$A$ solid copper sphere (density $\rho$ and specific heat capacity $c$) of radius $r$ at an initial temperature $200 \, K$ is suspended inside a chamber whose walls are at almost $0 \, K$. The time required (in $\mu s$) for the temperature of the sphere to drop to $100 \, K$ is:

$A$ black body emits energy at the rate of $1 \ cal/cm^2 \cdot s$ at a temperature of $127^{\circ}C$. Find the rate of energy emission at a temperature of $527^{\circ}C$ in $cal/cm^2 \cdot s$.

The energy spectrum of a black body exhibits a maximum around a wavelength '$\lambda$'. The temperature of the black body is now changed such that the energy is maximum around a wavelength $2\lambda/3$. The power radiated by the black body will now increase by a factor

The energy spectrum of a black body has maximum energy at a wavelength of $\lambda_0$. Now, the temperature of the black body is increased such that the maximum energy is obtained at a wavelength of $3\lambda_0/4$. By what factor will the power radiated by the black body increase?

$A$ human body has a surface area of approximately $1 \,m^2$. The normal body temperature is $10 \,K$ above the surrounding room temperature $T_0$. Take the room temperature to be $T_0=300 \,K$. For $T_0=300 \,K$, and the value of $\sigma T_0^4=460 \,W/m^2$ (where $\sigma$ is the Stefan-Boltzmann constant). Which of the following option(s) is/are correct?
[$A$] The amount of energy radiated by the body in $1 \,s$ is close to $60 \,J$.
[$B$] If the surrounding temperature reduces by a small amount $\Delta T_0 < < T_0$, then to maintain the same body temperature the same (living) human being needs to radiate $\Delta W = 4 \sigma T_0^3 \Delta T_0$ more energy per unit time.
[$C$] Reducing the exposed surface area of the body (e.g., by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation.
[$D$] If the body temperature rises significantly, then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths.

$A$ wire of length $L$ and diameter $d$ is used in a bulb. The temperature of the wire is $T$ and the power radiated by the wire is $P$. Its emissivity is $e$ ($\sigma$ = Stefan's constant). (Assume that the emissivity of the wire material is the same at all wavelengths).