--- Page BANSAL CLASSES
TARGET LIT JEE 2007 XI (PQRS )
CALORIMETRY & HEAT TRANSFER C O N T E N T S KEYCONCEPT EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY --- Page THERMAL EXP ANSION Definition of Heat:
Heat is a form of energy which is transferred between a system and it s surrounding as a result of
temperature difference only.
Thermal Expansion : Expansion due to increase in temperature . 1. Type of thermal expansion
Coefficient of expansion For temperature change
At change in
T . 1 A /
(i) Linear a = Lim length Al = l a At At—>0 / 0 A t 0 1 AA P = Lim (ii) Superficial A t — A 0 A t Area AA= A ^ A t 1 AV
(iii) Volume y = Lim volume AV = V yAt
At—>o v0 A t 0
(a) For isotropic solids otj = a 2 = a 3 = a (let) so P =2 a and y = 3 a
(b) For anisotropic solids p = otj + a 2 and y = a , + a 2 + a 3
Here , a 2 and a 3 are coefficient of linear expansion in X , Y and Z directions .
2. Variation in density : With increase of temperature volum e increases so density decreases and
vice-versa.
H d = -
(1 + yAt)
For solids values of y are generally small so we can write d = d (1-y At) (using bimomial expansion)
Note 0
(0 y for liquids are in order of 10~3 (ii)
For water density increases from 0 to 4°C so y is -v e (0 to 4° C) and for 4° C to higher temperature y
is +ve. At 4° C density is maximum. 3.
Thermal Stress: Aro d of length 1 is clamped between two fixed walls with distance 1 . If temperature
0 0
is changed by amount At then
stress: F (area assumed to be constant) A A/ strain = I F/A F/ 0 F so, Y = A///0 AA I AaAt or F = Y A a A t
(!l Bansal Classes Calorimetry & Heat Transfe r [3]
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(i) ( a varies with distance) Let a = ax+b
i
Total expansion = J expansion of length dx = |(a x + b)dxAt
(ii) ( a varies with tempearture) Let a = f (T) " x 1 ddxx T2 _ j"a/ d T A/ 0 Ti
Caution: If a is in °C then put Tj and T in °C. 2
similarly if a is in K then put Tj and T 2 in K. CAL ORIMETR
Y
Quantity of heat transfered and specific heat
The amount ofheat needed to incerase the temperature of 1 gmofwaterfr om 14.5°Cto 15.5°CatSTP
is 1 calorie dQ = mcdT 'h
Q = m [ C dT (be careful about unit of temperature, use uni ts according to the given units of C)
Ti
Heat transfer in phase change
Q = rnL L = latent heat of substance in cal/ gm/ °C or i n Kcal/ kg/ °C
L i c e = 80 cal/ gm for ice L = 5 4 0 C a l / m steam g
HEAT- TRANSF ER
(A) Conduction : Due t o vibration and collision of medium partic les .
(i) Steady State : In this state heat absorption stops and temperature gradient throughout the rod
dT
becomes constant i.e. — = constant . dx
(ii) Before steady state : Temp of rod at any point changes Note: If specific heat of any substance is zero, it can be considered a lways in steady state .
1. Ohm's law for Thermal Conduction in Steady St ate :
Let the two ends of rod of length 1 is maintained at temp Tj and T ( Tj > T ) 2 2 dQ T i ~ T 2 I Thermal current = L D 1 K-XH T 1 /
Where thermal resistance R T h = 1 1 KA
T-dT
2. Differential form of Ohm's Law dQ dT dT
— = K A — — = temperature gradient dT dx dx
dx
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4---(B) Convection: Heat transfer due to movement of medium particl es.
(Q Radiation: Every body radiates electromagnetic radiation of a ll possible wavelength at all temp>0 K.
2 4
1. Stefan's Law: Rate of heat emitted by a body at temp T K f rom per unit area E = GT J/sec/m
dQ 4 Radiation power — = P = o AT watt d l If a body is placed in a surrounding of temperature T s
dQ
4 4 ^ = c A (T - T s ) valid only for black body
heat from ge neral body
Emissmty or emmisive power e = ~ h e a t f r o m
If temp of body falls by dT in time dt
dT _ _ j4x
(dT/dt=ra te of cooling)
dt ~ m S s Newton's law of cooling
If temp difference of body with surrounding is small i.e. T = T s dT 4 e A a r r 3 / then, - T ( T - T ) dt m S dT so a ( T - T ) dt
Average form of Newtons law of cooling If a body cools from T j to T in time 51
2
T s - T 2 _ K T, +T , -T
(used gen erally in objective questions)
5t mS
(for bet ter results use this generally in subjective)
dt m S
4. Wein's black body radiation
At every temperature (>0K) a body radiates energy radiations of all wavelengths.
T >T >T, 3 2 According to Wein's displacement law if the wavelength corresponding t o
maximum energy is Xm
then X T = b where b = is a constant (Wein's constant) m
T=temperature of body ess
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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 system after 4 minutes (specific heat of
ice = 0.5 and L = 80 cal/gm, specific heat of A 1 = 0.2 cal/gm/°C) Q. 2 A U-tube filled with a liquid of volumetric coefficient of 10_ 5 /° C lies in a vertical plane. The height of
liquid column in the left vertical limb is 100 cm. The liquid in the left vertical limb is maintained at a
temperature = 0°C while the liquid in the right limb is maintained at a temperature = 100°C. Find the
difference in levels in the two limbs.
2 Q.3 A thin walled metal tank of surface area 5m is filled with water t ank and contains an immersion
heater dissipating 1 kW. The tank is covered with 4 cm thick laye r 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
Q.4 A glass flask contains some mercury at room temperature . It is foun d that at different temperatures
the volume of air inside the flask remains the same. If the volume of mercury in the flask is
3
300 cm , then find volume of the flask (given that coefficient of vo lume expansion of mercury and
_ 1 6 _ 1
and9 x 10~ (°C) respectively)
Q.5 A clock pendulum made of invar has a period of 0.5 sec at 20°C . If the clock is used in a climate
where average temperature is 30°C, aporoximately. How much fast or slow will the clock run in
6 - 6 10 sec. (a i l w a r =lxlO /°C )
Q.6 A pan filled with hot food cools from 50. 1 °C to 49.9 °C in 5 sec. Ho w long will it take to cool
from 40. 1 °C to 39.9°C if room temperature is 30°C?
