Two metal rods \((1)\) and \((2)\) of the same length 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. \(K_1 A_1=4K_2 {A}_2 \)
2. \(K_1 {A}_1=2 {K}_2 {A}_2 \)
3. \(4 {K}_1{A}_1={K}_2 {A}_2 \)
4. \({K}_1 {A}_1={K}_2 {A}_2\)

A slab of stone with an area \(0.36~\text{m}^{2}\)${\mathrm{}}^{}$ and thickness of \(0.1~\text{m}\) is exposed on the lower surface to steam at \(100​​^{\circ}\mathrm{C}\)$$. A block of ice at \(0^{\circ}\mathrm{C}\) rests on the upper surface of the slab. In one hour \(4.8~\text{kg}\) of ice is melted. The thermal conductivity of the slab will be: (Given latent heat of fusion of ice \(= 3.36\times10^{5}~\text{JKg}^{-1}\)${\mathrm{}}^{}$)
1. \(1.29~\text{J/m/s/}^{\circ}\text{C}\)
2. \(2.05~\text{J/m/s/}^{\circ}\text{C}\)
3. \(1.02~\text{J/m/s/}^{\circ}\text{C}\)
4. \(1.24~\text{J/m/s/}^{\circ}\text{C}\)

A black body at 1227 °C emits radiations with maximum intensity at a wavelength of 5000 Å. If the temperature of the body is increased by 1000 °C, the maximum intensity will be observed at:

1. 4000 Å

2. 5000 Å

3. 6000 Å

4. 3000 Å

A black body is at \(727^\circ\text{ C}.\) The rate at which it emits energy is proportional to:
1. \((727)^2\)
2. \((1000)^4\)
3. \((1000)^2\)
4. \((727)^4\)

Assuming the sun to have a spherical outer surface of radius \(r,\) radiating like a black body at temperature \(t^\circ \text{ C},\) the power received by a unit surface of the earth (normal to the incident rays) at a distance \(R\) from the centre of the sun is: (where \(\sigma\) is Stefan's constant)
1. \(\dfrac{4\pi r^2\sigma t^4}{R^2}\)

2. \(\dfrac{r^2\sigma(t+273)^4}{4\pi R^2}\)

3. \(\dfrac{16\pi^2r^2\sigma t^4}{R^2}\)

4. \(\dfrac{r^2\sigma(t+273)^4}{R^2}\)

On a new scale of temperature, which is linear and called the \(\mathrm{W}\) scale, the freezing and boiling points of water are \(39^\circ ~\mathrm{W}\)$$and \(239^\circ ~\mathrm{W}\) respectively. What will be the temperature on the new scale corresponding to a temperature of \(39^\circ ~\mathrm{C}\) on the Celsius scale?
1. \(78^\circ ~\mathrm{C}\)
2. \(117^\circ ~\mathrm{W}\)
3. \(200^\circ ~\mathrm{W}\)
4. \(139^\circ ~\mathrm{W}\)

The two ends of a rod of length L and a uniform cross-sectional area A are kept at two temperatures T₁ and T₂ (T₁> T₂). The rate of heat transfer $\frac{\mathrm{dQ}}{\mathrm{dt}}$ through the rod in a steady state is given by:

1. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{KL}({\mathrm{T}}_1-{\mathrm{T}}_2)}{\mathrm{A}}$

2. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{K}({\mathrm{T}}_1-{\mathrm{T}}_2)}{\mathrm{LA}}$

3. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\mathrm{KLA}({\mathrm{T}}_1-{\mathrm{T}}_2)$

4. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{KA}({\mathrm{T}}_1-{\mathrm{T}}_2)}{\mathrm{L}}$

A black body at \(227^{\circ}~\mathrm{C}\) radiates heat at the rate of \(7~ \mathrm{cal-cm^{-2}s^{-1}}\).  At a temperature of \(727^{\circ}~\mathrm{C}\), the rate of heat radiated in the same units will be:
1. \(60\)
2. \(50\)
3. \(112\)
4. \(80\)