Thermal Resistance

Q. Three rods of the same dimension have thermal conductivities $3K,2K$ and $K$. They are arranged as shown in figure given below, with their ends at ${100}^{\circ }C,{50}^{\circ }C$ and ${20}^{\circ }C$. The temperature of their junction is

1. ${70}^{\circ }C$
2. ${50}^{\circ }C$
3. ${35}^{\circ }C$
4. ${60}^{\circ }C$

Q. Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at $0^{\circ }\text{C}$ and ${90}^{\circ }\text{C}$ respectively as shown. The temperature of junction of the three rods will be

1. ${45}^{\circ }\text{C}$
2. ${60}^{\circ }\text{C}$
3. ${20}^{\circ }\text{C}$
4. ${30}^{\circ }\text{C}$

Q. Two rods of same length, cross sectional area and material transfer a given amount of heat in $12$ seconds, when they are joined in parallel. If they are joined in series with same temperature difference across the ends, then they will transfer same heat in

1. $24\text{s}$
2. $3\text{s}$
3. $1.5\text{s}$
4. $48\text{s}$

Q. A wall has two layers $A$ and $B$, each made of different materials. Both the layers have the same thickness and cross secional area. 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

1. $6^{\circ }C$
2. ${12}^{\circ }C$
3. ${18}^{\circ }C$
4. ${24}^{\circ }C$

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

1. ${75}^{\circ }\text{C}$
2. ${67}^{\circ }\text{C}$
3. ${33}^{\circ }\text{C}$
4. ${25}^{\circ }\text{C}$

Q. Statement 1 : Equivalent thermal conductivity of two rods of same physical dimensions and having same thermal conductivity $\left(k\right)$ connected to each other in parallel combination is equal to $k$.
Statement 2 : Equivalent thermal conductivity of two rods of same physical dimensions and having same thermal conductivity $\left(k\right)$, connected to each other in series combination is equal to $k$.

1. Statement 1 is true but statement 2 is false.
2. Statement 2 is true but statement 1 is false.
3. Both statement 1 and statement 2 are true.
4. Both statement 1 and statement 2 are false.

Q.

Which of the following cylindrical rods will conduct most heat, when their ends are maintained at the same steady temperature

1. Length 1 m; radius 1 cm

2. Length 2 m; radius 1 cm

3. Length 2 m; radius 2 cm

4. Length 1 m; radius 2 cm

Q.

Why electrical energy is transmitted at high voltage and low current?

Q. Four rods of same material having the same cross section and length have been joined as shown. The temperature of the junction of the rods will be

1. ${20}^{\circ }\text{C}$
2. ${30}^{\circ }\text{C}$
3. ${45}^{\circ }\text{C}$
4. ${60}^{\circ }\text{C}$

Q. The three rods shown in the figure have identical dimensions. Heat flows from the hot end at a rate of $40\text{W}$ in the arrangement (a). Find the rate of heat flow when the rods are joined as in the arrangement shown in (b).
[Assume $K_{AI}=200{\text{W/m}}^{\circ }\text{C}$ and $K_{cu}=400{\text{W/m}}^{\circ }\text{C}$]

1. $75\text{W}$
2. $400\text{W}$
3. $180\text{W}$
4. $4\text{W}$

Q. Two rods $P$ and $Q$ have equal lengths. Their thermal conductivities are $K_1$ and $K_2$ and cross-sectional areas are $A_1$ and $A_2$. When the temperature at ends of each rod are $T_1$ and $T_2$ respectively, the rate of flow of heat through $P$ and $Q$ will be equal, if

1. $\frac{A_1}{A_2}=\frac{K_2}{K_1}$
2. $\frac{A_1}{A_2}=\frac{K_2}{K_1}\times \frac{T_2}{T_1}$
3. $\frac{A_1}{A_2}=\sqrt{\frac{K_1}{K_2}}$
4. $\frac{A_1}{A_2}={\left(\frac{K_2}{K_1}\right)}^2$