Q.7 A composite rod made of three rods of equal length and cros s-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 fa ce A goes out of the end B there being
no loss of heat from the sides of the bar. Find the effective thermal conductivity of the bar
A B I I 1 1 K/2 5K K 1 1 2 6 3 2
Q.8 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 comp ressive force developed
inside the bar.
Q.9 A solid copper cube and sphere, both of same mass & emissivity are heated to same 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 cause melting of 0. 1 gm of
ice/sec. If the rod is replaced with another rod of half the length and double the radius of first and
thermal conductivity of second rod is 1/4 that of first, find the rate of ice melting in gm/sec
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6---Q.l l Three aluminium rods of equal length form an equilateral triangle AB C . Taking O (mid
point of rod BC) as the origin. Find the increase in Y-coordinate per unit change in
temperature of the centre of mass of the system. Assume the length of the each rod is 2m, and a d = 4 v 3 x10"6 /° C Q.12 Three conducting rod s of same material and cross-section are shown in figure .
Temperature of A, D and C are maintained at 20°C, 90°C an d 0°C . Find the 20°C 0°C
ratio of length BD and BC if there is no heat flow in AB
90'C
Q. 13 If two rod s of layer L and 2 L having coefficients of linear ex pansion a and 2 a respectively are
connected so that total length becomes 3 L, determine the average c oefficient of linear expansion
of the composite rod .
Q.14 A volume of 120 ml of drink (half alcohol + half water by mass) orig inally at a temperature of 25°C
is cooled by adding 20 gm ice at 0°C. If all the ice melts, find the f inal temperature of the drink,
(density of drink = 0.833 gm/cc, specific heat of alcohol = 0.6 ca l/gm/°C)
Q.15 A solid receives heat by radiation over its surface at the rate of 4 kW. The heat convection rate
from the surface of solid t o the surrounding is 5.2 kW, and heat i s generated at a rate of 1.7 kW
over the volume of the solid. The rate of ch ange of the average temperature of the solid is
o - 1
0.5 Cs . Find the heat capacity of the solid.
20°C 10°C E -5° C -10° C Q.16 The figure shows the face and interface temperature of a c omposite
slab containing of four layers of two materials having identical thi ckness. 2k 2k
Under steady state condition, find the value of temperature 6 . 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 4 (C°)_ 1 is dropped i nto A and a 5 gm piece of metal Y into B . The
equilibrium temperature in A is 22°C and in B 23°C . The initial temper ature of both the metals is
1 l
40°C. Find the specific heat of metal Y in cal g" (C°)~ -
1 and the ratio of their specific
heats are 1 : 4. If by radiation their rates of fall of temperatur e are same, then find the ratio of their
rates of losing heat .
Q.19 In the square frame of side I of metallic rods, the corners A and C are
maintained at Tj and T 2 respectively. The rate of hea t flow from A t o
Cisa . IfA and D are instead maintained Tj & T respectivley find, f ind the
2
total rate of heat flow.
Q.20 A hot liquid contained in a container of negligible heat capacity los es temperature at rate 3 K/min,
jus t before it begins t o solidify. The temperature remains constant for 30 min, Find the ratio of
1
specific heat capacity of liquid to specific latent heat of fusion is in Kr (given that rate of losing heat
is constant) .
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7---Q. 2 1 A thermostatted chamber at small height h above earth's surface maintai ned at 30°C has a clock fitted in it
with an uncompensated pendulum. The clock designer correctly designs it for height h, but for temperature of
20°C. If this chamber is taken to earth's surface, the clock in it would c lick correct time. Find the coefficient
of linear expansion of material of pendulum, (earth's radius is R) Q.22 The coefficient of volume expansion of mercury is 20 times the coe fficient of linear expansion of
glass. Find the volume of mercury that must be poured into a glass vesse l of volume 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. If t he water was initially at a temperature
of 25° C and th e temperatur e of ice -15°C . Find th e final temperatur e o f water ,
(specific heat of ice = 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 water heater at a rate of 70 litre per minute.
Natural gas with a density of 1.2 kg/m3 is used in the heater, which has a transfer efficiency of
32%. Find the gas consumption rate in cubic meters per hour, (heat co mbustion for natural gas is
Q.25 A metal rod A of 25cm lengths expands by 0.050cm . When its temperatu re is raised from 0°C to
100°C. Another rod B of a different metal of length 40cm expands by 0.04 0 cm for the same rise in
temperature . A third rod C of 50cm length is made up of pieces of rod s A and B placed end t o end
expands by 0.03 cm on heating from 0°C to 50°C. Find the lengths of each p ortion of the composite
rod.
Q.26 A substance is in the solid form at 0°C. The amount of heat added to this substance and its 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 of uniform cross-section and of length 1.5 mete rs is in contact with melting ice
and the other end with boiling water. At what point along its length s hould a temperature of200° C be
maintained, so that in steady state, the mass of ice melting is equal to that of steam produced in the same
interval of time? Assume that the whole system is insulated from the s urroundings.
Q.28 Two solids spheres are heated t o the same temperatur e and allowed t o cool under identical
conditions. Compare : (i) initial rates of fall of temperature, and (ii) initial rates of loss of heat .
Assume that all the surfaces have the same emissivity and ratios of th eir radii 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 15 minutes the
temperature of the water rises by 2°C. When an equal amount of ice is pl aced in the vessel, it melts in
10 hours. Calculate the specific heat of fusion ofice .
Q. 3 0 The maximum in the energy distribution spectrum of the sun is at 475 3 A and its temperature is 6050K.
What will be the temperature of the star whose energy distribution sho ws a maximum at 9506 A.
[3]
(!l Bansal Classes Calorimetry & Heat Transfer --- Page
EXERCISE -II
Q. 1 A copper calorimeter of mass 100 gm contains 200 gm of a m ixture of ice and water . Steam at
100°C under normal pressure is passed into the calorimeter and the t emperature of the mixture is
tents is now 330 gm, what wa s
the ratio of ice and water in the beginning? Neglect heat losses .
