1. ⇒ (MHT CET 2023 13th May Morning Shift )

A black body at temperature $127^{\circ} C$ radiates heat at the rate of $5 cal / {cm}^{2} s$ . At a temperature $927^{\circ} C$ , its rate of emission in units of $cal / {cm}^{2} s$ will be

A. 405

B. 35

C. 45

D. 350

Correct Option is (A)

From Stefan $-$ Boltzmann's Law

$\begin{aligned} E = σ T^{4} \\ \Rightarrow E \propto T^{4} \end{aligned}$

$\begin{aligned} ∴ \frac{E_{1}}{E_{2}} & = {(\frac{T_{1}}{T_{2}})}^{4} = {(\frac{400}{1200})}^{4} \\ \frac{E_{1}}{E_{2}} & = \frac{1}{81} \\ E_{2} & = 81 \times E_{1} \\ = 81 \times 5 \\ = 405 cal / {cm}^{2} s \end{aligned}$

2. ⇒ (MHT CET 2023 12th May Evening Shift )

A black body radiates maximum energy at wavelength ' $λ$ ' and its emissive power is ' $E$ '. Now due to a change in temperature of that body, it radiates maximum energy at wavelength $\frac{λ}{3}$ . At that temperature emissive power is

A. $16 : 1$

B. $256 : 1$

C. $81 : 1$

D. $128 : 1$

Correct Option is (C)

$∴$ Emissive power of black body is, From Wien's Law:

$\begin{aligned} E & = σ T^{4} \\ λ & = \frac{b}{T} \\ ∴ E & = \frac{σ b^{4}}{λ^{4}} \\ ∴ \frac{E_{2}}{E_{1}} & = \frac{λ_{1}^{4}}{λ_{2}^{4}} = \frac{λ^{4}}{{(\frac{λ}{3})}^{4}} = \frac{81}{1} \end{aligned}$

3. ⇒ (MHT CET 2023 12th May Evening Shift )

Compare the rate of loss of heat from a metal sphere at $627^{\circ} C$ with the rate of loss of heat from the same sphere at $327^{\circ} C$ , if the temperature of the surrounding is $27^{\circ} C$ . (nearly)

A. 6.2

B. 5.3

C. 4.8

D. 7.4

Correct Option is (B)

The heat loss is given as $R = e σ A (T^{4} - T_{0}^{4})$

Let surrounding temperature be denoted as $T$

Heat loss from the metal sphere at a temperature $T_{1}$

$R_{1} = e σ A (T_{1}^{4} - T_{0}^{4})$

Heat loss from the metal sphere at a temperature $T_{2}$

$R_{2} = e σ A (T_{2}^{4} - T^{4})$

$\begin{aligned} ∴ \frac{R_{1}}{R_{2}} & = \frac{T_{1}^{4} - T^{4}}{T_{2}^{4} - T^{4}} = \frac{900^{4} - 300^{4}}{600^{4} - 300^{4}} \\ \frac{R_{1}}{R_{2}} & = 5.3 \end{aligned}$

4. ⇒ (MHT CET 2023 12th May Morning Shift )

A black sphere has radius ' $R$ ' whose rate of radiation is ' $E$ ' at temperature ' $T$ '. If radius is made $R / 3$ and temperature ' $3 T$ ', the rate of radiation will be

A. E

B. 3E

C. 6E

D. 9E

Correct Option is (D)

$E = e A σ T^{4}$

But for sphere, $A = 4 π R^{2}$

$\begin{aligned} ∴ & E = e (4 π R^{2}) σ T^{4} \\ ∴ & \frac{E_{1}}{E_{2}} = \frac{R_{1}^{2} T_{1}^{4}}{R_{2}^{2} T_{2}^{4}} \\ ∴ & \frac{E}{E_{2}} = \frac{R^{2} T^{4}}{{(\frac{R}{3})}^{2} (3 T)^{4}} \\ ∴ & E_{2} = 9 E \end{aligned}$

5. ⇒ (MHT CET 2023 12th May Morning Shift )

About black body radiation, which of the following is the wrong statement?

A. For all wavelengths, intensity is same.

B. For shorter wavelengths, intensity is more.

C. For longer wavelengths, intensity is less.

D. All wavelengths are emitted by a black body.

Correct Option is (A)

Currently no explanation available

⇒Kinetic Theory of Gases