Q. Two ends of a conducting rod of varying cross-section are maintained at ${200}^{\circ }\text{C}$ and $0^{\circ }\text{C}$ respectively. In staedy state

1. $\text{temperature difference across}AB\text{and}CD\text{are equal}$
2. $\text{temperature difference across}AB\text{is greater than that of across}CD$
3. $\text{temperature difference across}AB\text{is less than that of across}CD$
4. $\text{temperature difference may be equal or different depending on the thermal conductivity of the rod}$

Q. A composite slab have two layers of different materials, having thermal conductivity $k_1$ and $k_2$. If each layer has same thickness, then what is the equivalent thermal conductivity of the slab?

1. $\frac{k_1{k}_2}{2(k_1+k_2)}$
2. $\frac{2k_1{k}_2}{3(k_1+k_2)}$
3. $\frac{2k_1{k}_2}{(k_1+k_2)}$
4. $\frac{k_1{k}_2}{(k_1+k_2)}$

Q. Two metal rods $1$ and $2$ of same lengths have the same temperature difference between their ends. Their thermal conductivities are $K_1$ and $K_2$, and cross-sectional areas $A_1$ and $A_2$ respectively. If the rate of heat conduction in $1$ is four times that in $2$, then

1. $K_1{A}_1=K_2{A}_2$
2. $K_1{A}_1=4K_2{A}_2$
3. $K_1{A}_1=2K_2{A}_2$
4. $4K_1{A}_1=K_2{A}_2$

Q. A composite block is made of slabs $A,B,C,D$ and $E$ of different thermal conductivities (given in terms of a constant $K$) and sizes (given in terms of length $L$ and area of cross-section $S$) as shown in the figure. All slabs are of the same width. Heat $Q$ flows only from left to right through the blocks. Then in the steady state:

1. heat flow through slabs $A$ and $E$ are same
2. heat flow through slab $B$ is minimum
3. temperature difference across slab $E$ is smallest
4. heat flow through $C=$ heat flow through $B+$ heat flow through $D$

Q.

consider an ice cube of edge 1.0 cm kept in a gravity -free hall. Find the surface area of the water when the ice melts. Neglect the difference in densities of ice and water.

Q. Two rods having same length and cross-sectional area are connected individually between temperature $T_1$ and $T_2$. If the rate of loss of heat due to condution is equal and specific heat of two rods are in the ratio $1:2$, then the ratio of their thermal conductivity will be

1. $1:1$
2. $1:\sqrt{2}$
3. $1:2$
4. $2:1$

Q. Find the equivalent capacitance across points $\text{A}$ and $\text{B}$. All the capacitors have capacitance $C$.

1. $\frac{7C}{8}$
2. $\frac{7C}{4}$
3. $\frac{9C}{2}$
4. $\frac{9C}{4}$

Q. A hollow cylindrical pipe of length $10\text{cm}$ has inner radius $r_1=3\text{cm}$ and outer radius $r_2=12\text{cm}$. The temperature difference between the inner and outer layer of cylindrical pipe is ${100}^{\circ }\text{C}$ and the thermal conductivity of cylindrical pipe is $150{\text{W/m}}^{\circ }\text{C}$. Find the rate of heat transfered (in $\text{kW}$) between inner and outer layers.
(Take $\ln 2=0.693$)

1. $3.4$
2. $6.8$
3. $10$
4. $13.6$

Q. Two rods having same length and cross-sectional area are connected individually between temperature $T_1$ and $T_2$. If the rate of loss of heat due to condution is equal and specific heat of two rods are in the ratio $1:2$, then the ratio of their thermal conductivity will be

1. $1:1$
2. $1:2$
3. $2:1$
4. $1:\sqrt{2}$

Q. Find the equivalent capacitance between points $\text{A}$ and $\text{B}$.

1. $\frac{20}{9}\mu \text{F}$
2. $9\mu \text{F}$
3. $2\mu \text{F}$
4. $\frac{26}{5}\mu \text{F}$

