature of the water rises by 2°C. 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.
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EXERCISE-II
Q. 1 A copper calorimeter of mass 100 gm contains 200 gm of a mixture of ice and water. Steam at 100°C under normal pressure is passed into the calorimeter and th e temperature of the mixture is allowed to rise to 50°C. 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 - 10°C was heated to - 2°C (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 1°C, while 900 ca lories was required to raise the temperature from -2°C to 3°C. 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 heatingfrom0°C. 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 10°C. How much less water it will have to take in when the sea is at 15°C? (Coefficient of cubic expansion of sea water = 2 x 10 "V°C, 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
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Q.8
Toluene liquid of volume 300 cm at 0°C is contained in a beaker an another quantit y of toluene of volume 110 cm at 100°C 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 -20°C isfilledupto height h = 10 cm in a uniform cylindr ical vessel. Water at temperature 9°C 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/gm°C 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 0°C. Ifthe vessel is exposed to an atmosphere havi ng constant temperature of- 0°C (< 0°C) 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
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Q.15 A lagged stick of cross section area 1 cm and length 1 m is initially at a
temperature of 0°C. It is then kept between 2 reservoirs of tempeature 100°C and 0°C. Specific heat capacity is 10 J/kg°C and linear mass density is 2 kg/m. Find 100°C o°c (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 risesfrom25°C to 30°C 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 32°C 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 25°C. Q.18 A solid copper sphere cools at the rate of 2.8°C per minute, when its temperature is 127°C. Find the rate at which another solid copper sphere oftwice the radius lose its t emperature at 327°C, ifin both the cases, the room temperature is maintained at 27°C . 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 55°C. The current is increased so th at the temperature rises slightly above 55°C, and then it is switched off. The cal orimeter and the content are found to cool at the rate of 3.6°C/min. (a) Find the specific heat of the liquid. (b) The room temperature during the experiment was 10°C. If the room temperature rises to 26°C, find the current required to keep the l iquid at 55°C. 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-rn°K. 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 ].
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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 coolfrom80°C to 50°C. How much time will it take to coolfrom60°C to 30°C ? The temperature of surrounding is 20°C. 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
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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/m°C and 200 W/m°C 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 27°C and 0°C 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
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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 - 10°C is slowiy heated and converted to steam at 100°C. 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 0°C and 90°C respectively. The temperature of the junction of th e three rods will be [JEE(Scr)2001 ] o°c(A) 45°C (B) 60°C (C) 30°C (D)20°C
,S0°C "90°C
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 0°C is placed in an is olated container which is at 227°C. 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 27°C, 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
[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 - 20°C is mixed with 5 kg water at 20°C. Thenfinalamount ofwater in the mixture would be; Given specific heat of ice = 0.5cal/g°C, specific heat ofwa ter = 1 cal/g°C, 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 <A (D) e = e ; Ay = A Q.18 Hot oil is circulated through a n insulated container with a wooden lid at the top whose conductivity K = 0.149 J/(m-°C-sec), thickness t = 5 mm, emissivity = 0.6. Temperature of the top of the lid in steady state is at T =27°C T, = 127°. If the ambient temperature T = 27°C. Calc ulate -=• Hot oil (a) rate ofheat loss per unit area due to radiationfromthe lid. 17 _ [JEE 2003] temperature ofthe oil. (Given a = — 10 ) (b)
x y x y x x y x y x y x y x y x x V.a
a x 8
Q.19 Three discs A, B, and C having radii 2 m, 4 m and 6 m respectively are coat ed with carbon black on their outer surfaces. The wavelengths corresponding to m aximum intensity are 300 nm, 400 nm and 500 nm respectively. The power radiated by them are QA, QB and QC respectively, (a) Q is maximum (B) QB is maximum [JEE' 2004 (Scr.)] (C) QC is maximum (D) QA = QB = QC Q.20 Two identical conducting r ods are first connected independently to two vessels, one containing water at 10 0°C and the other containing ice at 0° C. In the second case, the rods are joined en d to end and connected to the same vessels. Let qj and q g/s be the rate of melt ing of ice in the two cases respectively. The ratio q /q is (A) 1/2 (B) 2/1 (C) 4/1 (D) 1/4 [JEE'2004 (Scr.)] Q.21 Liquid oxygen at 50 K is heated to 300 K at c onstant pressure of 1 atm. The rate of heating is constant. Which of the followi ng graphs represents the variation of temperature with time?
