Spheres P and Q are uniformly constructed fro | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the radius of sphere $P$ be $r$ and the radius of sphere $Q$ be $3r$.The rate of heat loss by radiation is given by Stefan-Boltzmann Law: $\fr

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(C) Let the radius of sphere $P$ be $r$ and the radius of sphere $Q$ be $3r$.The rate of heat loss by radiation is given by Stefan-Boltzmann Law: $\frac{dQ}{dt} = \sigma e A T^4$.Also,the rate of heat loss is related to the rate of fall of temperature by the equation: $\frac{dQ}{dt} = -ms \frac{dT}{dt}$,where $m$ is mass,$s$ is specific heat capacity,and $\frac{dT}{dt}$ is the rate of fall of temperature.Equating the two: $ms \frac{dT}{dt} = \sigma e A T^4$.Since $m = \rho V = \rho (\frac{4}{3} \pi r^3)$,we have $\rho (\frac{4}{3} \pi r^3) s \frac{dT}{dt} = \sigma e (4 \pi r^2) T^4$.Thus,the rate of fall of temperature is $\frac{dT}{dt} = \frac{\sigma e (4 \pi r^2) T^4}{\rho (\frac{4}{3} \pi r^3) s} = \frac{3 \sigma e T^4}{\rho r s}$.This shows that $\frac{dT}{dt} \propto \frac{1}{r}$.Therefore,$\frac{(\frac{dT}{dt})_P}{(\frac{dT}{dt})_Q} = \frac{r_Q}{r_P} = \frac{3r}{r} = 3$.Given $(\frac{dT}{dt})_P = x (\frac{dT}{dt})_Q$,we find $x = 3$.

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:

Similar Questions

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

Two spherical black bodies have radii $R_1$ and $R_2$. Their surface temperatures are $T_1 \ K$ and $T_2 \ K$ respectively. If they radiate the same power,the ratio $\frac{R_1}{R_2}$ is

If the temperature of the body is increased by $10\%$,the percentage increase in the emitted radiation will be ....... $\%$

Two black bodies at temperatures $327^{\circ} C$ and $427^{\circ} C$ are kept in an evacuated chamber at $27^{\circ} C$. The ratio of their rates of loss of heat are :

Vedclass Test Series

Exam Paper Generator

Online Exam Module