PHYSICS HEAT TRANSFER

PHYSICS

HEAT TRANSFER

Board Level Exercise

1. The SI unit of thermal conductivity is

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2. Name the three modes of transfer of heat.

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3. De�ne coe�cient of thermal conductivity or thermal conductivity

of a substance.

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4. What are the basic requirements of a cooking utensil in respect of

speci�c heat, thermal conductivity and coe�cient of expansion?

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5. The ratio of thermal conductivites of two di�erent metals is . In

order to have the same thermal resistance in these metals of equal

thickness what should be the ratio of their length?

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5: 3

6. Two rods of length and coe�cients of thermal

conductivities are kept touching each other. Both have

the same area of cross-section. The equivalent of thermal

conductivity is

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l1 and l2

k1 and K2

7. Name the factor a�ecting the Centre of gravity of a body.

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8. What is the shift in the colour of light when the temperature

increase?

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9. Even when earth receives solar energy, why is it not getting

warmed up continuously?

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10. De�ne solar constant.

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11. Distinguish between Conduction and convections

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12. List the salient feastures of heat radiations.

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Exercise 1

13. State kirchho�'s law of blakc radiations.

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14. Show graphically the temperature variation with time associated

with a colling hot body.

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1. A uniform slab of dimension is kept between

two heat reservoir at temperatures and . The larger

surface areas touch the reservoirs. The thermal conductivity of the

material is . Find the amount of heat �owing

through the slab per minute.

W t h Vid S l ti

10cm × 10cm × 1cm

10∘C 90∘C

0.80Wm− 1C − 1

Exercise

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1. One end of a steel rod of length is

kept in ice at and the other end is kept in boiling water at

. The area of cross section of the rod is . Assuming

no heat loss to the atmosphere, �nd the mass of the ice melting per

second. Latent heat of fusion of ice .

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(K = 46Js− 1m− 1C − 1) 1.0m

0∘C

100∘C 0.04cm2

= 3.36 × 105Jkg− 1

2. A rod CD of thermal resistance is joined at the middle of

an identical rod AB as shown in �gure. The ends A, B and D are

maintained at and respectively. Find the heat current in

5.0K/W

100∘C 25∘C

CD.

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3. A semicircular rods is joined at its end to a straight rod of the

same material and the same cross-sectional area. The straight rod

forms a diameter of the other rod. The junctions are maintained at

di�erent temperatures. Find the ratio of the heat transferred

through a cross section of the straight rod in a given time.

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4. Three slabs of same surface area but di�erent conductivities

and di�erent thickness are placed in close contact.

After steady state this combination behaves as a single slab. Find its

e�ective thermal conductivity.

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k1, k2, k3 t1, t2, t3

5. A hollow metallic sphere of radius surrounds a concentric

metallic sphere of radius . The space between the two sphere is

�lled with a nonmetallic material. The inner and outer sphere are

maintained at and respectively and it is found that

of heat passes from the inner sphere to the outer sphere per second.

Find the thermal conductivity of the material between the sphere.

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20cm

5cm

50∘C 10∘C 100J

6. A hollow tube has a length l, inner radius and outer radius .

The material has a thermal conductivity K. Find the heat �owing

through the walls of the tube if (a) the �at ends are maintained at

temperature and (b) the inside of the tube is

maintained at temperature and the outside is maintained at .

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R1 R2

T1 T2(T2 > T1)

T1 T2

7. A metal rod of cross sectional area is being heated at one

end. At one time , the temperature gradient is at cross

section A and is at cross section B. Calculate the rate at

which the temperature is increasing in the part AB ,

thermal conductivity of the material of the rod.

^(@)C^(-1)` . Neglect any loss of heat to the atmosphere.

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1.0cm2

5.0∘Ccm− 1

2.5∘Ccm− 1

= 0.40J ∘C − 1

= 200Wm− 1

8. When joules of radiation is incident on a body if re�ects and

transmits total of joules. Find the emissivity of the body.

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q1

q2

9. A blackbody of surface area is placed inside an enclosure. The

enclosure has a constant temperature and the blackbody is

1cm2

27( ∘ )C

maintained at by heating it electrically. What electric power is

needed to maintain the temperature? .

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327∘C

σ = 6.0 × 10− 8Wm− 2K − 2

10. Estimate the temperature at which a body may appear blue or

red. The values of for these are and respectively.

[Given Wein's constant

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λmean 5000 7500Å

b = 0.3cmK]

11. The temperature of a hot liquid in a container of negligible heat

capacity falls at the rate of due to heat emission to the

surroundings, just before it begins to solidify. The temperature then

remains constant for , by the time the liquid has all solid�ed.

Find the ratio of speci�c heat capacity of liquid to speci�c latent heat

of fusion.

h id l i

3K/ min

30 min

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12. The earth receives its surface radiation from the sun at the rate of

1400 . The distance of the centre of the sun from the surface

of the earth is m and the radius of the sun is m.

Treating sun as a black body, it follows from the above data that its

surface temeperature is

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W /m2

1.5 × 1011 7.0 × 108

13. A solid copper sphere (density and speci�c heat c) of radius r at

an initial temperature 200 K is suspended inside a chamber whose

walls are at almost 0 K. The time required for the temperature of the

sphere to drop to 100 K is _________

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ρ

ρnc

14. A liquid cools from to in minutes. Calculate the time

taken by the liquid to cool from to , If the temperature

of the surrounding is constant at .

