A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at $0 K$. At time $t =0$, the temperature of the object is $200 K$. The temperature of the object becomes $100 K$ at $t = t _1$ and $50 K$ at $t = t _2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left( t _2 / t _1\right)$ is. . . . .
IIT 2021, Advanced
Download our app for free and get started
$\sigma A T^4=-m s \frac{d T}{d t}$
$\int_{200}^{100} \frac{d T}{T^4}=\int_0^{t_1} k dt$
$\left.\frac{1}{3 T ^3}\right|_{200} ^{100}= kt _1$
$\frac{1}{3}\left(\frac{1}{100^3}-\frac{1}{200^3}\right)= kt _1$
$\left.\frac{1}{3 T ^3}\right|_{200} ^{50}= kt _2$
$\frac{1}{3}\left(\frac{1}{50^3}-\frac{1}{200^3}\right)= kt t _2$
$\frac{ t _2}{ t _1}=\left(\frac{200^3-50^3}{200^3-100^3}\right) \frac{100^3}{50^3}=9$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1View SolutionWhich of the following cylindrical rods will conduct most heat, when their ends are maintained at the same steady temperature
- 2On a clear sunny day, an object at temperature $T$ is placed on the top of a high mountain. An identical object at the same temperature is placed at the foot of mountain. If both the objects are exposed to sun-rays for two hours in an identical manner, the object at the top of the mountain will register a temperatureView Solution
- 3The spectrum of a black body at two temperatures $27^oC$ and $327^oC$ is shown in the figure. Let $A_1$ and $A_2$ be the areas under the two curves respectively. The value of $\frac{{{A_2}}}{{{A_1}}}$ isView Solution
- 4One end of a metal rod of length $1.0 m$ and area of cross section $100c{m^2}$ is maintained at ${100^o}C.$If the other end of the rod is maintained at ${0^o}C$, the quantity of heat transmitted through the rod per minute is (Coefficient of thermal conductivity of material of rod =$100W/m-K$)View Solution
- 5A cylindrical rod having temperature ${T_1}$ and ${T_2}$ at its ends. The rate of flow of heat is ${Q_1}$ $cal/sec$. If all the linear dimensions are doubled keeping temperature constant then rate of flow of heat ${Q_2}$ will beView Solution
- 6Four identical rods of same material are joined end to end to form a square. If the temperature difference between the ends of a diagonal is ${100^o}C$, then the temperature difference between the ends of other diagonal will be ........ $^oC$View Solution
- 7Temperature of a black body increases from ${327^o}C\,{\rm{to}}\,{\rm{92}}{{\rm{7}}^{\rm{o}}}C$, the initial energy possessed is $ 2KJ$ , what is its final energy ..... $KJ$View Solution
- 8A 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 $P(\theta = {\theta _1})$ and $Q(\theta = {\theta _2})$. These tangents meet the time axis at angles of ${\varphi _2}$and ${\varphi _1}$, as shownView Solution
- 9Instantaneous temperature difference between cooling body and the surroundings obeying Newton's law of cooling is $\theta$. Which of the following represents the variation of $\ln \theta$ with time $t ?$View Solution
- 10A hot metallic sphere of radius $r$ radiates heat. It's rate of cooling isView Solution