a 2 9 T Temp.f Temp.f , Temp.f Temp. (A)
[JEE' 2004 (Scr.)] Q.22 A cube of coefficient of linear expansion a is floating in a bath containing a liquid of coefficient of volume expansion yt When the tem perature is raised by AT, the depth upto which the cube is submerged in the liqu id remains the same. Find the relation between a and y showing all the steps. [J EE 2004] Q.23 One end of a rod of length L and cross-sectional area A is kept in a furnace of temperature T The other end of the rod is kept at a temperature T . The thermal conductivity ofthe material of the rod is K and emissivity ofthe r od is e. It is given that T = T + AT where AT Insulated « T , T being the temperat ure ofthe surroundings. IfAT oc (Tj - T ), Furance T Rod * L * find the proporti onality constant. Consider that heat is lost only by radiation Insulated at the end where the temperature ofthe rod is T . [JEE 2004]
s s b r 2 2 s s s s f
Time (B) Time (C) Time (D) Time 2 (!l Bansal Classes
Calorimetry & Heat Transfer [3]
Q. 24 Three graphs marked as 1,2,3 representing the variation of maximum emissiv e power and wavelength of radiation of the sun, a welding arc and a tungsten fil ament. Which ofthe following combination is correct (A) 1-bulb, 2 —> welding arc,
3 —> sun (B) 2-bulb, 3 —» welding arc, 1 -» sun (C) 3-bulb, 1 —» welding arc, 2 —» sun (D) 2 b, 1 -> welding arc, 3 sun [JEE' 2005 (Scr)] Q. 25 In which of the following phe
nomenon heat convection does not take place (A) land and sea breeze (B) boiling ofwater (C) heating ofglass surface due to filament ofthe bulb (D) air around th e furance
[JEE' 2005 (Scr)]
Q.26 2 litre water at 27°C is heated by a 1 kW heater in an open container. On an average heat is lost to surroundings at the rate 160 J/s. The time required for the temperature to reach 77°C is (A) 8 min 20 sec (B)10min (C)7min (D)14min [JEE' 2005 (Scr)] Q.27 A spherical body of area A and emissivity e = 0.6 is kept insid e a black body. What is the rate at which energy is radiated per second at tempe rature T (A) 0.6 a AT (B)0.4aAT (C)0.8cAT (D)l.OaAT [JEE 2005 (Scr)] Q. 28 1 cal orie is the heat required to increased the temperature of 1 gm ofwater by 1 °C fro
m (A) 13.5°Cto 14.5°C at 76 mm of Hg (B) 14.5°Cto 15.5°Cat760mmofHg (C) 0°C to 1°C at 760 mm of Hg (D) 3°C to 4°C to 760 mm of Hg [JEE* 2005 (Scr)]
4 4 4 4
(!l Bansal Classes
Calorimetry & Heat Transfer [3]
ANSWER KEY Q.i Q.5 Q.9 Q.12 Q.16 Q.20 Q.24 Q.27 Q.I Q.4 Q.7 25.5°C 5 sec slow /6M/ 3 .71. 7/2 5°C 1/90 104.2 10.34 cm Q.2 Q.6 0.1 cm 10 sec EXERCISE -1 Q.3 Q.7 65°C 15K/16 Q.ll Q.14 Q.18 Q.22 Q.4 Q.8 2000 cm 10, 000 N 3 Q.10 0.2 Q.13 5 a/3 Q.17 27/85 Q.21 h/5R Q.25 10cm, Q.28 ctPy: 4 x 10 m/°C 4°C Q.15 1000 J (C )2:1 Q.19 (4/3)© 3Y/20 Q.23 0 °C -6 0 1 :a 2 Q.29 80 k cal/kg 1 1 1 -1 Q.30 3025 K
Q.3 800 cal kg" K , 1000 cal kg" K 1 : 1.26 (a) 37.8 J/s (Watts), (b) 2.005 N-m
Q.5 25 kJ 5000 J/°C kg Q.8 decrease by 0.75 cm ,25°C . K +K 2K K Q.10 K„ > 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 45°C 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 9.72°C/min Q.21 36A 0 x
Q.16 166.3 sec 9
1
Q.19 (a)0.42 cal/gm°C, (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;0.55°C 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
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
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
<§Bansal Classes
(i) The energy of a charged conductor resides outside the conductor in its EF, w here as in a condenser it is stored within the condenser in its EF. (ii) The ene rgy of an uncharged condenser = 0 . (iii) The capacitance of a capacitor depends only on its size & geometry & the di-electric between the conducting surface .( i.e. independent ofthe conductor, like, whether it is copper, silver, gold etc) REMEMBER :
Q.i A solid conducting sphere ofradius 10 cm is enclosed by a thin metallic shel l of radius 20 cm. A charge
Q.2
EXERCISE # I Q.3 Q.4
q = 20pC is given to the inner sphere. Find the heat generated in the process, t he inner sphere is connected to the shell by a conducting wire The capacitor eac h having capacitance C = 2pF are connected with a .CO, battery of emf 30 V as sh own infigure.When the switch S is closed. Find (a) the amount of chargeflownthro ugh the battery (b) the heat generated in the circuit (c) the energy supplied by the battery '30V (d) the amount of chargeflownthrough the switch S The plates o f a parallel plate capacitor are given charges +4Q and -2Q. The capacitor is the n connected across an uncharged capacitor of same capacitance asfirstone (= C). Find thefinalpotential difference between the plates of thefirstcapacitor. +i, I n the given network if potential difference between p and q is 2V and C = 3C The nfindthe potential difference between a&b.
