Thermal Resistance

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

1. $\frac{3(K_1+K_2)}{2}$
2. $K_1+K_2$
3. $2(K_1+K_2)$
4. $\frac{(K_1+K_2)}{2}$

Q. A wooden bowl conducts less heat than a copper bowl because

1. $k_{Wood}>k_{Copper}$
2. $k_{Wood}\le k_{Copper}$
3. $k_{Wood}=k_{Copper}$
4. $k_{Wood}<k_{Copper}$

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. 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. Figure shown below represents, parallel combination of two metallic slabs of same thickness having area of cross sections $2{\text{m}}^2$ and $3{\text{m}}^2$ respectively, connected to the same temperature difference. If the thermal conductivities of each slab is $300{\text{W/m}}^{\circ }\text{C}$, then equivalent thermal conductivity is (in ${\text{W/m}}^{\circ }\text{C}$)

1. $500$
2. $250$
3. $300$
4. $350$

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

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. 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. In the following figure, two insulating sheets with thermal resistances R and 3R as shown in figure. The temperature $\theta$ is

1. ${200}^{\circ }$ C
2. ${60}^{\circ }$ C
3. ${75}^{\circ }$ C
4. ${80}^{\circ }$ C

Q. A solid 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 two different temperatures. If there is no loss of heat across the cylindrical surface and the system is in steady state, the effective thermal conductivity of the system is:

1. $\frac{4K_1+3K_2}{4}$
2. $\frac{3K_1+4K_2}{4}$
3. $\frac{3K_1+K_2}{4}$
4. $\frac{K_1+3K_2}{4}$

Q. The diagram shown below represents a slab of conductivity $K$. Find the direction along which the thermal resistance is maximum.

1. Along $cd$
2. Along $lm$
3. Along $ab$
4. All the directions have equal thermal resistance

Q.

Three rods of material x and three of material y are connected as shown in figure. All the rods are identical in length and cross-sectional area. If the end A is maintained at 60${}^{\circ }$C and the junction E at 10 ${}^{\circ }$ C, calculate the temperature of the junction B. The thermal conductivity of x is 800 W/M- ${}^{\circ }$ C and

that of y is 400 W/M- ${}^{\circ }$ C

1. ${50}^{\circ }C$

2. ${20}^{\circ }C$

3. ${60}^{\circ }C$

4. ${40}^{\circ }C$

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. Thermal conductivity of the conductor shown in the figure is $0.92{\text{cal/cm s}}^{\circ }\text{C}$. Find the thermal resistance between points $P$ and $Q$ (in ${\text{s}}^{\circ }\text{C/cal}$).

1. $0.036$
2. $0.018$
3. $0.072$
4. $1$

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. 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

1. $\frac{3(K_1+K_2)}{2}$
2. $K_1+K_2$
3. $2(K_1+K_2)$
4. $\frac{(K_1+K_2)}{2}$

Q. Four rods of silver, copper, brass and wood are of same shape. They are heated together after wrapping a paper on it, the paper will burn first on

1. Silver
2. Copper
3. Brass
4. Wood

Q. The net thermal resistance of two identical rods connected in series is greater than that of resistance offered by each rod individually.

1. False
2. True

Q.
Find the heat current, i.

1. $\left(\frac{K_1{K}_{2}A}{l_1{K}_2+l_2{K}_1}\right)\Delta T$
2. $\left(\frac{K_1{K}_{2}A}{l_2{K}_2+l_1{K}_1}\right)\Delta T$
3. $\left(\frac{K_{1}A}{l_1+l_2}\right)\Delta T$
4. $\left(\frac{K_1^2{K}_2^{2}A}{l_1{K}_2^2+l_2{K}_1^2}\right)\Delta T$

Q. A spherical body of radius $b=\frac{4}{\pi }$ $\text{m}$ has a concentric cavity of radius $a=\frac{2}{\pi }$ $\text{m}$ as shown in the figure. Thermal conductivity of the material is $100{\text{W/m}}^{\circ }\text{C}$. If temperature of inner surface is kept at ${70}^{\circ }\text{C}$ and of the outer surface at ${30}^{\circ }\text{C}$, then find the rate of heat flow. (in $\text{kW}$)