3 _ 1 x
Given : Specific heat capacity of copper = 0.42 x 10 J kg K" ,
3 1
Specific heat capacity of water = 4.2 x 10 J kg^Kr ,
5 - 1
Specific heat of fusion of ice = 3.36 x 10 J kg
5 1
Latent heat of condensation of steam = 22.5 x 1Q Jkg"
Q. 2 A n isoscete s triangt e i s form ed w ith a ro d o f length l and coefficient of linear expansion OTJ for the
x
base and two thin rod s each of length l and coefficient of linear expansion a for the two pieces,
2 2
if the distance between the apex and the midpoint of the base remai n unchanged as the temperatures
/, varied show that 7 l2
Q.3 A solid substance of mass 10 gm at - 10°C was heated to - 2°C (still in the solid state) . The heat required was 64 calories. Another 880 calories was required to rais e the temperature of the substance
(now in the liquid state) t o 1°C, while 900 ca lories wa s required to raise the temperature from
-2°C to 3°C. Calculate the specific heat capacities of the subst ances in the solid and liquid state in
calories per kilogram per kelvin. Show that the latent h eat of fusion L is related to the melting
point temperature t m by L = 85400 + 200 t m .
Q.4 A steel drill making 180 rpm is used to drill a hole in a block of steel. The mass 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 (a) the rate of working of the drill in watts, and
(b) the torque required t o drive the drill.
Specific heat of steel = 0. 1 and J = 4.2 J/cal. Use ; P = i o
2
Q. 5 A brass rod of mass m = 4.25 kg and a cross sectional area 5 cm i ncreases its length by 0.3 mm upon
heating from 0°C. What amount of heat is spent for heating the rod? T he coefficient of linear expansic 1
- 5 3 3
for brass is 2xl0 /K , its specific heat is 0.39 kJ/k g.K and the density of brass is 8.5 x 10 kg/m .
9 8
Q.6 A submarine made of steel weighing 10 g has t o 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 t o take in when the sea is at
15°C? (Coefficient of cubic expansion of sea water = 2 x 10"V°C, c oefficient of linear expansion
5 of steel = 1.2 x 10- /°C)
Q. 7 A flow calorimeter is used to measure the specific heat of a liqu id. Heat is added at a known rate
to a stream of the liquid as it passes through the calorimeter at a known rate . Then a measurement
of the resulting temperature difference between the inflow an d the outflow points of the liquid
stream enables u s to compute the specific heat of the liquid. A liquid of density 0.2 g/cm3 flows
3
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 liqu id.
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3
Q.8 Toluene liquid of volume 300 cm at 0°C is contained in a beaker an another quantity of toluene of
3 3
mbined volume is 410 cm ) . Determine the
total volume of the mixture of the toluene liquids when they are mi xed 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 is filled upto height h = 10 cm in a uniform cylindrica l vessel. Water at temperature 9°C is
filled in another identical vessel upto the same height h= 10 c m. Now, water from second vessel is
poure d int o firs t vesse l an d it i s foun d tha t leve l o f uppe r surfac e fall s throug h
Ah = 0. 5 cm when thermal equilibrium is reached . Neglecting therm al capacity of vessels, change
in density of water due to change in temperature and loss of heat d ue t o radiation, calculate initial
temperature 0 of water .
Given, Density of water, p w = 1 gm cm - 3
Density of ice, p. =0.9gm/cm 3
Specific heat of water, s = 1 cal/gm °C
w
Specific heat of ice, s = 0.5 cal/gm°C
;
Specific latent heat of ice, L = 80 cal/gm
Q. 10 A composite body consists of two rectangular plates of the same di mensions but different thermal
conductivities K A and Kg. This body is used to trans fer heat between tw o objects maintained at
different temperatures . The composite body can be placed such that flow of heat takes place either
parallel to the interface or perpendicular to it. Calculate the eff ective thermal conductivities K . and
Kj Of the composite body for the parallel and perpendicular orie ntations . Which orientation will
have more thermal conductivity?
Q. 11 Two identical thermally insulated vessels, each containing n mole of an ideal monatomic gas, are
interconnected by a rod of length I and cross-sectional area 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 vessels is T, and T (< T ), neglecting thermal capac ity of the rod , calculate difference
2 } between temperature of gas in two vessels as a function of time. Q. 12 A highly conducting solid cylinder of radius a and length I i s surrounded by a co-axial layer of a
material having thermal conductivity K and negligible heat capacity . Temperature of surrounding space (out side the layer) is T , which is higher than temperature of the cylinder. If heat capacity
0
per unit volume of cylinder material is s and outer radius of the l ayer is b, calculate time required
to increase temperature of the cylinder from T t to T r Assume end faces t o be thermally insulated.
Q. 13 A vertical brick duct(tube) is filled with cast iron . The l ower 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
m
T which is less than the temperature of the melting point of cast iron. It is given that the conductivity of
2
liquid cast iron is equal to k times the conductivity of solid cas t iron. Determine the fraction of the duct
filled with molten metal.
Q.14 Water is filled in a non-conducting cylindrical vessel of unif orm cross-sectional area. Height of
water column is h and temperature is 0°C. If the vessel is exposed t o an atmosphere having constant
0
temperature of - 0°C (< 0°C) at t = 0, calculate total height h of th e column at time t .Assume thermal
conductivity ofice to be equal to K.Density ofwater is p and that of ice is p.. Latent heat of fusion of ice
f f i
isL.
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2
Q.15 A lagged stick of cross section area 1 cm and length 1 m is initial ly at a temperature of 0°C. It is then
kept between 2 reservoirs of tempeature 100°C and 0°C. Specific heat capa city 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 area of cross-section 0. 04m2 is placed coaxially on a thin
ace of the cylinder is maintained
at a constant temperature of 400K and the initial temperature of the disc is 300K . If the thermal
conductivity of the material of the cylinder is 10 watt/m-K and the s pecific 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 calculation, the thermal conductivity of th e disc to be very high and the
system to be thermally insulated except for the upper face of the c ylinder.