Q. Find the equivalent capacitance of given circuit between points $\text{A}$ and $\text{B}$ ? (Take $C=12\mu \text{F})$

1. $7\mu \text{F}$
2. $14\mu \text{F}$
3. $21\mu \text{F}$
4. $12\mu \text{F}$

Q. Three identical rods of length $1\text{m}$ each, having cross-sectional area of $1{\text{cm}}^2$ each and made of aluminium, copper and steel respectively are maintained at temperatures of ${12}^{\circ }\text{C},4^{\circ }\text{C}$ and ${50}^{\circ }\text{C}$ respectively at their separate ends. Find the temperature of their common junction.
$[K_{Cu}=400\text{W/m-K},K_{Al}=200\text{W/m-K},K_{\text{Steel}}=50\text{W/m-K}]$

1. ${10}^{\circ }\text{C}$
2. ${20}^{\circ }\text{C}$
3. ${30}^{\circ }\text{C}$
4. ${40}^{\circ }\text{C}$

Q.

Consider two rods of same length and different specific heats ($S_1$ and $S_2$), conductivities $K_1$ and $K_2$ and areas of cross section ($A_1$ and $A_2$) and both having temperature $T_1$ and $T_2$ at their ends. If the rate of heat loss due to conduction is equal, then

1. 

2. 

3. 

4. 

Q. An aluminium wire of length L = 60 cm and of cross-sectional area $0.01c{m}^2$ is connected to a steel wire of the same cross-sectional area. The compound wire is loaded with a block of mass 10 kg as shown in the figure. The distance $L_2$ from the joint to the supporting pulley is 86.6 cm. Transverse waves are set up in the composite wire at its lowest frequency, such that the joint in the wire is a node. What is the total number of nodes observed at the frequency excluding the two at the ends of the wire? Density of aluminium is $2.6gc{m}^{-3}$ and that of steel is $7.8gc{m}^{-3}$.___

Q. Two sheets of thickness $d$ and $3d$, are in contact with each other. The temperature just outside the thinner sheet is $T_1$ and thicker sheet is $T_3$ as shown in figure. The temperature $T_1,T_2$ and $T_3$ are in arithmetic progression such that $T_1>T_3$. The ratio of thermal conductivity of thinner sheet to thicker sheet is

1. $1:3$
2. $3:1$
3. $2:3$
4. $2:9$

Q. Five identical rods are joined as shown in figure. Point A and C are maintained at temperature ${120}^{\circ }C$ and ${20}^{\circ }C$, respectively. The temperature of junction B will be

1. ${100}^{\circ }C$
2. ${80}^{\circ }C$
3. $0^{\circ }C$
4. ${70}^{\circ }C$

Q. What is the equivalent capacitance of the system of capacitors shown in the figure below between points $\text{A}\&\text{B}$

1. $\frac{7C}{6}$
2. $1.6C$
3. $C$
4. None

Q. Five rods with identical geometries are arranged as shown. Their thermal conductivities are shown. Only A and C are maintained at ${100}^{\circ }C$ and $0^{\circ }C$ respectively. In other words, heat flows in from A and flows out of C. If temperature difference between ends D and B can be written as $10x^{\circ }C$, where $x$ is an integer. Then find $x$.

Q.

An electric heater is used in a room of total wall area $137m^2$ to maintain a temperature of ${20}^{\circ }C$ inside it, when the outside temperature is - ${10}^{\circ }C$.

The walls have three different layers of materials. The innermost layer is of wood of thickness 2.5 cm, the middle layer is of cement of thickness 1.0 cm and the outer most layer is of brick of thickness 25.0 cm. Find the power of the electric heater. Assume that there is no heat loss through the floor and the ceiling. The thermal conductivities of wood, cement and brick are

$0.125W/m^{\circ }C$, $1.5W/m^{\circ }C$ and $1.0W/m^{\circ }C$ respectively.

1. 900W.
2. 19000W.
3. 9000W.
4. 1000W.