Question

A small circular surface of area \(A_{1}=2 \mathrm{~cm}^{2}\) located at the center of a 2-m-diameter sphere emits radiation as a blackbody at \(T_{1}=1000 \mathrm{~K}\). Determine the rate at which radiation energy is streaming through a \(D_{2}=1\)-cm-diameter hole located \((a)\) on top of the sphere directly above \(A_{1}\) and \((b)\) on the side of sphere such that the line that connects the centers of \(A_{1}\) and \(A_{2}\) makes \(45^{\circ}\) with surface \(A_{1}\).

Short answer

Answer: The rates of radiation energy streaming through the hole are approximately (a) 0.2835 W when it is directly above surface A₁, and (b) 0.2005 W when it is on the side making a 45º angle with surface A₁.

Step-by-step solution

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the total power radiated by a blackbody is given by: \(P = \sigma A T^4\) where \(\sigma\) is the Stefan-Boltzmann constant (\(5.67\times10^{-8} \mathrm{W/(m^2.K^4)}\) ), A is the area, and T is the temperature.

Calculate \(P_{1}\), the total power emitted by surface \(A_{1}\)

Using the Stefan-Boltzmann Law, we can calculate the total power emitted by surface \(A_{1}\). The area of \(A_{1}\) is \(2\;cm^2 = 2\times10^{-4}\mathrm{m^2}\), and the temperature is \(1000\;\mathrm{K}\). Therefore, \(P_1 = \sigma \cdot A_1 \cdot T_1^4 = (5.67\times10^{-8}\mathrm{W/(m^2.K^4)})\times(2\times10^{-4}\mathrm{m^2})\times(1000\;\mathrm{K})^4 = 113.4\;\mathrm{W}\)

Calculate the solid angle subtended by the hole at the center of the sphere for both positions

We will find the solid angle \(\Omega_i\) subtended by the hole at the center of the sphere for each position using the formula: \( \Omega_i = \frac{A_i}{R^2} \) where \(A_i\) is the area of the hole and \(R\) is the radius of the sphere. (a) When the hole is directly above \(A_1\), the hole has an area \(A_2 = \frac{\pi D_2^2}{4} = \frac{\pi (1\;\mathrm{cm})^2}{4} = \frac{\pi}{4}\;\mathrm{cm^2} = \frac{\pi}{400}\;\mathrm{m^2}\). The radius of the sphere is \(R = 1\;\mathrm{m}\). Therefore, the solid angle is \( \Omega_1 = \frac{A_2}{R^2} = \frac{\pi}{400}\) (b) When the hole is on the side such that the line connecting the centers of \(A_1\) and \(A_2\) makes a \(45^\circ\) angle, the distance between the hole and the center of the sphere is also R due to symmetry. The projected area of the hole on the plane perpendicular to the line connecting the centers of \(A_1\) and \(A_2\) is \(A'_2 = A_2 \cos(45^\circ) = \frac{A_2}{\sqrt{2}} = \frac{\pi}{400\sqrt{2}}\;\mathrm{m^2}\). Therefore, the solid angle is \( \Omega_2 = \frac{A'_2}{R^2} = \frac{\pi}{400\sqrt{2}}\)

Calculate the rate of radiation energy streaming through the hole for both positions

Use the solid angle and the total power emitted by surface \(A_1\) to find the rate of radiation energy streaming through the hole for both positions. The rate of radiation energy leaving through the hole is given by the formula: \(P_i = \frac{\Omega_i}{4\pi} \cdot P_1\) (a) For the hole directly above \(A_1\): \(P_1 = \frac{\Omega_1}{4\pi} \cdot P_1 = \frac{\pi/400}{4\pi} \cdot 113.4\;\mathrm{W} = 0.2835\;\mathrm{W}\) (b) For the hole on the side making a \(45^\circ\) angle with surface \(A_1\): \(P_2 = \frac{\Omega_2}{4\pi} \cdot P_1 = \frac{\pi/(400\sqrt{2})}{4\pi} \cdot 113.4\;\mathrm{W} = 0.2005\;\mathrm{W}\) In conclusion, the rate at which radiation energy is streaming through the hole is approximately \((a)\) \(0.2835\;\mathrm{W}\) when the hole is directly above \(A_{1}\), and \((b)\) \(0.2005\;\mathrm{W}\) when the hole is on the side making a \(45^\circ\) angle with surface \(A_{1}\).

Key concepts

Understanding the Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in thermodynamics that provides insight into how objects at a certain temperature emit radiation. In its essence, it describes the power radiated from a blackbody in terms of its temperature. A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence, and emits radiation at a given temperature with a characteristic spectrum. The law can be mathematically expressed as:
\[ P = \sigma A T^4 \] where \(P\) represents the total power radiated per unit time, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67\times10^{-8} \mathrm{W/m^2.K^4}\)), \(A\) is the radiating surface area, and \(T\) is the temperature of the blackbody in Kelvin.
To provide a comprehensive understanding, imagine holding your hands near a fire. The heat you feel is radiation, which increases significantly with the temperature of the fire. The Stefan-Boltzmann Law quantifies this effect, explaining why hotter objects (like the sun) emit exponentially more radiant energy compared to cooler ones (like a candle flame).

Solid Angle: A Geometrical Perspective in Three Dimensions

The concept of solid angle is equivalent to the two-dimensional angle we are familiar with, but it extends into three dimensions. It measures the size of the 'cone' of space that an object covers as seen from a certain point. In technical terms, the solid angle \(\Omega\) subtended by an object as viewed from a given point is the area of the projection of that object on a sphere centered around the point, divided by the square of the sphere's radius.
Mathematically, the solid angle is given by: \[ \Omega = \frac{A}{R^2} \] where \(A\) is the area of the projected object and \(R\) is the radius of the sphere.
For instance, consider the hole at the top of the sphere in the given problem. It covers a specific part of our field of vision from the center of the sphere, which is quantified by the solid angle. Practically, photographers use similar calculations for the coverage of a camera lens, astronomers for the apparent size of celestial bodies, and engineers for coverage areas of antennas.

Radiation Energy Rate: Quantifying Energy Transmission

The radiation energy rate, also referred to as the power transfer by radiation, describes how much energy travels through a given area in a certain amount of time due to thermal radiation. This is central to understanding energy budgets in systems ranging from simple heaters to complex stars. The relationship between the radiation energy rate through an opening and the solid angle can be described as:
\[ P_i = \frac{\Omega_i}{4\pi} \cdot P \] where \(P_i\) is the power or radiation energy rate through the opening, \(\Omega_i\) is the solid angle subtended by the opening, and \(P\) is the total power emitted by the surface.
Using the case from the problem, when we have a hole on a sphere, not all of the energy emitted by a surface inside the sphere will exit through the hole. The fraction that does is proportional to the solid angle the hole subtends as seen from the radiating surface. This concept explains why the radiation energy rate through the hole is less than the total power emitted by the surface—we are capturing just a 'slice' of the radiated energy.