Two spheres of the same material have radii $1 \; m$ and $4 \; m$ and temperatures $4000 \; K$ and $2000 \; K$ respectively. The ratio of the energy radiated per second by the first sphere to that by the second is

$A$ black body is at a temperature $300 K$. It emits energy at a rate,which is proportional to

$A$ black coloured solid sphere of radius $R$ and mass $M$ is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature $T_0$. The initial temperature of the sphere is $3T_0$. If the specific heat of the material of the sphere varies as $\alpha T^3$ per unit mass with the temperature $T$ of the sphere,where $\alpha$ is a constant,then the time taken for the sphere to cool down to temperature $2T_0$ will be ($\sigma$ is Stefan-Boltzmann constant).

Two identical plates $P$ and $Q$,radiating as perfect black bodies,are kept in vacuum at constant absolute temperatures $T_P$ and $T_Q$,respectively,with $T_Q < T_P$,as shown in Fig. $1$. The radiated power transferred per unit area from $P$ to $Q$ is $W_0$. Subsequently,two more plates,identical to $P$ and $Q$,are introduced between $P$ and $Q$,as shown in Fig. $2$. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from $P$ to $Q$ (Fig. $2$) in the steady state is $W_S$,then the ratio $\frac{W_0}{W_S}$ is $.....$

The temperature of a body is $400^{\circ}C$. Assume the temperature of the surroundings is negligible. At what temperature will the body emit double the energy?

Spheres $P$ and $Q$ are uniformly constructed from the same material,which is a good conductor of heat,and the radius of $Q$ is thrice the radius of $P$. The rate of fall of temperature of $P$ is $x$ times that of $Q$ when both are at the same surface temperature. The value of $x$ is: