(C) Let $A_1$ and $A_2$ be the cross-sectional areas of the inner cylinder and the outer shell respectively.$A_1 = \pi R^2$$A_2 = \pi (2R)^2 - \pi R^2

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(C) Let $A_1$ and $A_2$ be the cross-sectional areas of the inner cylinder and the outer shell respectively.$A_1 = \pi R^2$$A_2 = \pi (2R)^2 - \pi R^2 = 3\pi R^2$Total area $A = A_1 + A_2 = 4\pi R^2$.Since the two ends are at different temperatures and the system is in a steady state,the heat flow through both parts occurs in parallel.The equivalent thermal resistance $R_{eq}$ is given by $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$.Using $R = \frac{l}{kA}$,we have $\frac{kA}{l} = \frac{k_1 A_1}{l} + \frac{k_2 A_2}{l}$.Substituting the values:$k(4\pi R^2) = k_1(\pi R^2) + k_2(3\pi R^2)$$4k = k_1 + 3k_2$$k = \frac{k_1 + 3k_2}{4}$

Class 11 Physics 2 Thermal Properties of Matter: Heat Transfer Thermal Resistance and it's Combination

Difficult

$A$ cylinder of radius $R$ made of a material of thermal conductivity $k_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of a material of thermal conductivity $k_2$. The two ends of the combined system are maintained at different temperatures. There is no loss of heat from the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is

A
$k_1 + k_2$
B
$\frac{k_1 k_2}{k_1 + k_2}$
C
$\frac{1}{4}(k_1 + 3k_2)$
D
$\frac{1}{4}(3k_1 + k_2)$

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Class 11

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Chapter

2 Thermal Properties of Matter: Heat Transfer

Subtopic

Thermal Resistance and it's Combination

$A$ copper rod of length $18 \ cm$ and a steel rod of length $6 \ cm$ are joined together to form a composite rod of uniform cross-section. The temperature of the free end of the copper rod is $100 \ ^\circ C$ and the temperature of the free end of the steel rod is $0 \ ^\circ C$. What is the temperature at the junction in $^\circ C$? (Thermal conductivity of copper is $9$ times that of steel. The rod is in a steady state.)

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The ratio of thermal conductivities of two different materials is $5:3$. If the thermal resistances of two rods of these materials having the same thickness are equal,then the ratio of the lengths of the rods will be:

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Two metal rods are arranged as shown in the figure. Their thermal conductivities are $K_1$ and $K_2$. What is their equivalent thermal conductivity?

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Three rods of same dimensions are arranged as shown in the figure. They have thermal conductivities ${K_1}, {K_2}$ and ${K_3}$. The points $P$ and $Q$ are maintained at different temperatures. For the heat to flow at the same rate along the path $PRQ$ and the path $PQ$,which of the following options is correct?

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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

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The ratio of thermal conductivities of two different materials is $5:3$. If the thermal resistances of two rods of these materials having the same thickness are equal,then the ratio of the lengths of the rods will be:

Two metal rods are arranged as shown in the figure. Their thermal conductivities are $K_1$ and $K_2$. What is their equivalent thermal conductivity?

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