2 r H M q C, C. c
Q.5 Find the equivalent capacitance of the circuit between point A and B.
c 2C 11 11 - C 11 11 2C 4C 11 11 :: C 11 11 4C : 8C 111 ! r 11 II 8C c \ \ \ \ \ Infinite / section/
y-Q.6 The two identical parallel plates are given charges as shown infigure.Ifthe plate area of either face of each plate is A and separation between plates is d, thenfindthe amount ofheat liberate after closing the switch. Q. 7 Find heat pro duced in the circuit shown infigureon closing the switch S.
+3q +q
Q.8 In the following circuit, the resultant capacitance between A and B is 1 pF. Find the value of C. Three capacitors of 2pF, 3pF and 5|iF are independently ch arged with batteries of emf's 5V, 20V and 10V respectively. After disconnectingf romthe voltage sources. These capacitors are connected as shown infigurewith the ir positive polarity plates are connected to A and negative polarity is earthed. Now a battery of 20V and an uncharged capacitor of4jaF capacitance are connecte d to the junction A as shown with a switch S. When switch is closed,find: (a) th e potential of the junction A. (b) final charges on all four capacitors. Q.9 T
T^ 2 f ^Slr 2\xV 5NF \ I —I I — ' 20V^ 4|I.— jF 4r (fe^Bansal Classes CAPACITANCE 121
Q.10 Find the charge on the capacitor C = 1 pF in the circuit shown in the figur e. Iph IjxIK C-luF l(iF IpF pF :pnF yUlF::IMF : Q.ll Find the capacitance ofthe system shown in figure. Q.12 Thefigureshows a circuit consisting offour capacito rs. Find the effective capacitance between X and Y. Q. 13 Five identical capacit or plates, each of area A, are arranged such that adjacent plates are at a dista nce'd* apart, the plates are connected to a source of emf V as shown infigure.Th e charge on plate 1 is and that on plate 4 is . Q.14 In the circuit shown in the figure,intially SW is open. When the switch is closed, the charge passing throug h the switch in the direction to Q.15 In the circuit shown infigure,findthe amou nt ofheat generated when switch s is closed. Q.16 Two parallel plate capacitors ofcapacitance C and 2C are connected in parallel then following steps are perfor med. (i) Abattery of voltage V is connected across points A and B. (ii) A dielec tric slab of relative permittivity k is slowly inserted in capacitor C. (iii) Ba ttery is disconnected. (iv) Dielectric slab is slowly removed from capacitor. Fi nd the heat produced in (i) and work done by external agent in step (ii) & (iv). Q.17 The plates of a parallel plate capacitor are separated by a distance d = 1 cm. Two parallel sided dielectric slabs ofthickness 0.7 cm and 0.3 cmfillthe sp ace between the plates. Ifthe dielectric constants ofthe two slabs are 3 and 5 r espectively and a potential difference of440V is applied across the plates. Find : (i) the electricfieldintensities in each ofthe slab s. (ii) the ratio of elect ric energies stored in thefirstto that in the second dielectric slab. Q.18 A 10 pF and 20 pF capacitor are connected to a 10 V cell in parallel for some time af ter which the capacitors are disconnectedfromthe cell and reconnected at t = 0 w ith each other, in series, through wires offinite resistance. The +ve plate of t hefirstcapacitor is connected to the -ve plate ofthe second capacitor. Draw the graph which best describes the charge on the +ve plate ofthe 20 pF capacitor wit h increasing time. List of recommended questions from LE. Irodov. 3.101, 3.102, 3.103, 3.113, 3.117, 3.121, 3.122, 3.123,3.124, 3.132,3.133, 3.141,3.142, 3.177, 3.184, 3.188. 3.199. 3.200,3.201. 3.203, 3.204. 3.205 121 AE60 V SW 7 k= 1 k=2 k=3 k=4 V- + 2 nF1 3 X 60 V I J (fe^Bansal Classes CAPACITANCE
EXERCISE # II
Q. 1 (a) For the given circuit. Find the potential difference across all the cap acitors, (b) How should 5 capacitors, each of capacities, lpF be connected so as to produce a total capacitance of 3/7 pF. Q.2
6oF,
Ih-H^f — I 8(xF —h ' 9|iF
+. 25V
The gap between the plates of a plane capacitor isfilledwith an isotropic insula tor whose di-electric constant varies in the direction perpendicular to the plat es according to the law K = K j 1 + sin 71 X — L d where d is the separation, betw een the plates & K is a constant. The area of the plates is S. Determine the cap acitance of the capacitor.
