If temperature if ideal black body is increased by 10% then percentage increase in quantity of radiation emitted from it`s surface will be ?

When considering how an ideal black body emits radiation, we can turn to the Stefan-Boltzmann Law, which tells us that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its absolute temperature. This relationship is crucial for understanding how changes in temperature affect radiation output.

The Stefan-Boltzmann Law Explained

The Stefan-Boltzmann Law can be expressed mathematically as:

J = σT⁴

Where:

• J is the total power radiated per unit area.
• σ is the Stefan-Boltzmann constant (approximately 5.67 × 10⁻⁸ W/m²K⁴).
• T is the absolute temperature in Kelvin.

Calculating the Effect of a 10% Increase in Temperature

If we increase the temperature of the black body by 10%, we can express this mathematically. If the original temperature is T, the new temperature will be:

T' = T + 0.1T = 1.1T

Now, we need to determine how the radiation output changes. According to the Stefan-Boltzmann Law:

J' = σ(T')⁴ = σ(1.1T)⁴

Expanding this gives:

J' = σ(1.1⁴T⁴)

Now, let's calculate 1.1⁴:

1.1⁴ ≈ 1.4641

Finding the Percentage Increase in Radiation

The percentage increase in radiation emitted can be calculated using the formula:

Percentage Increase = [(J' - J) / J] × 100%

Substituting our values into the formula gives:

Percentage Increase = [(σ(1.1⁴T⁴) - σ(T⁴)) / σ(T⁴)] × 100%

Since σ(T⁴) cancels out, we simplify to:

Percentage Increase = [(1.4641 - 1) / 1] × 100%

This results in:

Percentage Increase ≈ 46.41%

Summary of Findings

Thus, when the temperature of an ideal black body is increased by 10%, the quantity of radiation emitted from its surface will increase by approximately 46.41%. This significant increase emphasizes the sensitivity of black body radiation to temperature changes, illustrating the powerful impact of the Stefan-Boltzmann Law in thermodynamics.