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70∘C 60∘C 5

60∘C 50∘C

30∘C

15. A wall has two layers A and B, each made od=f di�erent material.

Both the layers have the same thickness. The thermal conductivity for

A is twice that B and, under steady condition, the temperature

di�erence across the wall is C. The temperature di�erence across

the layer A is

A.

B.

C.

D.

36∘

6∘C

12∘C

18∘C

24∘C

Answer: B

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16. Two metal cubes with 3 cm edges of copper and aluminium are

arranged as shown in �gure. Find

(i) the total thermal current from one reservoir to the other.

(ii) the ratio of the thermal current carried by the copper cube to

that carried by the aluminium cube . thermal conductivity of copper

is and that of aluminium is .

A.

401Wm− 1K 237Wm( − 1)K

1.42 × 103W

B.

C.

D.

Answer: A

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2.53 × 103W

1.53 × 104W

2.53 × 104W

17. Two metal cubes with 3 cm edges of copper and aluminium are

arranged as shown in �gure. Find

(i) the total thermal current from one reservoir to the other.

(ii) the ratio of the thermal current carried by the copper cube to

that carried by the aluminium cube . thermal conductivity of copper

is and that of aluminium is .

A.

B.

C.

D.

Answer: D

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401Wm− 1K 237Wm( − 1)K

1.79

1.69

1.54

1.84

18. A wall consists of alternating blocks with length 'd' and coe�cient

of thermal conductivity and . The cross sectional area of the

blocks are the same. The equivalent coe�cient of thermal

k1 k2

conductivity of the wall between left and right is

A. K1 + K2

B.

C.

D.

Answer: B

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(K1 + K2)

2

K1K2

K1 + K2

2K1 + K2

K1 + K2

19. A boiler is made of a copper plate thick with an inside

coating of a thick layer of tin The surface area exposed to

gases at is The maximum amount of steam that could

be generated per hour at atmospheric pressure is

.

A.

B.

C.

2.4mm

0.2mm

700∘C 400cm2

(Kcu = 0.9cal/cm − s −0 &ktin = 0.15cal / c /0 C

and Lsteam = 540cal/g)

m

s

5000Kg

1000Kg

4000Kg

D.

Answer: C

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200Kg

20. A lake surface is exposed to an atmosphere where the

temperature is less than , if the thickness of the ice layer formed

on the surface grows from 2 cm to 4 cm in 1 hour, the atomospheric

temperature will be

(Thermal conductivity of ice,

density of ice . Latent of density during of the state

change. Assume that the water below the ice has temperature

every where.)

A.

B.

C.

0∘C

K = 4 × 10− 3calcm− 1s− 1. ∘ C − 1

= 0.9gcc− 1

0∘C

−20∘C

0∘C

−30∘C

D.

Answer: C

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−15∘C

21. Heat �ows radially outward through a spherical shell of outside

radius and inner radius . The temperature of inner surface of

shell is and that of outer is .The radial distance from centre of

shell where the temperature is just half way between and is "

A.

B.

C.

D.

Answer: C

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R2 R1

θ1 θ2

θ1 θ2

R1 + R2

2

R1R2

R1 + R2

2R1R2

R1 + R2

R1 +R2

2

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22. A solid metallic sphere of diameter 20 cm and mass 10 kg is

heated to a temperature of and suspended in a box in which

a constant temperature of is maintained. Find the rate at

which the temperature of the Sphere will fall with time. Stefan's

constant and speci�c heat of metal

.

A.

B.

C.

D.

Answer: B

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327∘C

27∘C

= 5.67 × 10− 8W /m2 /K4

= 420J /kg/∘ C

0.55∘C /sec

0.66∘C /sec

0.44∘C /sec

0.03∘C /sec

23. Which of the law can be understood in terms of Stefan's law

A. Wien's displacement law

B. Kirho�'s law

C. Newton's law of cooling

D. Planck's law

Answer: C

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24. A hot liquid is kept in a big room. According to Newton's law of

cooling rate of cooling liquid (represented as y) is plotted against its

temperature . Which of the following curves may represent the

plot?

T

A.

B.

C.

D.

Answer: B

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25. Three metal rods made of copper, aluminium and brass, each

long in diameter, are placed end to end with aluminium

between the other two. The free ends of copper and brass are

maintained at and respectively. Assume that the thermal

conductivity of copper is twice that of aluminium and four times that

of brass. The approximately equilibrium temperatures of the copper-

aluminiu and aluminium-brass junctions are respectively.

A. and

B. and

C. and

D. and

Answer: D

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20cm 4cm

100 0∘C

68∘C 75∘C

75∘C 68∘C

57∘C 86∘C

86∘ 57∘C

26. A closed cubical box is made of perfectly insulating material and

the only way for heat to enter or leave the box is through two solid

cylindrical metal plugs, each of cross sectional area and

length 8cm �xed in the opposite walls of the box. The outer surface

of one plug is kept at a temperature of . while the outer

surface of the plug is maintained at a temperature of . The

thermal conductivity of the material of the plug is . A

source of energy generating 13W is enclosed inside the box. Find the

equilibrium temperature of the inner surface of the box assuming

that it is the same at all points on the inner surface.

A.

B.

C.

D.

12cm2

100∘C

4∘C

2.0Wm− 1C − 1

62∘C

46∘C

76∘C

52∘C

Answer: C

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27. Two models of a windowpane are made, two identical glass panes

of thickness are separated with an air gap of . This

composite system is �xed in the window of a room. The other model

consists of a single glass pane of thickness , the temperature

di�erence being the same as for �rst model. the ratio of the heat

�ow for the double pane to that for the single pane is

and

A.

B.

C.

D.

