If a black body is heated from 27°C to 927°C, then the ratio of the radiation emitted by the body at the two temperatures will be (a) 1: 4 (b) 1: 16 (c) 1: 256 (d) 1: 64

To answer your question about the radiation emitted by a black body as it heats from 27°C to 927°C, we can utilize the Stefan-Boltzmann Law. This law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. This means that if we know the temperatures, we can calculate the ratio of radiation emitted at those two temperatures.

Understanding the Temperatures

First, let's convert the temperatures from Celsius to Kelvin, since the Stefan-Boltzmann Law requires absolute temperatures. The conversion formula is:

• Kelvin = Celsius + 273.15

Applying this to our temperatures:

• Initial temperature: 27°C = 27 + 273.15 = 300.15 K
• Final temperature: 927°C = 927 + 273.15 = 1200.15 K

Applying the Stefan-Boltzmann Law

The Stefan-Boltzmann Law can be expressed mathematically as:

P = σT⁴

Where P is the power radiated, σ is the Stefan-Boltzmann constant, and T is the absolute temperature in Kelvin.

To find the ratio of the radiation emitted at the two temperatures, we set up the ratio of the powers:

Ratio = P₂ / P₁ = (σT₂⁴) / (σT₁⁴)

Since σ is constant, it cancels out:

Ratio = T₂⁴ / T₁⁴

Calculating the Ratio

Now substituting in our values for T₂ and T₁:

• T₂ = 1200.15 K
• T₁ = 300.15 K

Thus, the ratio becomes:

Ratio = (1200.15)⁴ / (300.15)⁴ = (1200.15 / 300.15)⁴

Calculating the ratio of the temperatures:

1200.15 / 300.15 ≈ 4

Now raising that to the fourth power:

Ratio = 4⁴ = 256

Finalizing the Result

This indicates that the radiation emitted at 927°C is 256 times greater than that emitted at 27°C. Therefore, when we express this ratio in the format of the options provided, we find:

Radiation ratio = 1:256

Hence, the correct answer is (c) 1: 256.

Understanding this relationship not only helps in solving this specific problem but also provides insights into the fundamental principles of thermal radiation and energy transfer, which are crucial in many fields, from astrophysics to engineering.