Q.17 A copper calorimeter of negligible thermal capacity is filled with a liquid. The mass of the liquid equals
250 gm. A heating element of negligible thermal capacity is immersed in the liquid. It is found that the
temperature of the calorimeter and its contents rises from 25°C to 30°C i n 5 minutes when a o r rent of
20.5 ampere is passed through it at potential difference of 5 volts. The liquid is thrown off and the heater is
again switched on. It is now found that the temperature of the calori meter alone is constantly maintained at
32°C when the current through the heater is 7A at the potential differe nce 6 volts. Calculate the specific heat
capacity of the 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 of twice the radius lose it s temperature at 327°C, if in both the
cases, the room temperature is maintained at 27°C.
Q.19 A calorimeter contains 100 cm 3 of a liquid o f density 0.8 8 g/cm3 in which are immersed a
thermometer and a small heating coil. The effective water equivalent of calorimeter, thermometer
and heater may be taken t o 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 tem perature is 55°C. The current is
increased so that the temperature rises slightly abo ve 55°C, and then it is switched off . The
calorimeter 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 wa s 10°C. If the roo m temperature rises to 26°C,
find the current required t o keep the liquid at 55°C . You may assu me that Newton's law is obeyed
and the resistance of the heater remains constant .
Q.20 End A of a rod AB of length L = 0.5 m and of uniform cross-sectional area is 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 the wavelength with maximum energy density emitted
from this end is X = 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 t hermally insulated .
3
Q.2 1 A wire of length 1.0 m and radius 10" m is carrying a heavy curr ent and is assumed to radiate as
a blackbody. At equilibrium temperature of wire is 900 K while that o f the surroundings is 300 K .
2 8
The resistivity of the material of the wire at 300 K is n x 10" O-m and its temperature coefficient
3 8 2 4
of resistance is 7.8 x 10' /°C . Find the current in the wire, [ a = 5.68 x 10" w/m K ].
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11---Q.22 The temperature distribution of solar radiation is more or less sa me as that of a black body whose
maximum emission corresponds to the wavelength 0.483 jam. Find the rate of change of mass due
8
to radiation . [Radius of Sun = 7.0 x 10 m]
Q.23 A black plane surface at a constant high temperature T , is paral lel to another black plane surface
h
at constant lower temperature T . Between the plates is vacuum. In order to reduce the heat flow due to
;
radiation, a heat shield consisting of two thin black plates, therm ally isolated from each other, it placed
between the warm and the cold surfaces and parallel to these. After some time stationary conditions are
obtained. By what factor r) is the stationary heat flow reduced du e to the presence of the heat shield?
Neglect end effects due to the finite size of the surfaces.
Q.24 The shell of a space station is a blackened sphere in which a tem perature T = 500K is maintained
due to operation of appliances of the station. Find t he temperature of the shell if the station is
enveloped by a thin spherical black screen of nearly the same radius as the radius of the shell.
Blacken ed
envelop
Q.25 A liquid takes 5 minutes to cool from 80°C to 50°C. How much time will it take to cool from 60°C to
30°C ? The temperature of surrounding is 20°C. Use exact method . Q .26 Find the temperature of equilibrium of a perfectly black disc exp osed normally to the Sun's ray on
the surface of Earth . Imagine that it has a nonconducting backing so that it can radiate only t o
s
hemisphere of space. Assume temperature of surface of Sun = 6200 K, radius of sun = 6.9 * 10 m,
1 1 s 2 4
distance between the Sun and the Earth = 1.5 x lo m. Stefan's constant = 5.7 x i0~ W/m .K .
What will be the temperature if both sides of the disc are radiate?
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EXERCISE - III
Q. 1 The temperature of 100 gm of water is to be raised from 24° C to 90° C by adding steam t o it.
Calculate the mass of the steam required for this purpose . [JEE '96] Q.2 Two metal cubes A & B of same size are arranged as shown in figu re.
The extreme ends of the combination are maintained at the ind icated
A B
temperatures . The arrangement is thermally insulated . The coeffici ents o
of thermal conductivity of A & B are 300 W/m° C and 200 W/m° C
respectively. After steady state is reached the temperature T of the interface will be . [JEE' 96]
Q.3 A double pane window used for insulating a room thermally from outside consists of two glass
2
thick stagnant air space. In the
steady state, the room glass interface and the glass outdoor interfa ce are at constant temperatures
of 27°C and 0°C respectively. Calculate the rate of heat flow through t he window pane . Also find
the temperatures of other interfaces . Given thermal c onductivities of glass and air as 0.8 and
0.08 W nr'K- 1 respectively. [JEE'97] Q. 4 The apparatus shown in the figure consists of four glass columns connected by horizontal sections . The height of two central
columns B & C are 49 cm each. The two outer columns A & D
are open t o th e atmosphere . A & C are maintained at a
temperature 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 & 5 1 cm respectively. Determine the coefficient A C
95° 95°
of thermal expansion of the liquid, [JEE '97]
Q.5 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 wou ld be : (A) 225 (B) 450 (C) 900 (D) 1800 2 2
Q.6 Earth receives 1400 W/m of solar power . If all the solar ener gy falling on a lens of area 0.2 m is
focussed on to a block of ice of mass 280 grams, the time take n t o melt the ice will be
5
minutes. (Latent heat of fusion of ice = 3.3 x 10 J/kg) [JEE '97] Q.7 A solid body X of heat capacity C is kept in an atmosphere whose temperature is T = 300K . At
A
time t = 0, the temperature of X is T = 400 K . It cools according t o Newton' s law of cooling. At
0
time tj its temperature is found to be 3 5 OK. At this time t p 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 conductivity K. The heat capacity of Y is so large that any variation in its temperature
may be neglected . The cross-sectional area A of the connecting rod is small compared to the
surface area of X . Find the temperature of X at time t = 3 t r [JEE' 98] Q.8 A black body is at a temperature of2880 K. The energy of radiation emitted by this 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
p 2
6
1500nmisU . TheWienconstantb = 2.88 x 10 nmK . Then [JEE' 98] 3 (A) Uj = 0 (B)U = 0 (C) Uj > U ( D ) U > U 3 2 2 1
(!l Bansal Classes Calorimetry & H eat Transfer [3] --- Page
13---Q.9 A bimetallic strip is formed out of two identical strips one of coppe r and the other of brass. The coefficient
of linear expansion of the two metals are a c and ct g. On heating, the temperature of the strip goes up by
AT and the strip bends to form an arc of radius of curvature R . Then R is :
(A) proportional at AT (B) inversely proportional t o AT [JEE' 99]
(C) proportional to lOg - a c | (D) inversely proportional t o |a B - a c |
Q.10 A block of ice at - 10°C is slowiy heated and converted to ste am at 100°C. Which of the following
curves represents the phenomenon qualitatively? [JEE (Scr) 2000]
(A) (B) \ (C) (D)
Heat supplied Heat supplied Heat supplied Heat supplied
Q. 11 The plots of intensity versus wavelength for three black bodies at
temperature T, , T 2 and T , respectively a re as shown . Then-
temperatures are such that [JEE (Scr) 2000]
( A ) T > T > T (B) T j > T > T
1 2 3 3 2 (C) T > T > T 2 3 1 (C) T . > T > T 2 t
Q . 1 2 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 len gth . The
left and right ends are kept at 0°C and 90°C respectively. The temperatu re ,S0°C
of the junction of the three rod s will be [JEE(Scr)200 1 ] o°c-
(A) 45°C (B) 60°C (C) 30°C (D)20° C "90°C
Q. 13 An ideal black body at room temperature is thrown into a furnace . It is observed that
(A) initially it is the darkest body and at later times the bright est .
