The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity \mathrm{K} and 2\mathrm{K} and thickness \mathrm{x} and 4\mathrm{x} , respectively are \mathrm{T}_{2} and \mathrm{T}_{1}(\mathrm{T}_{2}>\mathrm{T}_{1}) . The rate of heat transfer through the slab, in a steady state is \left ( \dfrac{A(T_{2}-T_{1})K}{X} \right )f, with f equals to

The thermal resistance of both the blocks will be \dfrac { x }{ KA } and \dfrac { 4x }{ 2KA } =\dfrac { 2x }{ KA } respectively. Since the two resistors are in series, (as same heat current will flow through both), equivalent thermal resistance will be { R }_{ eq. }=\dfrac { x }{ KA } +\dfrac { 2x }{ KA } =\dfrac { 3x }{ KA } Thus, heat current will be \dfrac { dQ }{ dt } =\dfrac { { T }_{ 2 }-{ T }_{ 1 } }{ { R }_{ eq. } } =\dfrac { 1 }{ 3 } \dfrac { KA({ T }_{ 2 }-{ T }_{ 1 }) }{ x } Hence, f=\dfrac { 1 }{ 3 }

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Conduction

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Advanced Knowledge of Conduction

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Q: A wall has two layers A and B , each made of different materials. Both the layers have the same thickness. The thermal conductivity of the material of A is twice that of B . Under thermal equilibrium, the temperature difference across the wall is 36^{\circ}C . The temperature difference across the layer A is

Consider two walls ' A ' and ' B ' of thickness ' t ' each, and thermal conductivity of ' 2K ' and ' K ' respectively.The temperatures at the left of A = T_{a} The temperatures at the right of B = T_{b} The temperature at the junction = T Heat flow is a constant in steady state: \dot{Q} = KA\dfrac{dT}{dx} = constant Equating heat flow for both walls we get: 2KA\dfrac{T-T_{a}}{t-0} = KA\dfrac{T_{b}-T}{2t -t} Which simplifies to: 3T = T_{b} + 2T_{a} Also given that the temperature difference between the walls is 36^{\circ}C T_{b} - T_{a} = 36 Combining the two equation in T,T_{a},T_{b} and eliminating T_{b} We get T - T_{a} = 12^{\circ}C

Q: Two rods A and B of different materials are welded together as shown in figure. Their thermal conductivities are K_1 and K_2 . The thermal conductivity of the composite rod will be :

R_1=\dfrac{d}{k_1A} R_2={d}{K_2A} Req=\dfrac{R_1R_2}{R_1+R_2} \dfrac{d}{keq(2A)} = \dfrac{(\dfrac{d}{k_1A})(\dfrac{d}{k_2A})}{\dfrac{d}{A}(\dfrac{1}{k_1}+\frac{1}{k_2})} \dfrac{1}{2keq}=\dfrac{\dfrac{1}{k_1k_2}}{\dfrac{k_1+k_2}{k_1k_2}}=\dfrac{1}{k_1+k_2} keq = \dfrac{k_1+k_2}{2}

Q: Three identical thermal conductors are connected as shown in figure. Considering no heat loss due to radiation, temperature at the junction will be

Let the temperature of junction be T .Also let the thermal conductivity, area and length of conductor be K, A and L respectively.Using Junction law, \dfrac{dq_1}{dt} = \dfrac{dq_2}{dt}+\dfrac{dq_3}{dt} \therefore \dfrac{KA (20-T)}{L} = \dfrac{KA (T-60)}{L}+ \dfrac{KA (T-70)}{L} Or 20 - T =T-60+T-70 \therefore T = \dfrac{150}{3} = 50^oC

Q: Consider a compound slab consisting of two different materials having equal lengths, thicknesses and thermal conductivities K and 2K respectively. The equivalent thermal conductivity of the slab is

The thermal resistances can be added as they are in series. But, note that the width of the combined slab is twice the width of individual slab. So, we have: \Rightarrow \dfrac { 2L }{ { K }_{ eq }A } =\dfrac { L }{ KA } +\dfrac { L }{ 2KA } \Rightarrow { K }_{ eq }=\dfrac { 4K }{ 3 }

Q: A marble tile would feel cold as compared to a wooden tile on a winter morning, because the marble tile

The correct answer is option (a). The 'marble tile' is a good conductor of heat as compared to the 'wooden tile'.

Q: The dimensional formula for coefficient of thermal conductivity is:

The for the coefficient of thermal conductivity is K_{th}=\dfrac{Q\Delta x}{A\Delta T t} Dimensional formula of K_{th} [K_{th}]=\dfrac{[Heat].[length]}{[Area].[Temperature].[time]} [K_{th}]=\dfrac{[ML^2T^{-2}].[L]}{[L^2].[K].[T]} [K_{th}]=[MLT^{-3}K^{-1}] The correct option is D.

Q: A slab consists of two parallel layers of two different materials of same thickness having thermal conductivities K_1 and K_2 . The equivalent thermal conductivity of the slab is :

At thermal equilibrium H = H_{1} + H_{2} Given that two slab are in parallel and have same dimensions. \dfrac{K\times A\times (T_{2} - T_{1})}{L} = \dfrac{K_{1}\times \dfrac{A}{2}\times (T_{2} - T_{1})}{L} + \dfrac{K_{2}\times \dfrac{A}{2}\times (T_{2} - T_{1})}{L} ,where K is the effective thermal conductivity, T_{2}-T_{1} is the temperature difference between the ends of the slab, A is area of the slab and L is the thickness of the each layer of the slabSo, \dfrac{K\times A}{L} = \dfrac{K_{1}\times \dfrac{A}{2}}{L} + \dfrac{K_{2}\times \dfrac{A}{2}}{L} or, K = \dfrac{K_{1} +K_{2}}{2}

Q: Which of the following are conductors and which are insulators ?Sulphur , Silver , Copper , Cotton , Aluminium , Air , Nichrome , Graphite , Paper , Porcelain , Mercury, Mica , Bakelite , Polythene , Manganin.

