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PHXI11:THERMAL PROPERTIES OF MATTER

366411 Consider a compound slab consisting of two different material having equal thickness and thermal conductivities $K$ and $2 K$ respectively. The equivalent thermal conductivity of the slab is

1

3 K

2

\frac{4}{3} K

3

\sqrt{2 K}

4

\frac{2}{3} K

Explanation:

The equivalent thermal resistance is
$\begin{aligned} R = R_{1} + R_{2} \\ \frac{2 l}{k^{'} A} = \frac{l}{K_{1} A} + \frac{l}{K_{2} A} \\ K^{'} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} = \frac{2 \cdot K \cdot 2 K}{K + 2 K} = \frac{4}{3} K . \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366412 A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2}$ . The effective thermal conductivity of the system is

1

\frac{1}{3} (K_{1} + 2 K_{2})

2

\frac{1}{2} (2 K_{1} + 3 K_{2})

3

\frac{1}{3} (3 K_{2} + 2 K_{1})

4

\frac{1}{4} (K_{1} + 3 K_{2})

Explanation:

Both the cylinders are in parallel, for the heat flow from one end as shown
supporting img
Hence $K_{eq} = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}}$
where, $A =$ area of cross-section of inner
cylinder $= π R^{2}$
and $A_{2} =$ area of cross-section of cylindrical
shell
$\begin{aligned} = π {(2 R)^{2} - (R)^{2}} = 3 π R^{2} \\ \Rightarrow K_{e q} & = \frac{K_{1} (π R^{2}) + K_{2} (3 π R^{2})}{π R^{2} + 3 π R^{2}} \\ = \frac{1}{4} (K_{1} + 3 K_{2}) \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366413 The ratio of the coefficient of thermal conductivity of two different materials is $5 : 3$ . If the thermal resistance of the two rods of these materials of same length is same, then the ratio of the area of these rods will be

1

5 : 3

2

3 : 5

3

9 : 25

4

25 : 9

Explanation:

The thermal resistance of a rod of length $l$ and area of cross section $A$ is
$R = \frac{l}{k A}$
$K$ - Thermal conductivity of the rod.
$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{k_{2}}{k_{1}} = \frac{3}{5}$

PHXI11:THERMAL PROPERTIES OF MATTER

366414 Assertion :
Greater the coefficient of thermal conductivity of a material, smaller is the thermal resistance of a rod of that material.
Reason :
Thermal resistance is the ratio of temperature difference between the ends of the conductor and rate of flow of heat.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

Resistance $\propto \frac{1}{conductivity}$
Resistance $= \frac{heat flow}{Δ T}$
So option (4) is correct.

PHXI11:THERMAL PROPERTIES OF MATTER

366411 Consider a compound slab consisting of two different material having equal thickness and thermal conductivities $K$ and $2 K$ respectively. The equivalent thermal conductivity of the slab is

1

3 K

2

\frac{4}{3} K

3

\sqrt{2 K}

4

\frac{2}{3} K

Explanation:

The equivalent thermal resistance is
$\begin{aligned} R = R_{1} + R_{2} \\ \frac{2 l}{k^{'} A} = \frac{l}{K_{1} A} + \frac{l}{K_{2} A} \\ K^{'} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} = \frac{2 \cdot K \cdot 2 K}{K + 2 K} = \frac{4}{3} K . \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366412 A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2}$ . The effective thermal conductivity of the system is

1

\frac{1}{3} (K_{1} + 2 K_{2})

2

\frac{1}{2} (2 K_{1} + 3 K_{2})

3

\frac{1}{3} (3 K_{2} + 2 K_{1})

4

\frac{1}{4} (K_{1} + 3 K_{2})

Explanation:

Both the cylinders are in parallel, for the heat flow from one end as shown
supporting img
Hence $K_{eq} = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}}$
where, $A =$ area of cross-section of inner
cylinder $= π R^{2}$
and $A_{2} =$ area of cross-section of cylindrical
shell
$\begin{aligned} = π {(2 R)^{2} - (R)^{2}} = 3 π R^{2} \\ \Rightarrow K_{e q} & = \frac{K_{1} (π R^{2}) + K_{2} (3 π R^{2})}{π R^{2} + 3 π R^{2}} \\ = \frac{1}{4} (K_{1} + 3 K_{2}) \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366413 The ratio of the coefficient of thermal conductivity of two different materials is $5 : 3$ . If the thermal resistance of the two rods of these materials of same length is same, then the ratio of the area of these rods will be

1

5 : 3

2

3 : 5

3

9 : 25

4

25 : 9

Explanation:

The thermal resistance of a rod of length $l$ and area of cross section $A$ is
$R = \frac{l}{k A}$
$K$ - Thermal conductivity of the rod.
$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{k_{2}}{k_{1}} = \frac{3}{5}$

PHXI11:THERMAL PROPERTIES OF MATTER

366414 Assertion :
Greater the coefficient of thermal conductivity of a material, smaller is the thermal resistance of a rod of that material.
Reason :
Thermal resistance is the ratio of temperature difference between the ends of the conductor and rate of flow of heat.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

Resistance $\propto \frac{1}{conductivity}$
Resistance $= \frac{heat flow}{Δ T}$
So option (4) is correct.

