A spherical black body radiator with an initial radius of 0.10\text{ m} emits thermal radi… — JAMB UTME

A spherical black body radiator with an initial radius of $0.10\text{ m}$ emits thermal radiation at a rate of $E_1$ when its surface temperature is $400\text{ K}$ . If the radius of the sphere is doubled to $0.20\text{ m}$ and its absolute temperature is reduced to $200\text{ K}$ , what is the value of the ratio of the new rate of heat radiation to the initial rate, $\frac{E_2}{E_1}$ ?

Cevap: 0.25

Cevap

The ratio of the new rate of heat radiation to the initial rate is

0.25

According to the Stefan-Boltzmann law, the rate of thermal radiation emitted by a black body is directly proportional to its surface area (

A \propto r^2

) and the fourth power of its absolute temperature (

T^4

). Doubling the radius increases the surface area by a factor of

2^2 = 4

, while halving the absolute temperature decreases the radiation rate per unit area by a factor of

(1/2)^4 = 1/16

. The net ratio of the new emission rate to the initial emission rate is

4 \times (1/16) = 0.25

Adım Adım Çözüm

Apply Stefan-Boltzmann's law of radiation to formulate the rate of heat emission.

The total power radiated by a sphere of radius

r

at thermodynamic temperature

T

is given by

P = \sigma A T^4 = 4\pi \sigma r^2 T^4

, where

\sigma

is the Stefan-Boltzmann constant.

Thermal radiation emission rate depends directly on surface area (

A = 4\pi r^2

) and the fourth power of absolute temperature (

T^4

Formulate the ratio

\frac{E_2}{E_1}

using the scaled physical parameters.

\frac{E_2}{E_1} = \frac{4\pi \sigma r_2^2 T_2^4}{4\pi \sigma r_1^2 T_1^4} = \left(\frac{r_2}{r_1}\right)^2 \left(\frac{T_2}{T_1}\right)^4

Constants

4\pi\sigma

cancel out when evaluating relative changes.

Substitute the given numerical ratios into the equation and compute the result.

Given

\frac{r_2}{r_1} = \frac{0.20}{0.10} = 2

and

\frac{T_2}{T_1} = \frac{200}{400} = 0.5

, we get

\frac{E_2}{E_1} = (2)^2 \times (0.5)^4 = 4 \times \frac{1}{16} = 0.25

Evaluating

2^2 = 4

and

(0.5)^4 = 1/16

yields a final ratio of

4/16 = 0.25

Anahtar Kavram

Stefan-Boltzmann Law of Radiation

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