3mm 3mm

6mm

(Kglass = 2.5 × 10− 4cal/s. m. ∘ C

Kair = 6.2 × 10− 6cal/s. m. ∘ C)

1/20

1/70

31/1312

31/656

Answer: D

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28. Heat is �owing through two cylindrical rods of the same material.

The diamters of the rods are in the ratio and the length in the

ratio . If the temperature di�erence between the ends is same

then ratio of the rate of �ow of heat through them will be

A.

B.

C.

D.

Answer: A

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1: 2

2: 1

1: 8

1: 4

1: 6

4: 1

29. The ends of a metre stick are maintain at and . One

end of a rod is maintained . Where should its other end be

touched on the metre stick so that there is no heat current in the

rod in steady state?

A. form the hot end

B. from the cold end

C. from the cold end

D. from the cold end

Answer: C

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100∘C 0∘C

25∘C

25cm

40cm

25cm

60cm

30. A spherical solid blakc body of radius 'r' radiates power 'H' and its

rate of cooling is 'C'. If density is constant then which of the following

is/are true.

A. and

B. and

C. and

D. and

Answer: B

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H ∝ r c ∝ r2

H ∝ r2 c ∝1

r

H ∝ r c ∝1

r2

H ∝ r2 c ∝ r2

31. Two rods of same dimensions, but made of di�erent materials are

joined end to end to end with their free end being maintained at

and respectively. The temperature of the junction is

. Then the temperature of the junction if the rods are

inerchanged will be equal to Find :

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100∘C 0∘C

70∘C

T ∘C T

32. Figure shows a steel rod joined to a brass rod. Each of the rods

has length of and area of cross-section . The junction

is maintained at a constant temperature and the two ends are

maintained at . The amount of heat taken out from the cold

junction in minutes after the steady state is reached in .

Find 'n' the thermal conductivities are and

.

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31cm 0.20cm2

50∘C

100∘C

10 n × 102J

Ksteel = 46W /m −∘ C

Kbrass = 109W /m −∘ C

33. Consider the situation shown in �gure. The frame is made of the

same material and has a uniform cross-section area everywhere.

Calculate the amount of heat �owing per second through a cross

section of the bent part if the total heat taken out per second from

the end at is .

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100∘C 130J

34. Four thin identical rods and made of the same

material are joined as shown. The free-ends and are

maintained at temperature and respectively. Assuming that

there is no loss of heat of the surroundings, the temperature at joint

when the steady state is attained is . Find

AB, AC, BD EF

C, D F

T1, T2 T3

E (2T1 + 2T2 + 3T3)1

K

is mid point of

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K(E AB)

35. One end of a copper rod of uniform cross section and length 1.5

m is kept in contact with ice and the other end with water at .

At what point along its length should a temperature of be

maintained so that in the steady state, the mass of ice melting be

100∘C

200∘C

equal to that of the steam produced in same interval of time. Assume

that the whole system is insulated from surroundings:

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[Lice = 80cal/g, Lsteam = 540cal/g]

36. A hollow spherical conducting sheel of inner radius

and outer radius is placed inside a heat reservoir of

temperature . The shell is initially �lled with water at

. The thermal conductivity of the material is

and its heat capacity is negligible. The time

required to raise the temperature of water to

. Find . Take speci�c heat of water

R1 = 0.25m

R2 = 0.50m

T0 = 1000∘C

0∘C

k = W /m − K102

4π

100∘Cis1100K ln. sec10

9K

, density of water

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s = 4.2kJ /kg. ∘ C dw = 1000kg/m3π =22

7

37. A cylindrical rod of length 50cm and cross sectional area is

�tted between a large ice chamber at and an evacuated

chamber maintained at as shown in �gure. Only small protions

of the rod are insid ethe chamber and the rest is thermally insulated

from the surrounding. The cross section going inti the evacuted

chamber is blackened so that it completely absorbe any radiation

falling on it. The temperatuere of the blackened end is when

1cm2

0∘C

27∘C

17∘C

steady state is reachhed. Stefan constant

. Find the thermal conductivity of the material of the rod.

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σ = 6 × 10−sWm− 2K − 4

38. A spherical tungsten pieces of radius is suspended in an

evacuated chamber maintained at . The pieces is maintained at

1000K by heating it electrically. Find the rate at which the electrical

energy must be supplied. The emissivity of tungsten is and the

Stefan constant is .

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1.0cm

300K

0.30

σ 6.0 × 10−sWm− 2K − 4

39. Assume transmitivity for all the cases:t → 0

A. bad absrober is bad emitter

B. bad absorber is good re�ector

C. bad re�ector is good emitter

D. bad emitter is good absorbed

Answer: A::B::C

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40. A solid sphere and a hollow sphere of the same material and of

equal radii are heated to the same temperature.

A. in the beginning both will emit equal amount of radiation per

unit time

B. in the beginning both will absorb equal amount of radiation

per unit time

C. both sphers will have same rate of fall of temperature

D. both spheres will have equal temperature at any constant

Answer: A::B

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(dT /dt)

41. Two bodies A and B have thermal emissivities of 0.01 and 0.81

respectively. The outer surface areas of the two bodies are same. The

two bodies emit total radiant power at the same rate. The

wavelength corresponding to maximum spectral radiancy from B

is shifted from the wavelength corresponding to maximum spectral

radiancy in the radiation from A by 1.0 . If the temperature of A is

5802 K, calculate (a) the temperature of B, (b) wavelength .

A. the temperature of

B.

