Three rods of identical area of cross-section and made from the same metal form the sides of an isosceles triangle ABC , right angled at B. The points A and B are maintained at temperatures T and $\sqrt{2}\mathrm{T}$ respectively. In the steady state the temperature of the point C is ${\mathrm{T}}_{\mathrm{C}}$. Assuming that only heat conduction takes place, $\frac{{\mathrm{T}}_{\mathrm{C}}}{\mathrm{T}}$ is equal to -
1. $\frac{1}{\left(\sqrt{2}+1\right)}$                   

2. $\frac{3}{\left(\sqrt{2}+1\right)}$
3. $\frac{1}{2\left(\sqrt{2}-1\right)}$                  

4. $\frac{1}{\sqrt{3}\left(\sqrt{2}-1\right)}$

A sphere, a cube and a thin circular plate, all made of the same material and having the same mass are initially heated to a temperature of 1000°C. Which one of these will cool first ?

1. Plate                     

2. Sphere

3. Cube                     

4. None of these

A black metal foil is warmed by radiation from a small sphere at temperature T and at a distance d where surrounding temperature is $T_o$. It is found that the power received by the foil is `P'. If both the temperature and the distance are doubled, the power received by the foil will be- [Assume $T>>T_o$]

1. 16P                 

2. 4P

3. 2P                   

4. P

Two metallic spheres ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are made of the same material and have identical surface finish. The mass of ${\mathrm{S}}_1$ is three times that of ${\mathrm{S}}_2$. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of ${\mathrm{S}}_1$ to that of ${\mathrm{S}}_2$ is 

1. $1/3$             

2. ${\left(1/3\right)}^{1/3}$

3. $1/\sqrt{3}$          

4. $\sqrt{3}/1$

The total energy radiated from a black body source is collected for one minute and is used to heat a quantity of water. The temperature of water is found to increase  from   20$°\mathrm{C}$ to 20.5$°\mathrm{C}$ . If the absolute temperature of the black body is doubled and the experiment is repeated with the same quantity of water at 20$°\mathrm{C}$, the temperature of water will be 

1. 21$°\mathrm{C}$               2. 22$°\mathrm{C}$ 
3. 24$°\mathrm{C}$               4. 28$°\mathrm{C}$ 

A solid sphere and a hollow sphere of the same material and size are heated to the same temperature and allowed to cool in the same surroundings. If the temperature difference between each sphere and its surroundings is same , then

1. The hollow sphere will cool at a faster rate for all values of T

2. The solid sphere will cool at a faster rate for all values of T

3. Both spheres will cool at the same rate for all values of T

4. Both spheres will cool at the same rate only for small values of T

A solid copper cube of edges 1 cm is suspended in an evacuated enclosure. Its temperature is found to fall from 100$°\mathrm{C}$ to 99$°\mathrm{C}$ in 100 s . Another solid copper cube of edges 2 cm, with similar surface nature, is suspended in a similar manner. The time required for this cube to cool from 100$°\mathrm{C}$ to 99$°\mathrm{C}$ will be approximately -

1. 25 s                 

2. 50 s  

3. 200 s                 

4. 400 s

A sphere and a cube of same material and same volume are heated upto same temperature and allowed to cool in the same surroundings. The ratio of the amounts of radiations emitted will be

1. 1 : 1                 

2. $\frac{4\mathrm{\pi }}{3}:1$

3. ${\left(\frac{\mathrm{\pi }}{6}\right)}^{1/3}:1$         

4. $\frac{1}{2}{\left(\frac{4\mathrm{\pi }}{3}\right)}^{\frac{2}{3}}:1$

The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, 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 $\left(\frac{\mathrm{A}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)\mathrm{K}}{\mathrm{x}}\right)\mathrm{f}$, with f which equals to:

1. 1

2. $\frac{1}{2}$

3. $\frac{2}{3}$

4. $\frac{1}{3}$

The graph shown in the adjacent diagram, represents the variation of temperature (T) of two bodies, x and y having same surface area, with time (t) due to the emission of radiation. Find the correct relation between the emissivity (e) and absorptivity (a) of the two bodies .

1. ${\mathrm{e}}_{\mathrm{x}}>{\mathrm{e}}_{\mathrm{y}} \& {\mathrm{a}}_{\mathrm{x}}<{\mathrm{a}}_{\mathrm{y}}$

2. ${\mathrm{e}}_{\mathrm{x}}<{\mathrm{e}}_{\mathrm{y}} \& {\mathrm{a}}_{\mathrm{x}}>{\mathrm{a}}_{\mathrm{y}}$

3. ${\mathrm{e}}_{\mathrm{x}}>{\mathrm{e}}_{\mathrm{y}} \& {\mathrm{a}}_{\mathrm{x}}>{\mathrm{a}}_{\mathrm{y}}$

4. ${\mathrm{e}}_{\mathrm{x}}<{\mathrm{e}}_{\mathrm{y}} \& {\mathrm{a}}_{\mathrm{x}}<{\mathrm{a}}_{\mathrm{y}}$