(B) it the darkest body at all times
(C) it cannot be distinguished at all times.
(D) initially it is the darkest body and at later times it cannot be distinguished . [JEE(Scr)2002]
Q. 14 An ice cube of mass 0. 1 kg at 0°C is placed in an isolated container which is at 227°C . The specific
heat S of the container varies with temperature T accordi ng the empirical relations = A + BT,
2 2
where A = 100 cal/kg-K and B = 2 x 10~ cal/kg-K . If the final tempe rature of the container is 27°C,
4
determine the mass of the container. (Latent heat of fusion for water = 8 x \ o cal/kg. Specific heat of
3
water = 10 cal/kg-K) [JEE' 2001]
Q.15 Two rods one of aluminium of length /, having coefficient of linear expansion a , and other steel of
a
length l having coefficient of linear expansion a are joined end t o end. The expansion in both the
2 s
h
, . r is [JEE (Scr) 2003] n +/2 ac a 0 (A) (B) (C) (D) None of these a a + a s a a - a s Otc
(!l Bansal Classes Calorimetry & Heat Transfer [3]
--- Page
14---Q.16 2 kg ice at - 20°C is mixed with 5 kg water at 20°C. Then final amount ofwater in the mixture would be;
Given specific heat of ice = 0.5cal/g°C, specific heat ofwater = 1 ca l/g°C,
Latent heat of fusion of ice = 80 cal/g. [JEE (Scr) 2003]
(A) 6 kg (B) 5 kg ( C) 4 kg (D) 2 kg
Q.17 If emissivity of bodies X and Y are e and e and absorptive power x y are A x and Ay then [JE F (Scr) 2003] (A) e > e ; Ay > A ( B) e < e ; A < A y x x y x y x V.- ( C ) e y > e x ; A y < A x ( D) e y = e x ; Ay = A x
Q.18 Hot oil is circulated through an insulated container with a wooden li d 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
a
T, = 127°. If the ambient temperature T a = 27°C . C alculate -=• Hot oil
(a) rate of heat loss per unit area due to radiation from the lid. 17 _8
(b) temperature of the oil. (Given a = — x 10 ) [JEE 2003]
Q.19 Three discs A, B, and C having radii 2 m, 4 m and 6 m respectively a re coated with carbon black
on their outer surfaces . The wavelengths corresponding to maximum in tensity are 300 nm, 400 nm
and 500 nm respectively. The power radiated by them are Q , Q and Q respectively,
A B C
B) Q is maximum [JEE' 2004 (Scr.)] a B (C) Q is maximum (D) Q = Q = Q C A B C
Q.20 Two identical conducting rod s are first connected independent ly to two vessels, one containing
water at 100°C and the other containing ice at 0° C. In the second case , the rod s are joined end to
end and connected to the same vessels. Let qj and q 2 g/s be the rate of melting of ice in the two
cases respectively. The ratio q /q is 9 T
(A) 1/2 (B) 2/ 1 (C) 4/ 1 (D) 1/4 [JEE'2004 (Scr.)]
Q.2 1 Liquid oxygen at 50 K is heated t o 300 K at constant pressure of 1 atm. The rate of heating is
constant. Which of the following graphs represents the variation of temperature with time?
Temp.f Temp.f , Temp.f Temp. (A) (B) ( C) (D) Time Time Time Time [JEE' 2004 (Scr.)]
