Wires A and B have identical lengths and have circular cross-sections. The radius of A is twice the radius of B i.e. ${\mathrm{r}}_{\mathrm{A}}=2{\mathrm{r}}_{\mathrm{B}}$ . For a given temperature difference between the two ends, both wires conduct heat at the same rate. The relation between the thermal conductivities is given by

(1) ${\mathrm{K}}_{\mathrm{A}}=4{\mathrm{K}}_{\mathrm{B}}$                     
(2) ${\mathrm{K}}_{\mathrm{A}}=2{\mathrm{K}}_{\mathrm{B}}$
(3) ${\mathrm{K}}_{\mathrm{A}}={\mathrm{K}}_{\mathrm{B}}/2$                   
(4) ${\mathrm{K}}_{\mathrm{A}}={\mathrm{K}}_{\mathrm{B}}/4$

Two identical plates of different metals are joined to form a single plate whose thickness is double the thickness of each plate. If the coefficients of conductivity of each plate are 2 and 3 respectively, then the conductivity of the composite plate will be:
1. 5
2. 2.4
3. 1.5                 
4. 1.2

The coefficients of thermal conductivity of copper, mercury and glass are respectively ${\mathrm{K}}_{\mathrm{c}}$, ${\mathrm{K}}_{\mathrm{m}}$ and ${\mathrm{K}}_{\mathrm{g}}$ such that ${\mathrm{K}}_{\mathrm{c}}>{\mathrm{K}}_{\mathrm{m}}>{\mathrm{K}}_{\mathrm{g}}$. If the same quantity of heat is to flow per second per unit area of each and corresponding temperature gradients are ${\mathrm{X}}_{\mathrm{c}}$, ${\mathrm{X}}_{\mathrm{m}}$ and ${\mathrm{X}}_{\mathrm{g}}$, then 
(1) ${\mathrm{X}}_{\mathrm{c}}={\mathrm{X}}_{\mathrm{m}}={\mathrm{X}}_{\mathrm{g}}$                 

(2) ${\mathrm{X}}_{\mathrm{c}}>{\mathrm{X}}_{\mathrm{m}}>{\mathrm{X}}_{\mathrm{g}}$

(3) ${\mathrm{X}}_{\mathrm{c}}<{\mathrm{X}}_{\mathrm{m}}<{\mathrm{X}}_{\mathrm{g}}$                 

(4) ${\mathrm{X}}_{\mathrm{m}}<{\mathrm{X}}_{\mathrm{c}}<{\mathrm{X}}_{\mathrm{g}}$

If two metallic plates of equal thicknesses and thermal conductivities ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ are put together face to face and a common plate is constructed, then the equivalent thermal conductivity of this plate will be

(1) $\frac{{\mathrm{K}}_1{\mathrm{K}}_2}{{\mathrm{K}}_1+{\mathrm{K}}_2}$     
               
(2) $\frac{2{\mathrm{K}}_1{\mathrm{K}}_2}{{\mathrm{K}}_1+{\mathrm{K}}_2}$

(3) $\frac{{\left({\mathrm{K}}_1^2+{\mathrm{K}}_2^2\right)}^{3/2}}{{\mathrm{K}}_1{\mathrm{K}}_2}$ 
           
(4) $\frac{{\left({\mathrm{K}}_1^2+{\mathrm{K}}_2^2\right)}^{3/2}}{2{\mathrm{K}}_1{\mathrm{K}}_2}$

The quantity of heat which crosses unit area of a metal plate during conduction depends upon

(1) The density of the metal

(2) The temperature gradient perpendicular to the area

(3) The temperature to which the metal is heated

(4) The area of the metal plate

Two spheres of different materials one with double the radius and one-fourth wall thickness of the other, are filled with ice. If the time taken for complete melting of ice in the larger sphere is 25 minutes and that for smaller sphere 16 minutes, the ratio of thermal conductivities of the materials of larger sphere to the smaller sphere is 
(1) 4 : 5                   

(2) 5 : 4

(3) 25 : 1                 

(4) 8 : 25

The mud houses are cooler in the summer and warmer in the winter because:

The temperature of the hot and cold ends of a 20 cm long rod in a thermal steady state is at \(100^{\circ}\mathrm{C}\) and \(20^{\circ}\mathrm{C}\) respectively. The temperature at the centre of the rod will be:
1. \(50^{\circ}\mathrm{C}\)
2. \(60^{\circ}\mathrm{C}\)
3. \(40^{\circ}\mathrm{C}\)
4. \(30^{\circ}\mathrm{C}\)

Two bars of thermal conductivities K and 3K and lengths 1 cm and 2 cm respectively have equal cross-sectional area, they are joined lengths wise as shown in the figure. If the temperature at the ends of this composite bar is 0$°\mathrm{C}$ and 100$°\mathrm{C}$ respectively (see figure), then the temperature $\varphi$ of the interface is

(1) 50$°\mathrm{C}$     
               
(2) $\frac{100}{3}°\mathrm{C}$ 

(3) 60$°\mathrm{C}$   
                 
(4) $\frac{200}{3}°\mathrm{C}$ 

A heat flux of 4000 J/s is to be passed through a copper rod of length 10 cm and area of cross-section 100 ${\mathrm{cm}}^2$. The thermal conductivity of copper is 400 W/m$°\mathrm{C}$. The two ends of this rod must be kept at a temperature difference of 
(1) 1$°\mathrm{C}$                

(2) 10$°\mathrm{C}$

(3) 100$°\mathrm{C}$             

(4) 1000$°\mathrm{C}$