λB

μm

λB

Bis1934K

λB = 1.5μm

C. the temperature of

D. the temperature of

Answer: A::B

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Bis1160K

Bis2901K

42. The solar constant is the amount of heat energy received per

second per unit area of a perfectly black surface placed at a mean

distance of the Earth form the Sun, in the absence of Earth's

atmosphere, the surface being held perpendicular to the direction of

Sun's rays. Its value is . If the solar constant for the earth

is 's'. The surface temperature of the sun is is the diameter of

the sun, is the mean distance of the Earth from the sun. The sun

subtends a small angle 'theta' at the earth. Then correct options

is/are:-

A.

1388W /m2

TK, D

R

s = σT 4( )2

D

R

B.

C.

D.

Answer: B::C

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s = ( )2

σT 4

4

D

R

s = θ2σT 4

4

s = ( )2

σT 4

4R

D

43. A heated body emits radiation which has maximum intensity at

frequency . If the temperature of the body is doubled :

A. the maximum intensity radiation will be at frequency

B. the maximum intensity radiation will be at frequency

C. the total emitted power will increase by a factor

D. the total emitted power will increase by a factor

Answer: A::C

vm

2vm

vm

16

2

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44. Figure shows in cross section a wall consisting of four layers with

thermal conductivities

.

The layer thickness are .

The temperature of interfaces is as shown in �gure. energy transfer

through the wall is in steady state. the temperature of the interface

between layer is:

A.

B.

C.

K1 = 0.06W /mK, K3 = 0.04W /mK and K4 = 0.10W /mK

L1 = 1.5cm, L3 = 2.8cm and L4 = 3.5cm

3 and 4

−1∘C

−3∘C

2∘C

D.

Answer: B

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0∘C

45.

Figure shows in cros section a wall consisting of four layers with

thermal conductivities

and . The layer thicknesses

are cm and the temperature of

interfaces is as shown in �gure. energy transfer through the wall is in

steady state.

Q. The temperature of the interface between layers 3 and 4 is

K1 = 0.06W /mK

K3 = 0.04W /mK K4 = 0.10W /mK

L1 = 1.5cm, L3 = 2.8 L4 = 3.5cm

A.

B.

C.

D.

Answer: A

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11∘C

8∘C

7.2∘C

5.4∘C

46. Figure shows in cross section a wall consisting of four layers with

thermal conductivities and

. The layer thickness are

and . The temperature of

interfaces is as shown in �gure. Energy transfer through the wall is

steady.

K1 = 0.06W /mK, K3 = 0.04W /mK

K4 = 0.10W /mK

L1 = 1.5cm, L3 = 2.8cm L4 = 3.5cm

The temperature of the interface between layers and is:

A.

B.

C.

D.

Answer: A

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3 4

2 × 10− 2

2 × 10− 3

4 × 10− 2

4 × 10− 3

47. A body cools in a surrounding of constant temperature Its

heat capacity is . Initial temperature of cooling is valid. The

30∘C

2J /∘ C

body of mass 1 kg cools to in 10 min

When the body temperature has reached , it is heated again so

that it reaches in 10 min. The heat required from a heater by

the body is

A.

B.

C.

D.

Answer: B

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38∘C

38∘C

40∘C

36∘C

36.4∘C

37∘C

37.5∘C

48. A body cools in a surrounding of constant temperature Its

heat capacity is . Initial temperature of cooling is valid. The

body of mass 1 kg cools to in 10 min

When the body temperature has reached , it is heated again so

30∘C

2J /∘ C

38∘C

38∘C

that it reaches in 10 min. The heat required from a heater by

the body is

A.

B.

C.

D.

Answer: A

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40∘C

49. A body cools in a surrounding of constant temperature Its

heat capacity is . Initial temperature of cooling is valid. The

body of mass 1 kg cools to in 10 min

When the body temperature has reached , it is heated again so

that it reaches in 10 min. The heat required from a heater by

the body is

A.

B.

C.

D.

Answer: C

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30∘C

2J /∘ C

38∘C

38∘C

40∘C

3.6J

7J

8J

4J

50. A metal ball of mass is heated means of a heater in a

room at . The temperature of the ball beomes steady at .

Find the rate of loss of heat to the surrounding when the ball is at

.

A.

B.

C.

D.

Answer: A

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2kg 40W

25∘C 60∘C

60∘C

40W

16W

96W

100W

51. A metal ball of mass is heated means of a heater in a

room at . The temperature of the ball becomes steady at .

2kg 40W

25∘C 60∘C

Find the rate of loss of heat to the surrounding when the ball is at

.

A.

B.

C.

D.

Answer: B

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39∘C

40W

16W

96W

100W

52. A metal ball of mass is heated means of a heater in a

room at . The temperature of the ball beomes steady at .

Assume that the temperature of the ball rises uniformly from

to minutes. Find the total loss of heat to the surrounding

during this period.

2kg 40W

25∘C 60∘C

25∘C

39∘Cin2

A.

B.

C.

D.

Answer: C

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900J

940J

960J

1000J

53. In which of the following process convection of does not take

place primarily ?

A. lead and sea breeze

B. boiling of water

C. heating of glass surface due to �lament of the bulb

D. air around the furnace

Answer: C

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54. A metal rod AB of length 10 x has its one end A in ice at and

the other end B in water at 100°C. If a point P on the rod is

maintained at 400°C, then it is found that equal amounts of water

and ice evaporate and melt per unit time. The latent heat of

evaporation of water is 540 cal/g and latent heat of melting of ice is

80 cal/g. If the point P is at a distance of from the ice end A, �nd

the value of (Neglect any heat loss to the surroundings)

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0∘C

λx

λ

55. Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at

temperature respectively. The maximum intensity in the

emission spectrum of A is at 500 nm and in that of B is at 1500 nm.