1. $16$
2. $32$
3. $64$
4. $40$

Q.
Find the $R_{equivalent}$.

1. $R_1$
2. $R_2$
3. $R_1+R_2$
4. $\frac{R_1{R}_2}{R_1+R_2}$

Q. Three rods of same dimensions have thermal conductivities $3\text{K}$, $\text{2K}$ and $\text{K}$. They are arranged as shown in the figure, with their ends at ${100}^{\circ }\text{C},{50}^{\circ }\text{C}$ and $0^{\circ }\text{C}.$ The temperature of their junction point is:

1. ${75}^{\circ }\text{C}$
2. ${\frac{200}{3}}^{\circ }\text{C}$
3. ${40}^{\circ }\text{C}$
4. ${\frac{100}{3}}^{\circ }\text{C}$

Q. For the figure shown below, choose the option that correctly represents the combination.

1. $K_1$ and $K_2$ are in series combination and together they are in parallel combination with $K_3$.
2. $K_1$ and $K_2$ are in parallel combination and together they are in series combination with $K_3$.
3. $K_1,K_2$ and $K_3$ all are in parallel combination with each other.
4. $K_1,K_2$ and $K_3$ all are in series combination with each other.

Q. A metallic cylindrical vessel whose inner and outer radii are $r_1$ and $r_2$ is filled with ice at $0^{\circ }\text{C}.$ The mass of the ice in the cylinder is $m$. Circular surfaces of the cylinder are sealed completely with adiabatic walls. The vessel is kept in air and the temperature of the air is ${50}^{\circ }\text{C}.$ How long will it take for the ice to melt completely?
[Thermal conductivity of the cylinder is $K$ and its length is $l$. Latent heat of fusion of ice is $L$].

1. $\frac{mL}{50\pi Kl}\ln \left(\frac{r_1}{r_2}\right)$
2. $\frac{mL}{200\pi Kl}\ln \left(\frac{r_1}{r_2}\right)$
3. $\frac{mL\ln \left(\frac{r_2}{r_1}\right)}{100\pi Kl}$
4. $\frac{mL\ln \left(\frac{r_2}{r_1}\right)}{50\pi Kl}$

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. ${70}^{\circ }C$
4. $0^{\circ }C$

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 spherical body of radius $b=\frac{4}{\pi }\text{m}$ has a concentric cavity of radius $a=\frac{2}{\pi }\text{m}$ as shown. Thermal conductivity of the material is $100{\text{W/m}}^{\circ }\text{C}$. Find the thermal resistance between inner and outer surface.

1. ${6.25\times {10}^{-4}}^{\circ }\text{C/W}$
2. ${1.25\times {10}^{-4}}^{\circ }\text{C/W}$
3. ${6.5\times {10}^{-4}}^{\circ }\text{C/W}$
4. ${3.75\times {10}^{-4}}^{\circ }\text{C/W}$

Q. A metallic cylindrical vessel whose inner and outer radii are $r_1$ and $r_2$ is filled with ice at $0^{\circ }\text{C}.$ The mass of the ice in the cylinder is $m$. Circular surfaces of the cylinder are sealed completely with adiabatic walls. The vessel is kept in air and the temperature of the air is ${50}^{\circ }\text{C}.$ How long will it take for the ice to melt completely?
[Thermal conductivity of the cylinder is $K$ and its length is $l$. Latent heat of fusion of ice is $L$].

1. $\frac{mL}{50\pi Kl}\ln \left(\frac{r_1}{r_2}\right)$
2. $\frac{mL}{200\pi Kl}\ln \left(\frac{r_1}{r_2}\right)$
3. $\frac{mL\ln \left(\frac{r_2}{r_1}\right)}{100\pi Kl}$
4. $\frac{mL\ln \left(\frac{r_2}{r_1}\right)}{50\pi Kl}$

Q. Two identical conducting rods are first connected independently to two vessels, one containing water at ${100}^{\circ }\text{C}$ and the other containing ice at $0^{\circ }\text{C}$. In the second case, the rods are joined end to end and connected to the same vessels. Let $q_1$ and $q_2$ gram per second be the rate of melting of ice in the two cases respectively. The ratio $\frac{q_1}{q_2}$ is

1. $\frac{1}{2}$
2. $2$
3. $4$
4. $\frac{1}{4}$

Q. Thermal conductivity of the conductor shown in the figure is $0.92{\text{cal/cm s}}^{\circ }\text{C}$. Find the thermal resistance between points $P$ and $Q$ (in ${\text{s}}^{\circ }\text{C/cal}$).

1. $0.036$
2. $0.018$
3. $0.072$
4. $1$