t Q.3
(i) (ii) Q.4 (j) (ii) (iii) Q.5
Five identical conducting plates 1,2,3,4 & 5 arefixedparallel to and equdistantf romeach other (seefigure).Plates 2 & 5 are connected by a conductor while 1 & 3 are joined by another conductor. The junction of 1 & 3 and the plate 4 are conne cted to a source of constant e.m.f. V . Find; the effective capacity of the syst em between the terminals ofthe source. the charges on plates 3 & 5. Given d = di stance between any 2 successive plates & A= area of either face of each plate . 5 0
Apotential difference of300 Vis applied between the plates of a plane capacitor spaced 1 cm apart. A plane parallel glass plate with a thickness of 0.5 cm and a plane parallel paraffin plate with a thickness of 0.5 cm are placed in the spac e between the capacitor platesfind: Intensity of electricfieldin each layer. The drop ofpotential in each layer. The surface charge density of the charge on cap acitor the plates. Given that: k = 6, k =2
glass paraffin
A charge 200pC is imparted to each of the two identical parallel plate capacitor s connected in parallel. At t =0, the plates of both the capacitors are 0.1 m ap art. The plates of first capacitor move towards each other with relative velocit y 0.001 m/s and plates of second capacitor move apart with the same velocity. Fi nd the current in the circuit at the moment. A parallel plate capacitor has plat es with area A & separation d . A battery charges the plates to a potential diff erence ofV . The battery is then disconnected & a di-electric slab of constant K & thickness d is introduced. Calculate the positive work done by the system (cap acitor + slab) on the man who introduces the slab.
0 Q.6 Q.7
A capacitor of capacitance C is charged to a potential V and then isolated. A sm all capacitor C is then chargedfromC , discharged & charged again, the process b eing repeated n times. The potential ofthe large capacitor has now fallen to V. Find the capacitance of the small capacitor. If V = 100 volt, V=35volt, find the value ofn for C = 0.2 pF & C = 0.01075 pF . Is it possible to remove charge on
C this way? 0 0 0 0 0 0 Q. 8
When the switch S in thefigureis thrown to the left, the plates of capacitors C, acquire a potential difference V. Initially the capacitors C C are uncharged. T hw switchis now thrown to the right. What are thefinalcharges q q & q on the cor responding capacitors. 2 3 p 2 3 .V TLPI Ic T (fe^Bansal Classes CAPACITANCE 121
Q.9
(1) (ii) (lii)
A parallel plate capacitor with air as a dielectric is arranged horizontally. Th e lower plate isfixedand the other connected with a vertical spring. The area of each plate is A. In the steady position, the distance between the plates is d . When the capacitor is connected with an electric source with the voltage V, a n ew equilibrium appears, with the distance between the plates as d Mass of the up per plates is m. Find the spring constant K. What is the maximum voltage for a g iven K in which an equilibrium is possible ? What is the angularfrequencyofthe o scillating system around the equilibrium value dj. (take amplitude of oscillatio n « d )
0 r {
Q.10 An insolated conductor initiallyfreefromcharge is charged by repeated conta cts with a plate which after each contact has a charge Q due to some mechanism. If q is the charge on the conductor after the first
Qq operation, prove that the maximum charge which can be given to the conductor in this way is ~
Q.ll A parallel plate capacitor is filled by a di-electric whose relative permit tivity varies with the applied voltage according to the law = aV, where a = 1 pe r volt. The same (but containing no di-electric) capacitor charged to a voltage V = 156 volt is connected in parallel to thefirst"non-linear" uncharged capacito r. Determine thefinalvoltage V across the capacitors.
f
Q.12 A capacitor consists oftwo air spaced concentric cylinders. The outer ofrad ius b isfixed,and the inner is of radius a If breakdown ofair occurs atfieldstre ngths greater than E^, show that the inner cylinder should have (i) radius a = b /e ifthe potential of the inner cylinder is to be maximum (ii) radius a = b/Ve i f the energy per unit length of the system is to be maximum.
,.JT 5V-r-46F =n 5V :d=6nf
Q. 13 Find the charge flown through the switchfromAto B when it is closed. Q.14 Figure shows three concentric conducting spherical shells with inner and outer s hells earthed and the middle shell is given a charge q. Find the electrostatic e nergy of the system stored in the region I and II.
6mF Jr~
Q.15 The capacitors shown infigurehas been charged to a potential difference of V volts, so that it carries a charge CV with both the switches Sj and S remainin g open. Switch Sj is closed at t=0. At t=R,C switch Sj is opened and S is closed . Find the charge on the capacitor at t=2RjC + R^C.
2 2 s, Hi s,
h is closed at t = 0. Find the charge on the rightmost capacitor as "yv a functi on oftime given that it was intially unchanged.