Q.22 A cube of coefficient of linear expansion a s is f loating in a bath containing a liquid of coefficient of
volume expansion y When the temperature is raised by AT, the depth upto which the cube is
t
submerged in the liquid remains the same. Find the relation between a s and y b showing all the
steps. [JEE 2004]
Q.23 One end of a rod of length L and cross-sectional area A is kept in a furnace of temperature T r The other end of the rod is kept at a
temperature T . The thermal conductivity of the material of the rod is K
2
and emissivity of the rod is e. It is given that T = T + AT where A T
2 s Insulated
« T , T being the temperature of the surroundings. If AT oc (Tj - T ) ,
s s s FurancFurancFuranceee T T Tfff Rod
find the proportionality constant. Consider that heat is lost onl y by radiation * L *
[JEE 2004] Insulated
2 (!l Bansal Classes Calorimetry & Heat T ransfer [3]
--- Page
15---Q. 24 Three graphs marked as 1,2, 3 representing the variation of maximum emi ssive
power and wavelength of radiation of the sun, a welding arc and a tungst en
filament. Which of the 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-bulb, 1 - > welding arc, 3 sun [JEE' 2005 (Scr)]
Q. 25 In which of the following phenomenon heat convection does not take plac e
(A) land and sea breeze (B) boiling of water
(C) heating of glass surface due to filament of the bulb
(D) air around the furance [JEE' 2005 (Scr)]
Q.26 2 litre water at 27°C is heated by a 1 kW heater in an open container . O n 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 inside a bl ack body. What is the rate at which
energy is radiated per second at temperature T
(A) 0.6 a AT4 (B)0.4aAT 4 (C)0.8cAT 4 (D)l.OaAT 4
[JEE 2005 (Scr)]
Q. 28 1 calorie is the heat required to increased the temperature of 1 gm of water by 1 °C from
(A) 13.5°Cto 14.5°C at 76 mm of Hg (B) 14.5°Cto 15.5°Cat760mmof H g
(C) 0°C to 1°C at 760 mm of Hg (D) 3°C to 4°C to 760 mm of H g
[JEE* 2005 (Scr)]
(!l Bansal Classes Calorimetry & Heat Transfer [3]
--- Page
ANSWER KEY
EXERCISE -1
Q.i 25.5°C Q.2 0. 1 cm Q.3 65°C Q.4 2000 cm3
Q.5 5 sec slow Q.6 10 sec Q.7 15 K/16 Q.8 10, 000 N /M/3 6 Q.9 Q.10 0.2 Q.l l 4 x 10 - 6 m/° C .71. 0 1 Q.12 7/2 Q.13 5 a/ 3 Q.14 4°C Q.15 1000 J (C )- Q.16 5°C Q.17 27/85 Q.18 2 : 1 Q.19 (4/3)© Q.20 1/90 Q.2 1 h/5R Q.22 3Y /2 0 Q.23 0 °C Q.24 104.2 Q.25 10cm, Q.27 10.34 cm Q.28 ctPy : : a 2 Q.29 80 k cal/kg Q.30 3025 K EXERCISE-II 1 1 1 - 1 Q.I 1 : 1.26 Q.3 800 cal kg" K , 1000 cal kg" K Q.4 (a) 37.8 J/s (Watts), (b) 2.005 N-m Q.5 25 kJ Q.6 9.02 x 105 gm 3 Q.7 5000 J/°C kg Q.8 decrease by 0.75 cm ,25°C Q.9 45°C T . K A + K R V 2 K A K B ( 4KAtN | Q.10 K„ > Kj_, K | = ; K x - Q.l l "\3nRi J 1 (T, ~T ) e B 2 2 k(Tt - T m ) a s . (-)l0g Q.12 ^ l o g e T T Q 1 3 I k ( T 1 - T m ) + ( T m - T 2 ) V. 0 ~ 2 J / \ 1 12k;6t 11 -- JBL Q.14 h 0 + - Q.15 (a) 100 °C/m, (b) 1000 J Q.16 166.3 sec L V \ P i f Q.17 21000 Jkg^Kr 1 Q.18 9.72°C/min
Q.19 (a)0.42 cal/gm°C, (b) 1.6A Q.20 T = 423 K 9 a Q.2 1 36 A Q.22 ~ = 5.06 x 10 kg/ s dt Q.23 r | = 3 Q.24 T " = x 500 = 600 K Q.25 10 minutes Q.26 T 0 = 420 K, T 0 = 353.6 K EXERCISE-III Q.I 12 gm Q.2 60° C Q.3 41.53 Watt; 26.4 8 °C;0.55° C Q.4 2 x 10^ C Q.5 D Q.6 5.5 min log 2 e Q.7 k = ; T = 300 + 50 exp. [LC tj Q.8 D Q.9 B, D Q.10 A Q.l l B Q.12 B Q.13 D Q.14 0.5 kg 2 Q.19 B Q.15 A Q.16 A Q.17 A Q.1 8 (a) 595 watt/m , ( b ) T 0 * 4 2 0 K K Q.24 A 4 e a L T f + K Q.20 D Q.2 1 C Q.22 y,= 2as Q.23 Q.25 C Q.26 A Q.27 A Q.2 8 B (!l Bansal Classes Calorimetry & Heat Tra nsfer [3]
--- Page 17---BA
TARGET IIT JEE 2007 XII (ALL) C O H T E N T S KEYCONCEPTS EXERCISE-1 EXERCISE-II EXERCISE-III ANSWER KEY --- Page KEY CONCEPTS
1. CAPACITANCE O F A N ISOLATED SPHERICA L CONDUCTO R :
C = 471 e e R in a medium C = 47C G „ R in air
0 (
This sphere is at infinite distance from all the conduct ors .
The Capacitance C = 47T ER exists between the surface o f the sphere & earth .
Q SPHERICAL CAPACITO R :
It consists of two concentric spherical shells as shown in figure. H ere capacitance of region
between the two shells is C and that outside the sh ell is C . We have t 2 471 e n ab C = and C = 471 e b 2 Q b - a
Depending on connection, it may have different combinations of C, an d -C .
2
3 . PARALLEL PLATE CAPACITOR :
( i ) UNIFOR M DI-ELECTRI C MEDIU M :
If tw o parallel plates each of area A & separated by a distance d are charged with
equal & opposite charge Q, then the system is called a parallel plat e capacitor & its capacitance is
given by,
^ S)6 A . r
C = — ; — in a medium C = with air as medium
U
This result is only valid when the electric field between plates of capacitor is constant,
So A
( i i ) M E D I U M PARTLY A I R : C =
d - l t - i
When a di-electric slab of thickness t & relative permit tivity e r is lll l
introduced between the plates of an air capacitor, then the distan ce between
PP 33
the plates is effectively reduced by irrespective of the position of BSSSSiiBSSSSii®®
V ^r J
the di-electric slab .
G A 0 ( i i i ) COMPOSIT E M E D I U M : c = I I -r l r2 r3 4 . CYLINDRICAL CAPACITO R :
It consist of two co-axial cylinders of radii a & b, the outer condu ctor is earthed .
The di-electric constant of the medium filled in the space between t he cylinder is
2n e e Farad
n
e . The capacitance per unit length is C = y - r
r
in m
(fe^Bansal Classes CAPACI TANCE 121
--- Page CONCEPT o r VARIATION OF PARAMETERS :
e 0 k A
As capacitance of a parallel plate capacitor isC = , if either of k, A or d varies in the region between
the plates, we choose a small dc in between the plates and for total c apacitance of system.