T1 and T2

Considering them to be black bodies, what will be the ratio of the

rate of total energy radiated by A to that of B?.

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56. A composite block is made of slabs A,B,C,D and E of di�erent

thermal conductivities (given in terms of a constant K and sizes

(given in terms of length, L) as shown in the �gure. All slabs are of

same width. Heat 'Q' �ows only from left to right through the blocks.

Then in steady state

A. heat �ow through and slabs are sameA E

B. heat �ow through slab is maximum

C. temperature di�erence across slab is smallest

D. heat �ow through heat �ow through heat �ow

through .

Answer: A::C::D

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E

E

C = B +

D

57. Three very large plates of same area are kept parallel and close to

each other. They are considered as ideal black surfaces and have very

high thermal conductivity. The �rst and third plates are maintained

at temperature 2T and 3T respectively. The temperature of the middle

(i.e. second) plate under steady state condition is

A.

B.

( ) T652

14

( ) T974

14

C.

D.

Answer: C

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( ) T972

14

(97) T14

58. Two rectangular blocks, having identical dimensions, can be

arranged either in con�guration I or in con�guration II as shown in

the �gure. One of the blocks has thermal conductivity k and the other

2 k. The temperature di�erence between the ends along the x-axis is

the same in both the con�gurations. It takes 9 s to transport a

certain amount of heat from the hot end to the cold end in the

con�guration I. The time to transport the same amount of heat in

the con�guration II is :

A. 2.0s

B.

C.

D.

Answer: A

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3.0s

4.5s

6.0s

59. Parallel rays of light of intensity are incident on a

spherical black body kept in surroundings of temperature 300 K. Take

Stefan’s constant and assume that the

energy exchange with the surrounding is only through radiation. The

�nal steady temperature of the black body is close to

A.

B.

C.

I = 912Wm− 2

σ = 5.7 × 10− 8Wm− 2K − 4

330K

660K

990K

D.

Answer: A

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1550K

60. Two spherical stars A and B emit black body radiation. The radius

of A is 400 times that of B and A emits times the power emitted

from B. The ratio of their wavelengths and at which

the peaks oc cur in their respective radiation curves is :

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104

(λA /λB) λA λB

61. The �gure shows a system of two concentric spheres of radii r 1

and r 2 and kept at temperature T 1 and T 2 , respectively. The radial

rate of �ow of heat in a substance between the two concentric

spheres, is proportional to:

A.

B.

C.

D.

Answer: C

Watch Video Solution

(r2 − r1)

(r1r2)

In(r2)

(r1)

r1r2

(r2 − r1)

(r2 − r1)

62. Assuming the sun to have a spherical outer surface of radius r,

radiating like a black body at temperature , the power received

by a unit surface, (normal to the incident rays) at distance R from the

centre of the Sun is:-

Where is the Stefan's Constant.

A.

B.

t∘C

σ

R2σT4

r2

4πr20R

2σT 4

r2

C.

D.

Answer: C

Watch Video Solution

πr20R

2σT 4

r2

r20R

2σT 4

4πr2

63. One end of a thermally insulated rod is kept at a temperature

and the other at . The rod is composed of two sections of lengths

and thermal conductivities respectively. The

temperature at the interface of the two sections is

A.

T1

T2

l1 and l2 K1 and K2

(K2L2T1 + K1L1T2)

(K1L1 + K2L2)

B.

C.

D.

Answer: C

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(K2L1T1 + K1L2T2)

(K1L1 + K1L2)

(K1L1T1 + K2L1T2)

(K1L2 + K2L1)

(K1L1T1 + K2L2T2)

(K1L1 + K2L2)

64. A long metallic bar is carrying heat from one its ends to the other

end under steady-state. The variation of temperature along the

length x of the bar from its end is best described by which of the

following �gure.

A.

B.

θ

C.

D.

Answer: A

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65. If a piece of metal is heated to temperature and then allowed to

cool in a room which is at temperature the graph between the

temperature T of the metal and time t will be closest to :

A.

θ

θ0

B.

C.

D.

Answer: C

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66. Three rods of Copper, Brass and Steel are welded together to

from a Y - –shaped structure. Area of cross – section of each rod

. End of copper rod is maintained at where as ends

of brass and steel are kept at . Lengths of the copper, brass and

steel rods are 46, 13 and 12 cms respectively. The rods are thermally

= 4cm2 100∘C

0∘C

insulated from surroundings except at ends. Thermal conductivities

of copper, brass and steel are 0.92, 0.26 and 0.12 CGS units

respectively. Rate of heat �ow through copper rod is :

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67. The two ends of a rod of length L and a uniform cross -

secontional area A are kept at two temperatures and

. The rate of heat tranfer, , through the rod in a steady state is

given by

A.

B.

C.

D.

Answer: D

Watch Video Solution

T1 T2(T1 > T2)

dQ

dt

=dQ

dt

KL(T1 − T2)

A

=dQ

dt

K(T1 − T2)

LA

= KLA(T1 − T2)dQ

dt

=dQ

dt

KA(T1 − T2)

L

Watch Video Solution

68. A cylindrical rod having temperature and at its end. The

rate of �ow of heat cal/sec. If all the linear dimension are doubled

keeping temperature remain const. then rate of �ow of heat will

be : -

A.

B.

C.

D.