(fe^Bansal Classes CAPACITANCE
Q.17 In the given circuit, the switch is closed in the position 1 at t = 0 and t hen moved , I V to 2 after 250 p,s. Derive an expression for current as a functi on oftime for J^ov [ t > 0. Also plot the variation of current with time. I X40V VL Q.18 Find the charge which flowsfrompoint Ato B, when switch is closed. 5(IF 5NF 5^F 5(.IF 5(IF 2 :500FJ :0.5 NF EXERCISE # III 2 2 120V
Q. 1 Two parallel plate capacitors A&B have the same separation d=8.85 x lO^m be tween the plates. The plate areas of A & B are 0.04 m & 0.02 m A B respectively. A slab of di-electric constant (relative permittivity) K=9 has dimensions such that it can exacdy 10V fill the space between the plates ofcapacitor B. (i) the di-electric slab is placed inside A as shown in thefigure(i) Ais then charged to a potential difference of 110 volt. Calculate the capacitance ofA and the energ y stored in it. (ii) the battery is disconnected & then the di-electric slab is removedfromA. Find the work done by the external agency in removing the slabfrom A. (iii) the same di-electric slab is now placed inside B,fillingit completely. The two capacitors A& B are then connected as shown in figure (iii). Calculate t he energy stored in the system. [ JEE '93,7] Q.2 Two square metallic plates of 1 m side are kept 0.01 m apart, like a parallel plate capacitor, in air in such a way that one oftheir edges is perpendicular, to an oil surface in a tankfilledw ith an insulating oil. The plates are connected to a battery of e.m.f. 500 volt. The plates are then lowered vertically into the oil at a speed of 0.001 m/s. Ca lculate the current drawn from the battery during the process, [di-electric cons tant of oil = 11, e = 8.85 x 10" C /N m ] [ JEE '94, 6 ] Q.3 A parallel plate ca pacitor C is connected to a battery & is charged to a potential difference V. An other capacitor of capacitance 2C is similarly charged to a potential difference 2V volt. The charging batteiy is now disconnected & the capacitors are connecte d in parallel to each other in such a way that the positive terminal of one is c onnected to the negative terminal of other. Thefinalenergy ofthe configuration i s: 25 (B) - CV (D) - CV [JEE'95, 1 ] (A) zero (C) — CV
0 12 2 2 2 2 2
2 Q.4
The capacitance of a parallel plate capacitor with plate area 'A' & separation d is C. The space between the plates isfilledwith two wedges of di-electric const ant Kj & K respectively. Find the capacitance ofthe resulting capacitor. [JEE'96 , 2]
2 Q.5 ©
(fe^Bansal Classes
Two capacitors A and B with capacities 3 pF and 2 pF are charged to a potential difference of 100 V and 180 V respectively. The plates of the capacitors are con nected as shown in figure with one wire from each 2nF capacitorfree.The upper pl
ate of a is positive and that of B is negative, an 1-. uncharged 2 pF capacitor C with lead wires falls on thefreeends to complete IOOV B 180V the circuit. Calc ulate: thefinalcharges on the three capacitors The amount of electrostatic energ y stored in the system before and after the completion ofthe circuit. [JEE'97 (c ancelled)]
CAPACITANCE 121
Q.6
An electron enters the region between the plates of a parallel plate capacitor a t a point equidistant from eitherplate. The capacitor plates are 2* 10 mapart& 1 0 m long. A potential difference of300 volt is kept across the plates. Assuming that the initial velocity of the electron is parallel to the capacitor plates, c alculate the largest value ofthe velocity ofthe electron so that they do notflyo ut ofthe capacitor at the other end. [ JEE '97, 5 ]
_2 -1 Q. 7
For the circuit shown, which ofthe following statements is true ? (A) with S, cl osed, Vj = 15 V, V = 20 V (B) with S closed, Vj = V = 25 V (C) with & S closed, Vj = V = 0 (D) with Sj & S closed, V = 30 V, V = 20 V
2 3 2 2 2 2 l 2 V, =30V
[JEE'99, 2] Q.8
Calculate the capacitance of a parallel plate condenser, with plate area A and d istance between plates d, whenfilledwith a medium whose permittivity varies as; 0<x< | e (x)= e + P x [REE2000, 6] 4 < x < d. S(X)=G + P (d-x)
0 0 Q. 9
Two identical capacitors, have the same capacitance C. One of them is charged to potential V and the other to V . The negative ends ofthe capacitors are connect ed together. When the positive ends are also connected, the decrease in energy o f the combined system is [ JEE 2002 (Scr), 3 ]
t 2 (A) Mvf-vl) (B)Mv,2+v22) (qIc^-vJ 0 0 (D^cfa+vJ 1—m— s/ + c^ " v T
Q.10 In the given circuit, the switch S is closed at time t = 0. The charge Q on the capacitor at any instant t is given by Q (t) = Q (l-e" *). Find the value o f Q and a in terms of given parameters shown in the circuit. [JEE 2005]
0
Q.ll Given: Rj = ID , R2 = 2Q, Cx = 2pF, C = 4pF The time constants (in pS) for the circuits I, n, HI are respectively .C, !!—i— R,: K "C2 hi r . - T T-r. k v ,vV ' h-—
2
(A) 18, 8/9, 4 (C) 4, 8/9, 18 (II.)
