A spherical black body of radius r at an initial temperature of 27^\circ\text{C} emits the… — JAMB UTME

A spherical black body of radius $r$ at an initial temperature of $27^\circ\text{C}$ emits thermal radiation at a rate of $W$ . If the radius of the sphere is doubled and its temperature is increased to $327^\circ\text{C}$ , what is the new rate of thermal radiation emitted by the sphere in terms of $W$ ?

$64W$ Answer
B
$32W$
C
$16W$
D
$8W$

Answer

The new rate of thermal radiation emitted by the sphere is

64W

According to the Stefan-Boltzmann law, the rate of energy radiation from a black body is given by

P = \sigma A T^4

. Doubling the radius of a sphere increases its surface area by a factor of

2^2 = 4

. Converting temperatures from Celsius to Kelvin gives

T_1 = 27 + 273 = 300\text{ K}

and

T_2 = 327 + 273 = 600\text{ K}

, showing that the absolute temperature doubles (

T_2 / T_1 = 2

). Raising this temperature ratio to the fourth power yields

2^4 = 16

. Combining the area factor of

4

and the temperature factor of

16

results in a total radiation rate increase of

4 \times 16 = 64

times the original rate

W

Step-by-Step Solution

Express initial radiation rate using Stefan-Boltzmann law and sphere surface area formula

The total power radiated by a black body is given by

P = \sigma A T^4

. For a sphere of radius

r

A_1 = 4\pi r^2

. Absolute temperature

T_1 = 27 + 273 = 300\text{ K}

. Thus,

W = \sigma (4\pi r^2) (300)^4

Stefan's law requires absolute temperature in Kelvin and total surface area of the radiator.

Determine the scaled surface area and absolute temperature for the final state

New radius

r_2 = 2r \implies A_2 = 4\pi (2r)^2 = 4 A_1

. New temperature

T_2 = 327 + 273 = 600\text{ K} = 2 T_1

Surface area of a sphere scales quadratically with radius, and temperatures must be converted to Kelvin.

Calculate the ratio of the new radiation rate to the initial radiation rate

\frac{P_2}{P_1} = \frac{A_2}{A_1} \times \left(\frac{T_2}{T_1}\right)^4 = 4 \times (2)^4 = 4 \times 16 = 64

Radiated power is directly proportional to surface area and to the fourth power of absolute temperature.

State the new power in terms of

W

P_2 = 64 W

Multiplying the initial rate

W

by the overall scaling factor of 64 gives the final answer.

Key Concept

Stefan-Boltzmann Law of Radiation (

P = \epsilon \sigma A T^4

)

Question

Answer

Step-by-Step Solution

Key Concept

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