1 dx
If all dC's are in series -, If al l dC's are in parallel C T = } dC
6. COMBINATION O F CAPACITOR S : ( i ) CAPACITOR S I N SERIE S :
In this arrangement all the capacitors when uncharged get the same cha rge Q Q Q
Q but the potential difference across each will differ (if the capaci tance are rIMHh
C | C2 C3 unequal). v, v, v, 1 1 1 1 1 — + — + — + + C3
(ii) CAPACITORS I N PARALLE L :
When one plate of each capacitor is connected t o the pos itive % Cj.V
terminal of the battery & the other plate of each capacitor is 1
connected t o the negative terminals of the bat tery , then the c,,v
s 1
capacitors are said to be in parallel connection . jC ,y
% 3
The capacitors have the same potential difference, V but the 1
charge on each one is different (if the capacitors are uneq ual) . Q +v
C I + C 2 + C 3 + + c eq.
ENERGY STORE D I N A CHARGE D CAPACITOR :
Capacitance C, charge Q & potential difference V ; then energy stor ed is
1 1 1 Q2 2
U = - CV = — QV = - — . This energy is stored in the electrostatic field set up in the di-electric
medium between the conducting plates of the capacitor . HEA T PRODUCED IN SWITCHING IN CAPACITIVE CIRCUIT
Due to charge flow always some amount of heat is produced when a swit ch is closed in a circuit which
can be obtained by energy conservation as -
Heat = Work done by battery - Energy absorbed by capacitor. 9. SHARING O F CHARGE S :
When two charged conductor s of capacitance C s & C 2 at potential V } & V 2 respectively are
connected by a conducting wire, the charge flows from higher potentia l conductor to lower potential
conductor, until the potential of the two condensers become s equal . The common potential (V)
after sharing of charges ; net charg e _ q j + q C,V C V V = 2 1 + 2 2 net capacitance C, + C C + C C, + C 2 t 2 2
charges after sharing qj = C,'V & q = C V . In this proces s energy is lost in the connecting wire
2 2 C C ( V , - V ) 2 2
as heat . This loss of energy is U i n i t i a l - U e a l = ^ r ^ g
10
REMEMBER :
(i) The energy of a charged conductor resides outside the conductor in its EF, where as in a condenser
it is stored within the condenser in its EF. (ii) The energy of an uncharged condenser = 0 .
(iii) The capacitance of a capacitor depends only on its siz e & geometry & the di-electric between the
conducting surface .(i.e. independent of the conductor, like, whether it is copper, silver, gold etc)
<§Bansal Classes CAPACITANCE --- Page
EXERCISE # I
Q.i A solid conducting sphere of radius 10 cm is enclosed by a thin met allic shell of radius 20 cm. A charge
q = 20pC is given to the inner sphere. Find the heat generated in the process, the inner sphere is
connected to the shell by a conducting wire
Q.2 The capacitor each having capacitance C = 2pF are connected with a battery of emf 30 V as shown in figure. When the switch S is closed. Find .CO,
(a) the amount of charge flown through the battery (b) the heat generated in the circuit
(c) the energy supplied by the battery
(d) the amount of charge flown through the switch S ' 3 0 V
Q.3 The plates of a parallel plate capacitor are given charges +4Q and -2 Q . The capacitor is then connected
across an uncharged capacitor of same capacitance as first one (= C). Find the final potential difference
between the plates of the first capacitor. +i , -
Q.4 In the given network if potential difference between p and q is 2V an d
H Mq C, C. c 2 C 4 C 8C 11 11 11 11 11 ! 1 \ \ \ \
Q.5 Find the equivalent capacitance of the circuit - C : : C : c \
r y-
between point A and B. Infinite / 11 11 11 11 11 II section/ c 2C 4 C 8C + 3 q + q
Q.6 The two identical parallel plates are given charges as shown in figur e. If the
plate area of either face of each plate is A and separation between p lates is
d, then find the amount of heat liberate after closing the switch. Q. 7 Find heat produced in the circuit shown in figure on closing the switc h S.
Q.8 In the following circuit, the resultant capacitance between A and B is 1 pF. Find the value of C.
T T 2 ^ f
Q.9 Three capacitors of 2pF, 3pF and 5|iF are independently charged with batteries of emf' s 5V, 20V and 10V res pectively . After
disconnecting from the voltage sources. These capacitors are connecte d
as shown in figure with their positive polarity plates are connected to ^Slr
A and negative polarity is earthed . Now a battery of 20V and an 2\xV
uncharged capacitor of 4jaF capacitance are connected to the junction I'—^—II—I
20V 4|j.F 4r
A as shown with a switch S. When switch is closed, find : 5NF \
(a) the potential of the junction A.
(b) final charges on all four capacitors.
(fe^Bansal Classes CAPACITANCE 121
--- Page
7
Q.10 Find the charge on the capacitor C = 1 pF in the circuit shown in th e figure. Iph IjxK
IpF IpF C-luF l(iF
:IMF : :pnF yUlF: k = 1 k = 2
Q.l l Find the capacitance of the system shown in figure.
k = 3 k = 4
Q.12 The figure shows a circuit consisting of four capacitors. Find the effective capacitance between X and Y.
Q. 13 Five identical capacitor plates, each of area A, are arranged such th at
adjacent plates are at a distance'd* apart, the plates are connected t o a
source of emf V as shown in figure. The charge on plate 1 V-
is and that on plate 4 is . +
Q.14 In the circuit shown in the figure, intially SW is open. X 60 V 2 nF1
When the switch is closed, the charge passing through AE- SW 3 J
the switch in the direction 60 V
to I
Q.15 In the circuit shown in figure, find the amount of heat generated when switch s is closed.
Q.16 Two parallel plate capacitors of capacitance C and 2C are connected in parallel then following steps are performed.
(i) Abattery of voltage V is connected across points A and B .
(ii) A dielectric slab of relative permittivity k is slowly inserted i n capacitor C.
(iii) Battery is disconnected.
(iv) Dielectric slab is slowly removed from capacitor.
(ii) & (iv).
Q.17 The plates of a parallel plate capacitor are separated by a distance d = 1 cm. Two parallel sided dielectric
slabs of thickness 0.7 cm and 0.3 cm fill the space between the plates . If the dielectric constants of the two
slabs are 3 and 5 respectively and a potential difference of440V is ap plied across the plates. Find :
(i) the electric field intensities in each of the slab s.
