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A spherical shell of inner and outer radii $r$ and $R$ has variable thermal conductivity $K = \frac{ρ}{x}$ , where $k$ is coefficient of thermal conductivity and $x$ is the distance from the centre of spherical shell and $ρ$ is a positive constant. If inner and outer walls are maintained at different constant temperatures $T_{1}$ and $T_{2}$ respectively, then

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Important Questions on Thermal Properties of Matter

MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Three rods of identical cross-section and length are made of three different materials of thermal conductivity $K_{1}, K_{2}$ and $K_{3}$ , respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at $100 ° C$ and the other at $0 ° C$ (see figure). If the joints of the rod are at $70 ° C$ and $20 ° C$ in steady and there is no loss of energy from the surface of the rod, the correct relationship between $K_{1}, K_{2}$ and $K_{3}$ is :

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HARD

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Three rods of Copper, Brass and Steel are welded together to form a Y-shaped structure. Area of cross-section of each rod is

4 {cm}^{2}

. End of copper rod is maintained at

100 ° C

. Where as ends of brass and steel are kept at

0 ° C

. Lengths of the copper, brass and steel rods are

46, 13

and

12 cms

respectively. The rods are thermally 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 flow through copper rod is :

MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Two materials having coefficients of thermal conductivity

3 K

and

K

and thickness

d

and

3 d

respectively, are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are

θ_{2}

and

θ_{1}

respectively,

(θ_{2} > θ_{1})

. The temperature at the interface is
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MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Three rods of same dimensions have thermal conductivities $3 K, 2 K$ and $K$ . They are arranged as shown in the figure below. Then in the steady state the temperature of the junction $' P'$ is

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EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

A heat source at

T = 10^{3} K

is connected to another heat reservoir at

T = 10^{2} K

by a copper slab which is

1 m

thick. Given that the thermal conductivity of copper is

0 .1 W K^{- 1} m^{- 1}

, the energy flux through it in the steady-state is:

EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Three rods of identical cross-sectional area and made from the same metal, form the sides of an isosceles triangle

A B C

right angled at

B

as shown in figure. The point

A

and

B

are maintained at temperature

T

and

\sqrt{2} T

respectively, in the steady state. Now, assuming that only heat conduction takes place. The temperature of point

C

will be

EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

The unit of thermal conductivity is

MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Two thin metallic spherical shells of radii

r_{1}

and

r_{2} (r_{1} < r_{2})

are placed with their centres coinciding. A material of thermal conductivity

K

is filled in the space between the shells. The inner shell is maintained at temperature

θ_{1}

and the outer shell at temperature

θ_{2} (θ_{1} < θ_{2}) .

The rate at which heat flows radially through the material is :

MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

A cylinder of radius

R

is surrounded by a cylindrical shell of inner radius

R

and outer radius

2 R

. The thermal conductivity of the material of the inner cylinder is

K_{1}

and that of the outer cylinder is

K_{2}

. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

The two ends of a metal rod are maintained at temperatures

100 ° C

and

110 ° C

. The rate of heat flow in the rod is found to be

4.0 J s^{- 1}

. If the ends are maintained at temperatures

200 ° C

and

210 ° C

, the rate of heat flow will be:

HARD

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

In steady state heat conduction, the equations that determine the heat current

j (r)

[heat flowing per unit time per unit area] and temperature

T (r)

in space are exactly the same as those governing the electric field

E (r)

and electrostatic potential

V (r)

with the equivalence given in the table below.

Heat flow	Electrostatics
$T (r)$	$V (r)$
$j (r)$	$E (r)$

We exploit this equivalence to predict the rate

Q

of total heat flowing by conduction from the surfaces of spheres of varying radii, all maintained at the same temperature. If

Q \propto R^{n},

where

R

is the radius, then the value of

n

HARD

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Temperature difference of $120^{o} C$ is maintained between two ends of a uniform rod $A B$ of length $2 L .$ Another bent rod $P Q,$ of same cross-section as $A B$ and length $\frac{3 L}{2},$ is connected across $A B$ (See figure). In steady state, temperature difference between $P$ and $Q$ will be close to:
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EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

The temperature $θ$ at the junction of two insulating sheets, having thermal resistances $R_{1}$ and $R_{2}$ as well as top and bottom temperatures $θ_{1}$ and $θ_{2}$ (as shown in figure) is given by :

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HARD

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Consider a pair of insulating blocks with thermal resistances $R_{1}$ , and $R_{2}$ as shown in the figure. The temperature $θ$ at the boundary between the two blocks is

Question Image

EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Two identical metal wires of thermal conductivities

K_{1}

and

K_{2}

respectively are connected in series. The effective thermal conductivity of the combination is:

EASY

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Two metallic blocks $M_{1}$ and $M_{2}$ of same area of cross-section are connected to each other (as shown in figure). If the thermal conductivity of $M_{2}$ is $K$ then the thermal conductivity of $M_{1}$ will be : [Assume steady state heat conduction]

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MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

In a steady state, the temperature at the end

A

and end

B

of a

20 cm

long rod

A B

are

100 ° C

and

0 ° C

. The temperature of a point,

9 cm

from

A

HARD

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Two rectangular blocks, having identical dimensions, can be arranged either in configuration

I

or in configuration

I I

as shown in the figure. One of the blocks has thermal conductivity

K

and the other

2 K

. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes

9 s

to transport a certain amount of heat from the hot end to cold end in the configuration

I

. The time to transport the same amount of heat in the configuration

II

HARD

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of

1 m

10 ° C .

Now the end P is maintained at

10 ° C,

while the end S is heated and maintained at

400 ° C .

The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is

1.2 \times 10^{- 5} K^{- 1},

the change in length of the wire PQ is

MEDIUM

Physics>Thermal Physics>Thermal Properties of Matter>Heat Conduction in One Dimension

Six identical conducting rods are joined as shown in the figure. Points

A

and

D

are maintained at

200^{o} C

and

20^{o} C

, respectively. The temperature of junction

B

will be

Important Questions on Thermal Properties of Matter

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