MCQ
Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume$E=$ $\frac{U}{V} \propto {T^4}$ and pressure $P = \frac{1}{3}\left( {\frac{U}{V}} \right)$ If the shell now undergoes an adiabatic expansion the relation between $T$ and $R$ is
  • A
    $T \propto {e^{ - 3R}}\;\;\;\;\;\;\;\;\;\;\;\;$
  • $\;T \propto \frac{1}{R}$
  • C
    $\;T \propto \frac{1}{{{R^3}}}$
  • D
    $\;T \propto {e^{ - R}}$

Answer

Correct option: B.
$\;T \propto \frac{1}{R}$
b
$As,\,P = \frac{1}{3}\left( {\frac{U}{V}} \right)$

$But\frac{U}{V} = K{T^4}$

$So,P = \frac{1}{3}K{T^4}$

$or\,\frac{{uRT}}{V} = \frac{1}{3}K{T^4}\,\,\,\left[ {As\,PV = u\,RT} \right]$

$\frac{4}{3}\pi {R^3}{T^3} = constant$

$Therfore,T \propto \frac{1}{R}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Similar questions

Which of the following most closely depicts the correct variation of the gravitational potential $V(r)$ due to a large planet of radius $R$ and uniform mass density ? (figures are not drawn to scale)
Three rods of identical area of cross-section and made from the same metal form the sides of an isosceles triangle $ABC$, right angled at $B$. The points $A$ and $B$ are maintained at temperatures $T$ and $\sqrt 2 T$ respectively. In the steady state the temperature of the point C is ${T_C}$. Assuming that only heat conduction takes place, $\frac{{{T_C}}}{T}$ is equal to
A particle is projected vertically upwards from a point $A$ on the ground. It takes time $t_1$ to  reach a point $B$, but it still continues to move up. If it takes further $t_2$ time to reach the  ground from point $B$. Then height of point $B$ from the ground is-
A hot liquid is kept in a big room. The logarithm of the numerical value of the temperature difference between the liquid and the room is plotted against time. The plot will be very nearly:
A stone is dropped from a certain height which can reach the ground in $5\, second$. If the stone is stopped after $3\, second$ of its fall and then allowed to fall again, then the time taken by the stone to reach the ground for the remaining distance is........$sec$
Three vectors $\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{OR}}$ each of magnitude $A$ are acting as shown in figure. The resultant of the three vectors is $A \sqrt{x}$. The value of $x$ is. . . . . . . . .
To what temperature should the hydrogen at $327°C$ be cooled at constant pressure, so that the root mean square velocity of its molecules become half of its previous value ....... $^oC$
Out of the given four waves $(1), (2), (3)$ and $(4)$

$y = a\sin (kx + \omega t)$   ......$(1)$

$y = a\sin (\omega t - kx)$   ......$(2)$

$y = a\cos (kx + \omega t)$   ......$(3)$

$y = a\cos (\omega t - kx)$   ......$(4)$

emitted by four different sources ${S_1},\,{S_2},\,{S_3}$ and ${S_4}$ respectively, interference phenomena would be observed in space under appropriate conditions when

A satellite is revolving round the earth with orbital speed $v_0$. If it stops suddenly, the speed with which it will strike the surface of earth would be : ($v_e =$ escape velocity of a particle on earth's surface)
One litre of Helium gas at a pressure $76\, cm$ of $Hg$ and temperature ${27}^\circ$ is heated till its pressure and volume are doubled. The final temperature attained by the gas is ..... $^oC$