Answer: B

Watch Video Solution

T1 T2

Q1

Q2

4Q1

2Q1

Q1

4

Q1

2

69. A wall has two layers A and B, each made od=f di�erent material.

Both the layers have the same thickness. The thermal conductivity for

A is twice that B and, under steady condition, the temperature

di�erence across the wall is C. The temperature di�erence across

the layer A is

A.

B.

C.

D.

Answer: B

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36∘

6∘C

9∘C

18∘C

27∘C

70.

All the rods have same conductance and same area of cross

section A. if ends A and C are maintained at temperature and

respectively then which of the following is/are correct

A. Rate of heat �ow through ABC, AOC and ADC is same

B. Rate of heat �ow through BO and OD is not same

C. Total Rate of �ow from A to C

K

2T0 T0

3KAT0

2a

D. Temperature at junction B,O and D are same

Answer: D

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71. Consider two rods of same length and di�erent speci�c heats (

), conductivities ( ) and area of cross-sections ( )

and both having temperature at their ends. If rate of loss

of heat due to conduction is equal, then :-

A.

B.

C.

D.

Answer: D

Watch Video Solution

S1, S2 K1, K2 A1, A2

T1 and T2

K1 = K2

K1S1 = K2S2

=K1

A1S1

K2

A2S2

K1A1 = K2A2

Watch Video Solution

72. Two plates each of area A, thickness thermal

conductivities respectively are joined to form a single

plate of thickness . If the temperatures of the free surfaces

are . Calculate.

(a) rate of �ow of heat

(b) temperature of interface and

(c) equivalent thermal conductivity.

A.

B.

C.

L1 and L2

K1 and K2

(L1 + L2)

T1 and T2

K1K2

K1 + K2

2K1K2

K1 + K2

(K21 + K2

2 )3 / 2

K1K2

D.

Answer: B

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(K21 + K2

2 )3 / 2

2K1K2

73. Consider a composite slab consisting of two di�erent materials

having equal thickness and thermal conductivities K and 2K

respectively. The equivalent thermal conductivity of the slab is

A.

B. 3k

C.

D.

Answer: C

Watch Video Solution

√2

k43

k2

3

74. A square is made of four rods of same material one of the

diagonal of a square is at temperature di�erence , then the

temperature di�erence of second diagonal:

A.

B.

C.

D.

Answer: A

Watch Video Solution

100∘C

0∘C

100

l

100

2l

100∘C

75. If the temperature di�erence on the two sides of a wall increases

from to , its thermal conductivity

A. remains unchanged

100∘C 200∘C

B. is doubled

C. is halved

D. becomes four times

Answer: A

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76. The colour of a star indicates its

A. temperature

B. distance

C. velocity

D. size

Answer: A

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77. The means of energy transfer in vacuum is:

A. irradiation

B. convection

C. radiation

D. conduction

Answer: C

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78. Which of the following is nearest to blackbody-

A. An enclosure with a small hole

B. carbon black

C. Abonite

D. none of these

Answer: A

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79. Which of the following is true statement?

A. A good absorber is bad conductor

B. Each body emits and absorb radiation at each temperature

C. In a black body energy of emitted radiation is equal for all

wavelength

D. Plank's law gives between maximum wavelength of black body

radiation and its temperature

Answer: B

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80. For a black body at temperature . If the temperature of the

black body is changed to , then its radiating power will be

A. 304 W

B. 320 W

C. 240 W

D. 120 W

Answer: B

Watch Video Solution

727∘C

1227∘C

81. According to Wien's law

A. Wavelength corresponding to maximum energy and

temperature

B. Radiation energy and wavelength

C. Temperature and wavelength

D. Colour of light and temperature

Answer: A

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82. If denotes the wavelength at which the radiative emission

from a black body at a temperature T K is maximum, then

A.

B. is independent of T

C.

D.

Answer: D

λm

λm ∝ T 4

λm

λm ∝ T

λm ∝ T − 1

Watch Video Solution

83. A black body at emits radiations with maximum intensity

at a wavelength of 5000 A. If the temperature of the body is

increased by , the maximum intensity will be observed at:

A. 4000 Å

B. 5000 Å

C. 6000 Å

D. 3000 Å

Answer: D

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1227∘C

1000∘C

84. The energy radiated by a black body is directly proportional to :

A.

B.

C.

D. T

Answer: C

Watch Video Solution

T 2

T − 2

T 4

85. If temperature becomes double, the emitted radiation will be :

A. 16 times

B. 8 times

C. times

D. 32 times

Answer: A

2√2

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86. Prevost's theory of heat exchange tells that a body radiates

thermal energy-

A. At temperature higher than that of surrounding only.

B. At temperature lower than that of surrounding only

C. At temperature equal to that of surrounding only

D. At all temperature

Answer: D

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87. Re�ection and absorption coe�cient of a given surface at for

a �xed wavelength are 0.5 (each). At the same temperature and

wavelength the transmission (coe�cient) of surface will be-

0∘C

A. 0.5

B. 1

C. zero

D. in between zero and one

Answer: C

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88. A black metal foil is warmed by radiation from a small sphere at

temperature 'T' and at a distance d'. It is found that the power

received by the foil is P. If both the temperature and distance are

doubled, the power received by the foil will be

A. P

B. 2P

C. 8P

D. 4P

Answer: D

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89. A body takes 10 min to cool douwn from to . If the

temperature of surrounding is then in the next 10 minutes

temperature of the body will be

A.

B.

C.

D.

Answer: C

Watch Video Solution

62∘C 50∘C

26∘C

38∘C

40∘C

42∘C

44∘C

90. Two circular disc and with equal radii are blackened. They are

heated to same temperature and are cooled under identical

conditions. What inference do your draw from their cooling curves?