(B) 18, 4, 8/9 (D) 8/9, 18,4 (in) [JEE 2006] (fe^Bansal Classes CAPACITANCE 121
ANSWER KEY EXERCISE # I Q.l Q.3 Q.7 Q.9 9J 3Q/2C 0 100 Q.2 Q.4 Q.8 (a) 20 pC, (b) 0.3 mJ, (c) 0.6 mJ. (d) 60 [iC 30 V 32 -MF Q.5 C Q-6 1 qd iZT 2
(a) — volts; (b) 28.56 |iC, 42.84 pC, 71.4 jnC, 22.88 pC Q.10 10 pC 2A e V 25 e„A Q. 13 A G V Q.14 60 (ic,AtoB Q.ll 24 d Q.12 ^ F 0 0 Q.15 150 mJ 4 Q.16 (i) | C V ; (ii) - ~ CV2(K- 1); ^ (K + 2) (K - l ^ V ; 2 2 q(nC), Q. 17 (i) 5 X 10 V/m, 3 x 10 V/m; (ii) 3 5/9 4 Q.18 200 EXERCISE # II HHI Q.l (a) 12 V, 9 V, 3 V, 13 V, 16 V, (b) GSTIK, m T TT ,Q5 = t 7 2 Q.2 C = 2d Q.5 2[iA 5 fe A^ Q.3 (i) 3 v " y ;(ii)Q3=T 0 4 4 IAV„ ,AV
Q.4 (i) 1.5 x 10 V/m, 4.5 x 10 V/m, (ii) 75 V, 225 V, (iii) 8 x 10" C/m Q.6 W = \ C V
0 0 2 q.8 q i
-Ci2V(C2+C3) CiC +C C +Cj c 2 2 3 2 0 0 3 c c +c c +c c 1 2 2 3 3 K CCCV 1 2 3 1-Q.7 C = C 1 f\v T n V V o \ -1 = 0.01078 |iF,n = 20 1 / n SpAV 2d (d -d!)'v As ^3 2 \3/2 Kdf-e AV 0 2 1/2 MDJ Q.ll 12 volt r 2 Q.13 69 mC Q.14 U, 3kg, lOr 2 where q, = ~ ; Uu O CV = 2K(q + ) / 3 5 r Q.15 q = CE 1 — + qi
v e y Q.16 q CV 1-—e~t/RC 2 (fe^Bansal Classes CAPACITANCE 121
I(ajnp)
Q.17 For t < 250 ps, I = 0.04 e^° amp ; For t > 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. <M
QUESTION FOR SHORT ANSWER
Q.l The electric strength of air is about 30,000 V/cm. By this we mean that when the electricfieldintensity
Q.2 Q.3 Q.4
exceeds this value, a spark will jump through the air. We say that "electric bre akdown" has occurred. Using this value, estimate the potential difference betwee n two objects where a spark jumps. Atypical situation might be the spark that ju mps between your body and a metal door handle after you have walked on a deep ca rpet or slid across a plastic car seat in very dry weather. Ifyou grasp the two wires leadingfromthe two plates of a charged capacitor, you may feel a shock. Th e effect is much greater for a 2-pF capacitor than for a 0.02p,F capacitor, even though both are are charged to the same potential difference. Why? <T(+) a(-) T hree infinite nonconducting sheets, with uniform surface charge densities a, 2a and 3ct are arranged to be parallel like the two sheets in Fig. What is their or der,fromleft to right, if the electricfieldE produced by the arrangement has mag nitude E = 0 in one region and E = 2a/e in another region? As shown in the figur e plots of charge versus potential difference for three parallel plate capacitor s, which have the plate areas and separations given in the table. Which of the p lots goes with which ofthe capacitors?
0
Capacitor 1 2 3 Q.5
Area Separation A d 2A d A 2d t 2 1 t
Q.6
Initially, a single capacitance C is wired to a battery. Then capacitance C is a dded in parallel. Are (a) the potential difference across C and (b) the charge q j on C now more than, less than, or the same as previously? (c) Is the equivalen t capacitance C of Cj and C more than, less than, or equal to Cj? (d) Is the tot al charge stored on C^ and C , together more than, less than, or equal to the ch arge stored previously on Cj? As shown in thefigurethree circuits, each consisti ng of a switch and two capacitors, initially charged as 6q_ indicated. After the switches have been closed, in which 6q_ _ Jq 6q__ C 3C 2C 2C circuit (ifany) wi ll the charge on the left-hand capacitor 2 C (a) increase, (b) decrease and (c) remain the same?
12 2 == :=
(fe Bansal Classes
Cap-monster maze. Inthe Figure all the capacitors have a capacitance -- I I h f¥T of 6.0 pF, and all the b atteries have an emf of 10V What is the charge | J_ J ^ on capacitor C? (Ifyou canfindthe proper loop through this maze, you T ~ , 4 , 4 4 T , , = I can answer the question with a few seconds of mental calculation.) -r I c 1 H H Q8 An oilfilledcapacitor has been designed to have a capacitance C and to operate safely at or below a certain maximum potential difference V with out arcing over. However, the designer did not do a good job and the capacitor o ccasionally arcs over. What can be done to redesign the capacitor, keeping C and V unchanged and using the same dielectric? Q.9 One of the plates of a capacitor connected to battery is earthed. Will the potential diffrence between the plate s change if the earthing wire is removed?
m m Q.7
Question Bank on Capacitance [13]
ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. Cond.
Diele.
