Two spheres of radii in the ratio 1 : 2 and densities in the ratio 2 : 1 and of same specific heat, are heated to same temperature and left in the same surrounding. Their rate of cooling will be in ratio

Q: Two spheres of radii in the ratio 1 : 2 and densities in the ratio 2 : 1 and of same specific heat, are heated to same temperature and left in the same surrounding. Their rate of cooling will be in ratio

(a) 2 : 1

(b) 1 : 1

(c) 1 : 2

(d) 1 : 4

Ans: (b)

Sol: Rate of cooling , $\large \frac{dT}{dt} = \frac{e \sigma A}{m s}(T^4 – T_o^4 ) $

$\large \frac{(dT/dt)_1}{(dT/dt)_2} = \frac{A_1}{A_2} . \frac{m_2}{m_1} $

$\large = \frac{r_1^2}{r_2^2} . \frac{\rho_2 . \frac{4}{3}\pi r_2^3}{\rho_1 . \frac{4}{3}\pi r_1^3} $

$\large \frac{(dT/dt)_1}{(dT/dt)_2} = \frac{\rho_2}{\rho_1} . \frac{r_2}{r_1} $

$\large \frac{(dT/dt)_1}{(dT/dt)_2} = \frac{1}{2} . \frac{2}{1} $

= 1:1

Two bodies A and B having equal surface areas are maintained at temperature 10°C and 20°C. The thermal radiation emitted in a given time by A and B are in the ratio

Q: Two bodies A and B having equal surface areas are maintained at temperature 10°C and 20°C. The thermal radiation emitted in a given time by A and B are in the ratio

(a) 1 : 1.15

(b) 1 : 2

(c) 1 : 4

(d) 1 : 16

Ans: (a)

Sol: $\large E \propto T^4 $

$\large \frac{E_1}{E_2} = (\frac{T_1}{T_2})^4$

$\large \frac{E_1}{E_2} = (\frac{273+10}{273 +20})^4$

$\large \frac{E_1}{E_2} = (\frac{283}{293})^4 = (\frac{293-10}{293})^4$

$\large \frac{E_1}{E_2} = (1- \frac{10}{293})^4 = 1- \frac{40}{293}$

$\large \frac{E_1}{E_2} = \frac{253}{293} = \frac{1}{293/253} $

$\large \frac{E_1}{E_2} = \frac{1}{1.15} $

Two spherical black bodies of radii R1 and R2 and with surface temperature T1 and T2 respectively radiate the same power. R1/R2 must be equal to

Q: Two spherical black bodies of radii R1 and R2 and with surface temperature T1 and T2 respectively radiate the same power. R1/R2 must be equal to

(a) $\large (\frac{T_1}{T_2})^2$

(b) $\large (\frac{T_2}{T_1})^2$

(c) $\large (\frac{T_1}{T_2})^4 $

(d) $\large (\frac{T_2}{T_1})^4 $

Ans: (b)

Sol: $\large E = e \sigma A T^4 $

$\large E = e \sigma (4 \pi R^2) T^4 = constant $

$\large R_1^2 T_1^4 = R_2^2 T_2^4 $

$\large \frac{R_1}{R_2} = (\frac{T_2}{T_1})^2 $

Three very large plates of same are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures 2T and 3T respectively. The temperature of the middle (i. e., second) plate under steady condition is

Q: Three very large plates of same are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures 2T and 3T respectively. The temperature of the middle (i. e., second) plate under steady condition is

(a) $(\frac{65}{2})^{1/4} T $

(b) $(\frac{97}{4})^{1/4} T $

(c) $(\frac{97}{2})^{1/4} T $

(d) $(97)^{1/4} T $

Ans: (c)

Sol: Let temperature of middle plate is T’

In steady state,

Energy absorbed by middle plate = Energy released by middle plate

$\large \sigma A [(3T)^4 – (T’)^4] = \sigma A [(T’)^4 – (2T)^4]$

$\large T’ = (\frac{97}{2})^{1/4} T $

A spherical black body with a radius of 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature double, the power radiated in watt would be

Q: A spherical black body with a radius of 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature double, the power radiated in watt would be

(a) 225

(b) 450

(c) 900

(d) 1800

Ans: (d)

Sol: $\large E = e \sigma A T^4 $

$\large E = e \sigma (4\pi r^2) T^4 $

$\large E \propto r^2 T^4 $

$\large \frac{E}{E’} = \frac{r^2}{(r/2)^2} (\frac{T}{2T})^2 $

$\large \frac{E}{E’} = \frac{1}{4} $

$\large \frac{450}{E’} = \frac{1}{4} $

E’ = 1800 W