PHXI11:THERMAL PROPERTIES OF MATTER

366411 Consider a compound slab consisting of two different material having equal thickness and thermal conductivities $K$ and $2 K$ respectively. The equivalent thermal conductivity of the slab is

1

3 K

2

\frac{4}{3} K

3

\sqrt{2 K}

4

\frac{2}{3} K

Explanation:

The equivalent thermal resistance is
$\begin{aligned} R = R_{1} + R_{2} \\ \frac{2 l}{k^{'} A} = \frac{l}{K_{1} A} + \frac{l}{K_{2} A} \\ K^{'} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} = \frac{2 \cdot K \cdot 2 K}{K + 2 K} = \frac{4}{3} K . \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366412 A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2}$ . The effective thermal conductivity of the system is

1

\frac{1}{3} (K_{1} + 2 K_{2})

2

\frac{1}{2} (2 K_{1} + 3 K_{2})

3

\frac{1}{3} (3 K_{2} + 2 K_{1})

4

\frac{1}{4} (K_{1} + 3 K_{2})

Explanation:

Both the cylinders are in parallel, for the heat flow from one end as shown
supporting img
Hence $K_{eq} = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}}$
where, $A =$ area of cross-section of inner
cylinder $= π R^{2}$
and $A_{2} =$ area of cross-section of cylindrical
shell
$\begin{aligned} = π {(2 R)^{2} - (R)^{2}} = 3 π R^{2} \\ \Rightarrow K_{e q} & = \frac{K_{1} (π R^{2}) + K_{2} (3 π R^{2})}{π R^{2} + 3 π R^{2}} \\ = \frac{1}{4} (K_{1} + 3 K_{2}) \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366413 The ratio of the coefficient of thermal conductivity of two different materials is $5 : 3$ . If the thermal resistance of the two rods of these materials of same length is same, then the ratio of the area of these rods will be

1

5 : 3

2

3 : 5

3

9 : 25

4

25 : 9

Explanation:

The thermal resistance of a rod of length $l$ and area of cross section $A$ is
$R = \frac{l}{k A}$
$K$ - Thermal conductivity of the rod.
$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{k_{2}}{k_{1}} = \frac{3}{5}$

PHXI11:THERMAL PROPERTIES OF MATTER

366414 Assertion :
Greater the coefficient of thermal conductivity of a material, smaller is the thermal resistance of a rod of that material.
Reason :
Thermal resistance is the ratio of temperature difference between the ends of the conductor and rate of flow of heat.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

Resistance $\propto \frac{1}{conductivity}$
Resistance $= \frac{heat flow}{Δ T}$
So option (4) is correct.

PHXI11:THERMAL PROPERTIES OF MATTER

366411 Consider a compound slab consisting of two different material having equal thickness and thermal conductivities $K$ and $2 K$ respectively. The equivalent thermal conductivity of the slab is

1

3 K

2

\frac{4}{3} K

3

\sqrt{2 K}

4

\frac{2}{3} K

Explanation:

The equivalent thermal resistance is
$\begin{aligned} R = R_{1} + R_{2} \\ \frac{2 l}{k^{'} A} = \frac{l}{K_{1} A} + \frac{l}{K_{2} A} \\ K^{'} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} = \frac{2 \cdot K \cdot 2 K}{K + 2 K} = \frac{4}{3} K . \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366412 A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2}$ . The effective thermal conductivity of the system is

1

\frac{1}{3} (K_{1} + 2 K_{2})

2

\frac{1}{2} (2 K_{1} + 3 K_{2})

3

\frac{1}{3} (3 K_{2} + 2 K_{1})

4

\frac{1}{4} (K_{1} + 3 K_{2})

Explanation:

Both the cylinders are in parallel, for the heat flow from one end as shown
supporting img
Hence $K_{eq} = \frac{K_{1} A_{1} + K_{2} A_{2}}{A_{1} + A_{2}}$
where, $A =$ area of cross-section of inner
cylinder $= π R^{2}$
and $A_{2} =$ area of cross-section of cylindrical
shell
$\begin{aligned} = π {(2 R)^{2} - (R)^{2}} = 3 π R^{2} \\ \Rightarrow K_{e q} & = \frac{K_{1} (π R^{2}) + K_{2} (3 π R^{2})}{π R^{2} + 3 π R^{2}} \\ = \frac{1}{4} (K_{1} + 3 K_{2}) \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366413 The ratio of the coefficient of thermal conductivity of two different materials is $5 : 3$ . If the thermal resistance of the two rods of these materials of same length is same, then the ratio of the area of these rods will be

1

5 : 3

2

3 : 5

3

9 : 25

4

25 : 9

Explanation:

The thermal resistance of a rod of length $l$ and area of cross section $A$ is
$R = \frac{l}{k A}$
$K$ - Thermal conductivity of the rod.
$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{k_{2}}{k_{1}} = \frac{3}{5}$

PHXI11:THERMAL PROPERTIES OF MATTER

366414 Assertion :
Greater the coefficient of thermal conductivity of a material, smaller is the thermal resistance of a rod of that material.
Reason :
Thermal resistance is the ratio of temperature difference between the ends of the conductor and rate of flow of heat.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

Resistance $\propto \frac{1}{conductivity}$
Resistance $= \frac{heat flow}{Δ T}$
So option (4) is correct.