Q. 1 The distance between plates of a parallel plate capacitor is 5d. Let the po sitively charged plate is at x=0 and negatively charged plate is at 1 * x=5d. Tw o slabs one of conductor and other of a dielectric of equal thickness d are inse rted between the plates as shown in figure. Potential versus distance graph will look like: v A) Q.2 w v
(B) (C)
x=0 x=d x=2d x=3d x=4d x=5d VI (D)
A parallel plate capacitor has two layers of dielectric as shown in figure. This capacitor is connected across a battery. The graph which shows the variation of electricfield(E) and distance (x)fromleft plate. y y y
(A) k=2 k=4 (d,0) (3d,0) (D) (d,0) (3d,0) " 1 (B) (d,0) (3d,0) " (C) (d,0) (3d,0)~ (d,0) (3d,0) Q.3
The distance between the plates of a charged parallel plate capacitor is 5 cm an d electricfieldinside the plates is 200 Vcnr . An uncharged metal bar of width 2 cm is fully immersed into the capacitor. The length of the metal bar is same as that of plate of capacitor. The voltage across capacitor after the immersion of the bar is (A) zero (B)400V (C)600V (D)100V Three large plates are arranged as s hown. How much charge will flow through the key k if it is closed? 5Q 4Q 3Q (D) none (A) (C) ( B ) F 2Q Q.4 2d L Q.5 1 E
placed as shown in the figure. Plate number 2 and 4 are connected wire and betwe en point A and B, a cell of emfE is connected. The charge flown through the cell is u 3 e AE 2 s AE 4s AE e AE (A) (C) (D) (B) 4 d 3 d 2d
0 0 0 0 5 Q.6 ->
If charge on left plane of the 5 pF capacitor in the circuit segment shown in th e figure is -20pC, the charge on the right plate of 3 pF capacitor is (A) +8.57 pC (B) -8.57 pC (C)+11.42pC (D)-11.42pC Five identical capacitor plates are arra nged such that they make capacitors each of Q.7 2 pF. The plates are connected t o a source of emf 10 V. The charge on plate C is (B) + 40 pC (C) + 60 pC (D) + 8 0pC u (A) + 20 pC
|3nF UjiF w
|ffH 2(iF
(fe Bansal Classes
Question Bank on Capacitance [13]
Q.8 A capacitor of capacitance C is charged to a potential difference V from a c ell and then disconnected
L +
from it. Acharge +Q is now given to its positive plate. The potential difference across the capacitor is now (C)v Q f (D) V - ^ , if V < CV (B) V + (A) V of cap acity 5 pf is (A) 60 pC (C) 30 pC
M^lfP^lf. 100 V
Q.9 In the circuit shown infigurecharge stored in the capacitor (B) 20 pC (D) zero
2 3
Q.10 A conducting body 1 has some initial charge Q, and its capacitance is C. Th ere are two other conducting bodies, 2 and 3, having capacitances : C = 2C and C -» Q . Bodies 2 and 3 are initially uncharged. O "Body 2 is touched with body 1. Then, body 2 is removed from body 1 and touched with body 3, and then removed." This process is repeated N times. Then, the charge on body 1 at the end must be (A) Q / 3 (B) Q / 3 (C)Q/N (D) None
N N _ 1 3
Q.ll Condenser A has a capacity of 15 pF when it isfilledwith a medium of dielec tric constant 15. Another condenser B has a capacity 1 pF with air between the p lates. Both are charged separately by a battery C of 100V. After charging, both are connected in parallel without the battery and the dielectric material being removed. The common potential now is
(A) 4 0 0 V (B) 800V (C) 1200V (D) 1600V
Q.12 In the adjoiningfigure,capacitor (1) and (2) have a capacitance C' each. Wh en the dielectric of dielectric consatnt K is inserted between the plates of one ofthe capacitor, the total charge flowing through battery is ,c KCE KCE h from B to C from C to B (B) K + l t (A) K + l (K-l)CE (K-l)CE (C) ^ FROM B to C (D) f rom C to B
A +
2 ( K + 1)
-WUr Q. 13 Two identical capacitors 1 and 2 are connected in series toabatteryas shown in 1 figure. Capacitor 2 contains a dielectric slab of dielectric constan t k as shown. Q and Q are the charges stored in the capacitors. Now the dielectr ic slab is removed and the corresponding charges are Q' j and Q' . Then H^ Q^_k +1 k q; = k+i 2 (A) (B) Q _ k + l ( C ) Q ~ 2k (D) 01 2 Qi k Qi
t 2 2 2
Q. 14 The area ofthe plates of a parallel plate capacitor is A and the gap betwe en them is d. The gap is filled with a non-homogeneous dielectric whose dielectr ic constant varies with the distance 'y'fromone plate • as : K = ^sec(7ty/2d), whe re X is a dimensionless constant. The capacitance ofthis capacitor is (A) 7ie ^ A/2d (B)7rs XA/d (C) 27te k A/d (D)none
0 0 0