A. A and B have same speci�c heats

B. speci�c heat of A is less

A B

C. Speci�c heat of B is less

D. Nothing can be said

Answer: B

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91. According to Newton’s law of cooling, the rate of cooling of a body

is proportional to , where is the di�erence of the

temperature of the body and the surrounding, and n is equal to :

A. 2

B. 3

C. 4

D. 1

Answer: D

h id l i

(Δθ)n Δθ

Exercise 2

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1. Two identical square rods of metal are welded end to end as shown

in �gure (a). Assume that cal of heat �ows through thr rod in

. Now the rods are welded as shown in �gure (b). The time it

would take for cal to �ow through the rods now, is

A.

B.

C.

D.

Answer: B

10

2 min

10

0.75 min

0.5 min

1.5 min

1 min

Exercise 3

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1. Variation of radiant energy emitted by sun, �lament of

tungsten lamp and welding arc as a function of its wavelength as

shown in the �gure. Which of the following option is the correct

match ?

A. 1-buld, 2 welding arc, 3 sun

B. 2-buld, 3 welding arc, 1 sun

C. 3-bulb,1 welding arc, 2 sun

D. 2-bulb, 1 welding arc, 3 sun

Answer: A

Watch Video Solution

Eλ

→ →

→ →

→ →

→ →

Advancel Level Problems

1. Seven rods A, B, C, D, E, F and G are joined as shown in �gure. All the

rods have equal cross-sectional area A and length l. The thermal

conductivities of the rods are , ,

, and . The rod E is kept at a

constant temperature and the rod G is kept at a constant

temperature . (a) Show that the rod F has a uniform

temperature . (b) Find the rate of heat �owing

from the source which maintains the temperature .

Watch Video Solution

KA = KC = K0 KB = KD = 2K0

KE = 3K0 KF = 4K0 KG = 5K0

T1

T2(T2 > T1)

T = (T1 + 2T2) /3

T2

2. Find the rate of heat �ow through a cross section of the rod shown

in �gure . Thermal conductivity of the material of the rod is

K.

Watch Video Solution

(θ2 > θ1)

3. A solid aluminium sphere and a solid copper sphere of twice the

radius are heated to the same temperature and are allowed to cool

under identical surrounding temperatures. Assume that the

emisssivity of both the spheres is the same. Find ratio of (a) the rate

of heat loss from the aluminium sphere to the rate of heat loss from

the copper sphere and (b) the rate of fall of temperature of the

aluminium sphere to the rate of fall of temperature of copper sphere.

The speci�c heat capacity of aluminium . and that

of copper . The density of copper times the

density of aluminium.

Watch Video Solution

= 900Jkg− 1C − 1

= 390Jkg− 1C − 1 = 3.4

4. A hot body placed in a surrounding of temperature obeys

Newton's law of cooling . Its temperature at

is the speci�c heat capacity of the body is and its mass is

. Find

(a) the maximum heat that the body can lose and

(b) the time starting from in which it will lose of this

maximum heat.

Watch Video Solution

θ0

= − k(θ − θ0)dθ

dt

t = 0 θ1 s

m

t = 0 90 %

5. Find the total time elapsed for a hollow copper sphere of inner

radius outer radius , density , speci�c

heat and emissivity to cool from

to when the surrounding temperature is (for inner

surface Stefan's constant

Watch Video Solution

3cm 6cm ρ = 9 × 103kg/m3

s = 4 × 103J /kgK e = 0.4 727∘C

227∘C 0. K

e = 1 σ = 5.6 × 10− 8W /m2K4)

6. A metal block of heat capacity placed in a room at

is heated electrically. The heater is switched o� when the

temperature reaches . The temperature of the block rises at the

rate of just after the heater is switched on and falls at the

rate of just after the heater is switched o�. Assume

Newton's law of cooling to hold (a) Find the power of the heater. (b)

Find the power radiated by the block just after the heater is switched

o�. (c ) Find the power radiated by the block when the temperature

of the block is . (d) Assuming that the power radiated at

90J /. ∘ C 25∘C

35∘C

2∘C /s

0.2∘C /s

30∘C 30∘C

respresents the average value in the heating process, �nd the time

for which the heater was kept on.

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7. A hollow tube has a length l, inner radius and outer radius .

The material has a thermal conductivity K. Find the heat �owing

through the walls of the tube if (a) the �at ends are maintained at

temperature and (b) the inside of the tube is

maintained at temperature and the outside is maintained at .

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R1 R2

T1 T2(T2 > T1)

T1 T2

8. Calculate thermal conductance for radial �ow of an annular

cylinder of length and inner and outer radius and . Assumel r1 r2

that thermal conductivity of the material is

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K

9. Calculated thermal conductance for radial �ow of a spherical sheel

of inner and outer radius and . Assume that thermal conductivity

of the material is

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r1 r2

K

10. A metallic cylindrical vessel whose inner and outer radii are

is �lled with ice at . The mass of the ice in the cylinder

is m. Circular portions of the cylinder is sealed with completely

adiabatic walls. The vessel is kept in air. Temperature of the air is

. How long will it take for the ice to melt completely. Thermal

conductivity of the cylinder is K and its length is l. Latent heat of

fusion of L.

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r1 and r2 0∘C

50∘C

11. A uniform cylinder of length and thermal conductivity is

placed on a metal plate of the same area of mass and in�nite

conductivity. The speci�c heat of the plate is . The top of the cylinder

is maintained at . Find the time required for the temperature of

the plate to rise from to .

