Question

A small circular plate with a surface \(A_{1}\) can be approximated as a blackbody. A radiometer with a surface \(A_{2}\) is placed normal to the direction of viewing from the plate at a distance \(L\). Would the irradiation on the radiometer from the plate decrease if the distance \(L\) is doubled, and if so, by how much?

Short answer

Answer: Yes, the irradiation on the radiometer decreases when the distance L is doubled. It decreases by a factor of 1/4.

Step-by-step solution

Determine the irradiance at the original distance L

Find the irradiance E₁ received by the radiometer from the plate at the original distance L using the equation E₁ = σ * T^4 * (A₁ * A₂) / (4 * π * L^2).

Determine the irradiance at the doubled distance 2L

Find the irradiance E₂ received by the radiometer from the plate at the doubled distance 2L using the equation, E₂ = σ * T^4 * (A₁ * A₂) / (4 * π * (2L)^2).

Compare the irradiance at the original and doubled distances

Determine if the irradiance decreases by finding the ratio of E₂ to E₁. If the ratio is less than 1, then the irradiance has decreased. Calculate the ratio as follows: E₂ / E₁ = {σ * T^4 * (A₁ * A₂) / (4 * π * (2L)^2)} / {σ * T^4 * (A₁ * A₂) / (4 * π * L^2)}.

Simplify the ratio

Simplify the ratio E₂ / E₁ by canceling out the common terms: E₂ / E₁ = (L^2) / (4L^2).

Calculate the factor by which the irradiance decreases

Calculate the factor by which the irradiance decreases when the distance L is doubled: Factor = (L^2) / (4L^2) = 1/4. So, the irradiation on the radiometer from the plate decreases by a factor of 1/4 when the distance L is doubled.

Key concepts

Blackbody Radiation

Blackbody radiation refers to the theoretical concept where a body or surface emits thermal radiation at maximum intensity across all wavelengths. This idealized body perfectly absorbs all incident radiation and re-emits the energy according to its temperature. The amount of energy emitted is described by the Stefan-Boltzmann law, which states that the energy radiated per unit area of the body is proportional to the fourth power of the absolute temperature. The formula can be expressed as:

• Energy emitted per unit area: \( E = \sigma T^4 \) where \( \sigma \) is the Stefan-Boltzmann constant.

In the context of the original exercise, the small circular plate acts as a blackbody. This means it emits radiation efficiently, and its heat transfer characteristics can be easily predicted using the established blackbody formula. The radiometer measures this radiation for various distances to understand how emitted radiation spreads out over space.

Inverse Square Law

The inverse square law is a physical principle that describes how a specified physical quantity or strength decreases with the distance from its source. Specifically, this law dictates that intensity is inversely proportional to the square of the distance from the source. Mathematically, if you double the distance from a source, its intensity becomes four times less.

• Expression: Intensity \( I \propto \frac{1}{d^2} \)

In the heart of our exercise, this law helps us understand why the irradiance on the radiometer decreases significantly when the distance \( L \) is increased. When the radiometer moves two times further, the intensity of radiation that it detects is reduced to a quarter of its original value (\( \frac{1}{4} \)). This is a typical scenario illustrating the inverse square law in action.

Irradiance Calculation

Irradiance is a measure of the power of electromagnetic radiation per unit area. In practical terms, it determines how much energy from radiation is received by a surface. We calculate irradiance to understand the levels of energy exchange involved in radiation heat transfer.

• For a radiometer initially at distance \( L \), the irradiance \( E_1 \) received from the blackbody plate can be expressed using: \(E_1 = \sigma T^4 \frac{A_1 A_2}{4 \pi L^2}\)
• When the distance is doubled (to \( 2L \)), the irradiance \( E_2 \) changes to: \( E_2 = \sigma T^4 \frac{A_1 A_2}{4 \pi (2L)^2} \)

By comparing these two expressions, we can determine the change in irradiance. Specifically, the ratio of \( E_2 \) to \( E_1 \) helps us understand the change, demonstrating that the irradiance decreases by a factor of 1/4, perfectly aligning with our understanding from the inverse square law. This calculation is crucial for applications involving thermal efficiency and energy distribution.