Q.15 A capacitor stores 60pC charge when connected across a battery. When the ga p between the plates is filled with a dielectric, a charge of 120pC flows throug h the battery. The dielectric constant of the material inserted is: i (A) 1 (B) 2 . (C) 3 (D) none
Question Bank on Capacitance [13]
Q.16 In the above question, if the initial capacitance ofthe capacitor was 2pF, the amount of heat produced when the dielectric is inserted. £ • (A) 3600pJ (B) 2700 pJ (C) 1800pJ (D)none Q.17 A capacitor of capacitance C is initially charged to a potential difference of V volt. Now it is connected •j to a battery of 2V with o pposite polarity. The ratio of heat generated to the final energy stored in the capacitor will be (A) 1.75 (B) 2.25 (C) 2.5 (D) 1/2
AQ.18 Three plates A B and C each of area 0.1 m are separated by 0.885 Bmmfromea ch other as shown in the figure. A10 V battery is used to Ccharge the system. Th e energy stored in the system is (A)lpJ (B) 10 pj (C) 10' pJ (D) 10" pJ
2 _1 2 3 hH 10V
Q.19 A parallel plate capacitor of capacitance C is connected to a battery and i s charged to a potential difference V. Another capacitor of capacitance 2C is si milarly charged to a potential difference 2V. The charging battery is now discon nected and the capacitors are connect in parallel to each other in such a way th at the positive terminal of one is connected to the negative terminal of the oth er. Thefinalenergy I ofthe configuration is 2 „ 5 (A)zero (D)-CV ( B ) - CV (C)yCV 2 2 2
Q.20 A 2 pF capacitor is charged to a potential = 10V. Another 4 pF capacitor is charged to a potential = 20V. The two capacitors are then connected in a single loop, with the positive plate of one ; connected with negative plate of the oth er. What heat is evolvecl in the circuit? (A) 300 pj (B) 600 pJ (C) 900 pj (D)45 0pJ Q.21 The plates S and T of an uncharged parallel plate capacitor are connect ed across a battery. The battery is then disconnected and the charged plates are now connected in a system as shown in thefigure.The system shown is in equilibr ium. All the strings are insulating and massless. The magnitude of charge on one ofthe capacitor plates is: [Area ofplates=A] -^svtvw 4mgA (A) pmgA (B) 77777777 7777777777ininiii (C)VmgA
(D) 2mgA e m 0
Q.22 In the circuit shown, the energy stored in 1 pF capacitor is (A) 40 pJ (B) 64 pJ (D)none (C) 32 pJ
3nF I^ HF
Q.23 Four metallic plates arearranged as shown in thefigure.Ifthe distance betwe en each plate then capacitance of the given system between points A and B is (Gi ven d « A) 2s A I - , _ S pA sA / , I (A) d (B) 3s A 4s o A (C) d (D)
n 0n 1 0
(fe Bansal Classes
Question Bank on Capacitance [13]
£
Q.24 What is the equivalent capacitance of the system of capacitors between A & B
W~6c .A 2.
B Q.25 From a supply ofidentical capacitors rated 8 pF, 250 V, the minimum numbe r of capacitors required to form a composite 16 pF, 1000 Vis : (A) 2 (B) 4 (C) 1 6 (D) 32 cP
(B) 1.6 C (C)C (D) None £.
Q.26 The minimum number ofcapacitors each of3 pF required to make a circuit with an equivalent capacitance 2.25 pF is (A) 3 (B)4 (C)5 (D)6
0
Q.29 A capacitor of capacitance 1 pF withstands the maximum voltage 6 kV while a capacitor of 2 pF withstands the maximum voltage 4 kV. What maximum voltage wil l the system of these two capacitor withstands if they are connected in series? £ (A) lOkV (B)12kV (C) 8 kV (D)9kV Q.30 Four identical plates 1,2,3 and 4 are plac ed parallel to each other at equal distance as shown in the figure. Plates 1 and 4 are joined together and the space between 2 and 3 is filled with a dielectric of dielectric constant k = 2. The capacitance of the system between 1 and 3 & 2 and 4 are Cj and C2 C, respectively. The ratio — is : C-2 3 (B)l (A)-: (D) 5 ( C )
£
Q.27 The capacitance (C) for an isolated conducting sphere of radius (a) is give n by 47ts a. Ifthe sphere is enclosed with an earthed concentric sphere. The rat io ofthe radii of the spheres being n then the (n-1) Icapacitance of such a sphe re will be increased by a factor n (n-1) (A)n (D) a. n (B) (n-1) (C) n -' Q.28 T wo capacitor having capacitances 8 pF and 16 pF have breaking voltages 20 V and 80 V. They are combined in series. The maximum charge they can store individuall y in the combination is (A) 160 pC (B) 200 p,C (C) 1280 p,C (D) none ofthese y
Q.31 #
In the circuit shown infigure,the ratio ofcharges on 5pF and 4pF capacitor is: ( A) 4/5 (B)3/5 (C) 3/8 (D) 1/2
3jiF •JL— 5(iF 4nF