Watch Video Solution

L k

S m

c

T0

T1 T2(T1 < T2 < T0)

12. Assume that the total surface area of a human body is and

that it radiates like an ideal radiator. Calculate the amount of energy

radiates per second by the body if the body temperature is .

Stefan constant is .

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1.6m2

37∘C

σ 6.0 × 10− 8Wm− 2K − 4

13. The surface of a household radiator has an emissivity of and

an area of .

(a) At what rate is radiation emitted by the radiator when its

temperature is ?

(b) At what rate is the radiation absorbed by the radiator when the

walls of the room are at ? (c ) What is the net rate of radiation

from the radiator? (stefan constant

Watch Video Solution

0.55

1.5m2

50∘C

22∘C

σ = 6 × 10− 8W /m2 − K4)

14. A man, the surface area of whose skin is , is sitting in a room

where air temperature is if his skin temperature is and

emissivity of his skin equals 0.97, �nd the rate at which his body loses

heat.

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2m2

20∘C 28∘C

15. An electric heater is used in a room of total wall area to

maintain a temperature of inside it, when the outside

temperature is .The walls have three di�erent layers of

materials. The innermost layer is of wood of thickness 2.5cm, the

middle layer is of cement of thickness 1.0cm and the outermost layer

is of brick of thickness 25.0cm. Find the power of the electric heater.

Assume that there is no heat loss through the �oor and the celling.

The thermal conductivities of wood, cement and brick are

, . and respectively.

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137m2

20∘C

−10∘C

0.125Wm− 1C − 1 1.5Wm− 1C − 1 1.0Wm− 1C − 1

16. A rod of length l with thermally insulated lateral surface consists

of material whose heat conductivity coe�cient varies with

temperature as , where a is a constant. The ends of the rod

are kept at temperatures . Find the function T(x), where x

is the distance from the end whose temperature is .

Watch Video Solution

k = a/T

T1 and T2

T1

17. Two block with heat capacities and are connected by a rod

of length l, cross-sectional area A and heat conductivity K. Initial

temperature di�erence between the two blocks I . Assuming the

entire system to be isolated from surroundings , heat capacity of the

rod of be negligible. The temperature di�erence between the blocks

as a function of time is.

Watch Video Solution

C1 C2

T0

Solved Example

Example

1. One face of a copper cube of edge 10 cm is maintained at

and the opposite face is maintained at . All other surfaces are

covered with an insulating material. Find the amount of heat �owing

per second through the cube. Thermal conductivity of copper is

.

Watch Video Solution

100∘C

0∘C

385Wm− 1C − 1

1.

Three identical rods of length 1 m each, having cross-sectional area of

each and made of aluminium, copper and steel, respectively, are

maintained at temperatures of , and , respectively, at

their separate ends. Find the teperature of their common junction.

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1cm2

12∘C 4∘C 50∘C

[KCu = 400W /m − K, KAl = 200W /m − K, Ksteel = 50W /m − K]

2. The �gure shows the cross-section of the outer wall of a house buit

in a hill-resort to keep the house insulated from the freezing

temperature of outside. The wall consists of teak wood of thickness

and brick of thickness , sandwitching two layers of an

unknown material with identical thermal conductivites and thickness.

The thermal conductivity of teak wood is and that of brick is

. Heat conducion through the wall has reached a steady

state with the temperature of three surfaces being known.

and . Find the interface

temperature and .

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L1 (L2 = 5L1)

K1

(K2 = 5K1)

(T1 = 25∘C, T2 = 20∘C T5 = − 20∘C)

T4 T3

3. Three copper rods and three steel rods each of length

and area of cross-section are connected as shown

If ends and are maintained at temperatures and

respectively, calculate the amount of heat �owing per second from

the hot of cold function.

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l = 10cm

1cm2

A E 125∘C 0∘C

[KCu = 400W /m − K, Ksteel = 50W /m − K]

4. Two thin conectric shells made of copper with radius and

have a material of thermal conductivity �lled between

them. The inner and outer spheres are maintained at temperature

r1

r2(r2 > r1) K

and respectively by keeping a heater of power at the centre

of the two sphers. Find the value of .

Watch Video Solution

TH TC P

P

5. A container of negligible heat capacity contains of water. It is

connected by a steel rod of length and area of cross-section

to a large steam chamber which is maintained at . If

initial temperature of water is , �nd the time after which it

beomes . (Neglect heat capacity of steel rod and assume no

loss of heat to surroundings) (use table , take speci�c heat of

water

Watch Video Solution

1kg

10m

10cm2 100∘C

0∘C

50∘C

3.1

= 4180J /kg. ∘ C)

6. The solar radiation spectrum reveals that the intensity

corresponding to a wavelength of is maximum Estimate the4750Å

surface temperature of the sun.

(given wien's cosntant )

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= 2.89 × 10− 3m − K

7. A body of emissivity , surface area of and

temperature is kept in a room at temperature .

Calculate the initial value of net power emitted by the body.

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(e = 0.75) 300cm2

227∘C 27∘C

8. A hot black body emits the enegy at the rate of 16 and

its most intense radiation corresponds to 20000 . When the

temprerature of this body is further increased and its most intense

radiation corresponds to , then �nd the value of energy

radiated in .

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Jm− 2s− 1

Å

10000Å

Jm− 2s− 1

9. A body at temperature is kept in a surrounding of constant

temperature . It is observed that its temperature falls to

in minutes. Find how much more time will it taken for the body to

attain a temperature of .

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40∘C

20∘C 35∘C

10

30∘C