(ii) the ratio of electric energies stored in the first to that in the seco nd dielectric slab.
Q.18 A 10 pF and 20 pF capacitor are connected to a 10 V cell in parallel for some time after which the
capacitors are disconnected from the cell and reconnected at t = 0 wit h each other , in series, through
wires of finite resistance. The +ve plate of the first capacitor is co nnected to the -v e plate of the second
capacitor. Draw the graph which best describes the charge on the +ve p late of the 20 pF capacitor with
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, 3.14 1,3.142, 3.177,3.184, 3.188. 3.199. 3.200,3.201 . 3.203, 3.20 4. 3.205 121
(fe^Bansal Classes CAPACITANCE --- Page EXERCISE # II 6oF, Ih-H^f
Q. 1 (a) For the given circuit. Find the potential difference across all the capacitors, —'—I h
(b) How should 5 capacitors, each of capacities, lp F be connected s o 9|iF 8(xF as to produce a total capacitance of 3/7 pF. + . -
25V
Q.2 The gap between the plates of a plane capacitor is filled with an i sotropic insulator whose di-electric
71 constant varies in the direction perpendicular to the plates accordi ng to the law K = Kj 1 + sin — X L d where d is the separation, between the plates & K t is a constant. The area of the plates is S. Determine
the capacitance of the capacitor.
5
to and equdistant from each other (see figure). 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 connected to a source of constant e.m.f. V . Find ;
0
(i) the effective capacity of the system between the terminals of the so urce.
(ii) the charges on plates 3 & 5.
Given d = distance between any 2 successive plates & A = area of e ither face of each plate .
Q.4 Apotential difference of30 0 Vi s applied between the plates of a p lane capacitor spaced 1 cm apart. A
plane parallel glass plate with a thickness of 0.5 cm and a plane par allel paraffin plate with a thickness of
0.5 cm are placed in the space between the capacitor plates find : (j) Intensity of electric field in each layer.
(ii) The drop of potential in each layer.
(iii) The surface charge density of the charge on capacitor the plates. G iven that : k g l a s s = 6, k p a r a f f i n = 2
Q.5 A charge 200pC is imparted to each of the two identical parallel p late capacitors connected in parallel.
At t =0, the plates of both the capacitors are 0. 1 m apart . The pla tes of first capacitor move towards
each other with relative velocity 0.00 1 m/s and plates of second ca pacitor move apart with the same
velocity. Find the current in the circuit at the moment.
Q.6 A parallel plate capacitor has plates with area A & separation d . A battery charges the plates to a
potential difference of V . The battery is then disconnected & a di-electric slab of constant K & thickness
0
d is introduced . Calculate the positive work done by the system (ca pacitor + slab) on the man who
introduces the slab.
Q.7 A capacitor of capacitance C is charged to a potential V and then isolated. A small capacitor C is then
0 0
charged from C , discharged & charged again, the process being repea ted n times. The potential of the
0
large capacitor has now fallen to V. Find the capacitance of the smal l capacitor. If V 0 = 100 volt, V=35volt, find the value of n for C = 0.2 pF & C = 0.01075 pF . Is i t possible to remove charge on
0 C this way?
Q. 8 When the switch S in the figure is thrown to the left, the plates o f capacitors . V
TLP I C, acquire a potential difference V. Initially the capacitors C C are
2 3
uncharged. Thw switchis now thrown to the right. What are the final c harges I c q q & q on the corresponding capacitors. T p 2 3 (fe^Bansal Classes C APACITANCE 121 --- Page
23---Q.9 A parallel plate capacitor with air as a dielectric is arranged hori zontally. The lower plate
is fixed and 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 capa citor is connected
0
with an electric source with the voltage V, a new equilibrium appears , with the distance
between the plates as d r Mass of the upper plates is m. (1) Find the spring constant K.
(ii) What is the maximum voltage for a given K in which an equilibrium is possible ?
(lii) What is the angular frequency of the oscillating system around the e quilibrium value dj .
(take amplitude of oscillation « d ) {
Q.10 An insolated conductor initially free from charge is charged by repe ated contacts with a plate which after
each contact has a charge Q due to some mechanism . If q is the charg e on the conductor after the first
operation, prove that the maximum charge which can be given to the co nductor in this way is ~
Q.l l A parallel plate capacitor is filled by a di-electric whose relative permittivity varies with the applied
voltage according to the law = aV , where a = 1 per volt . The sa me (but containing no di-electric)
capacitor charged to a voltage V = 156 volt is connected in parallel to the first "non-linear" uncharged
capacitor. Determine the final voltage V across the capacitors. f
er of radius b is fixed, and the inner is of
radius a If breakdown of air occurs at field strengths greater t han E^, show that the inner cylinder should have
(i) radius a = b/e if the potential of the inner cylinder is to be maxim um
(ii) radius a = b/V e if the energy per unit length of the system is to be maximum.
,.JT 4=6nF
5 V - r -
Q. 13 Find the charge flown through the switch from At o B when it is clos ed. 6 m F Jr~
5V :d=6nf
Q.14 Figure shows three concentric conducting spherical shells with inner and outer shells earthed and the middle shell is given a charge q. Find the electrostatic energy of the system stored in the region I and II.
Q.15 The capacitors shown in figure has been charged to a potential differ ence
of V volts, so that it carries a charge CV with both the switches Sj and S 2
remaining open. Switch Sj is closed at t=0. At t=R,C switch Sj is opened s,
and S is closed. Find the charge on the capacitor at t=2Rj C + R^C. 2
Hi
s,
Q.16 In the figure shown initially switch is open for a long time. Now the switch is closed at t = 0. Find the charge on the rightmost capacito r as "y v
a function of time given that it was intially unchanged.
121
(fe^Bansal Classes CAPACITANCE --- Page
24---Q.17 In the given circuit, the switch is closed in the position 1 at t = 0 and then moved , I V
:500FJ
to 2 after 250 p,s. Derive an expression for current as a function of time for J^ov [ 2
t > 0. Also plot the variation of current with time. I X40V :0.5 NF
VL
