A black body emits radiation at the rate $P$ when its temperature is $T$. At this temperature the wavelength at which the radiation has a maximum intensity is ${\lambda }_0$. If at another temperature $T^{\text{'}}$ the power radiated is $P^{\text{'}}$ and wavelength at maximum intensity is ${\lambda }_0/2$ then

A metallic sphere having radius $0.08 \mathrm{m}$ and mass $\mathrm{m}=10 \mathrm{kg}$ is heated to a temperature of $227°\mathrm{C}$ and suspended inside a box whose walls are at a temperature of $27°\mathrm{C}.$ The maximum rate at which its temperature will fall is:-

(Take $e=1,$ Stefan's constant $\sigma =5.8\times {10}^{-8} {\mathrm{Wm}}^{-2}{\mathrm{K}}^{-4}$ and specific heat of the metal $\mathrm{s}=90 {\mathrm{cal}}^{-1}{\mathrm{kg}}^{-1}\mathrm{deg}$ $\mathrm{J}=4.2$ Joules/Calorie)

A body cools in a surrounding which is at a constant temperature of ${\theta }_0$. Assume that it obeys Newton's law of cooling. Its temperature $\theta$ is plotted against time $t$. Tangents are drawn to the curve at the points $P\left(\theta ={\theta }_2\right)$ and $Q\left(\theta ={\theta }_1\right)$. These tangents meet the time axis at angles of ${\varphi }_2$ and ${\varphi }_1$, as shown.

A spherical body with an initial temperature $T_1$ is allowed to cool in surroundings at temperature $T_0\left(<T_1\right)$. The mass of the body is $m$, its gram specific heat is $c$, density $\rho$, area $A$. If $\sigma$ be the Stefan's constant then the temperature $T$ of the body at time $t$ can be best represented by:-

A ring consisting of two parts $\mathrm{ADB}$ and $\mathrm{ACB}$ of same conductivity $k$ carries an amount of heat $H$. The $\mathrm{ADB}$ part is now replaced with another metal keeping the temperatures $T_1$ and $T_2$ constant. The heat carried increases to $2H$. What should be the conductivity of the new $\mathrm{ADB}$ part? Given $\frac{\mathrm{ACB}}{\mathrm{ADB}}=3$