\textbf{Conductors:} The materials, which have free electrons are good conductor of electricity. In general all the metals and their Alloys are good conductors of electricity.Hence among the given materials, following are conductors: \text{Silver, Copper, Aluminium, Nichrome, Graphite, Mercury, Manganin} \textbf{Note: } Graphite is not a metal, but still a conductor of electricity, as it has delocalized(free) electrons, which help in conduction. \textbf{Insulators:} Insulators are the materials which do not have free electrons, therefore do not conduct electricity. Following are insulators: \text{Sulphur, Cotton, Air, Paper, Porcelain, Mica, Bakelite, Polythene}

Q: The coefficient of thermal conductivity of copper is 9 times that of steel. In the composite cylindrical bar shown in the figure, what will be the temperature at the junction of copper and steel?

heat current along the rod will be same \dfrac{dq}{dt}=\dfrac{9kA(100-T)}{18}=\dfrac{kA(T-0)}{6} T=75^oC

Conduction

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Thermal Conductivity

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Important Questions

A wall has two layers $A$ and $B$ , each made of different materials. Both the layers have the same thickness. The thermal conductivity of the material of $A$ is twice that of $B$ . Under thermal equilibrium, the temperature difference across the wall is $3 6^{\circ} C$ . The temperature difference across the layer $A$ is

Medium

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Two rods $A$ and $B$ of different materials are welded together as shown in figure. Their thermal conductivities are $K_{1}$ and $K_{2}$ . The thermal conductivity of the composite rod will be :

Medium

NEET

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Three identical thermal conductors are connected as shown in figure. Considering no heat loss due to radiation, temperature at the junction will be

Hard

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The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $K$ and $2 K$ and thickness $x$ and $4 x$ , respectively are $T_{2}$ and $T_{1} (T_{2} > T_{1})$ . The rate of heat transfer through the slab, in a steady state is $(\frac{A ( T _{2} - T _{1} ) K}{X}) f,$ with $f$ equals to

Hard

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Consider a compound slab consisting of two different materials having equal lengths, thicknesses and thermal conductivities $K$ and $2 K$ respectively. The equivalent thermal conductivity of the slab is

Medium

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A marble tile would feel cold as compared to a wooden tile on a winter morning, because the marble tile

Medium

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The dimensional formula for coefficient of thermal conductivity is:

Medium

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A slab consists of two parallel layers of two different materials of same thickness having thermal conductivities $K_{1}$ and $K_{2}$ . The equivalent thermal conductivity of the slab is :

Medium

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Which of the following are conductors and which are insulators ?
Sulphur , Silver , Copper , Cotton , Aluminium , Air , Nichrome , Graphite , Paper , Porcelain , Mercury, Mica , Bakelite , Polythene , Manganin.

Medium

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The coefficient of thermal conductivity of copper is $9$ times that of steel. In the composite cylindrical bar shown in the figure, what will be the temperature at the junction of copper and steel?

Medium

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Revise with Concepts

Advanced Knowledge of Conduction

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Conduction using Kinetic Model

Thermal Conductivity

Important Questions

Two rods A and B of different materials are welded together as shown in figure. Their thermal conductivities are K1​ and K2​. The thermal conductivity of the composite rod will be :

Three identical thermal conductors are connected as shown in figure. Considering no heat loss due to radiation, temperature at the junction will be

Consider a compound slab consisting of two different materials having equal lengths, thicknesses and thermal conductivities K and 2K respectively. The equivalent thermal conductivity of the slab is

A marble tile would feel cold as compared to a wooden tile on a winter morning, because the marble tile

The dimensional formula for coefficient of thermal conductivity is:

A slab consists of two parallel layers of two different materials of same thickness having thermal conductivities K1​ and K2​. The equivalent thermal conductivity of the slab is :

Which of the following are conductors and which are insulators ? Sulphur , Silver , Copper , Cotton , Aluminium , Air , Nichrome , Graphite , Paper , Porcelain , Mercury, Mica , Bakelite , Polythene , Manganin.

The coefficient of thermal conductivity of copper is 9 times that of steel. In the composite cylindrical bar shown in the figure, what will be the temperature at the junction of copper and steel?

Two rods $A$ and $B$ of different materials are welded together as shown in figure. Their thermal conductivities are $K_{1}$ and $K_{2}$ . The thermal conductivity of the composite rod will be :

Consider a compound slab consisting of two different materials having equal lengths, thicknesses and thermal conductivities $K$ and $2 K$ respectively. The equivalent thermal conductivity of the slab is

A slab consists of two parallel layers of two different materials of same thickness having thermal conductivities $K_{1}$ and $K_{2}$ . The equivalent thermal conductivity of the slab is :

Which of the following are conductors and which are insulators ?
Sulphur , Silver , Copper , Cotton , Aluminium , Air , Nichrome , Graphite , Paper , Porcelain , Mercury, Mica , Bakelite , Polythene , Manganin.

The coefficient of thermal conductivity of copper is $9$ times that of steel. In the composite cylindrical bar shown in the figure, what will be the temperature at the junction of copper and steel?