Problem 44 - Physics

Question

Concept Questions Part (a) of the drawing shows a rectangular bar whose dimensions are \(L_{0} \times 2 L_{0} \times 3 L_{0}\). The bar is at the same temperature as the room (not shown) in which it is located. (a) Is the net radiant power emitted by the bar greater than zero, equal to zero, or less than zero? Provide a reason for your answer. (b) The bar is then cut, lengthwise, into two equal pieces, as shown in part \((b)\) of the drawing. The temperature of the bars does not change. Which situation, if either, emits more power into the room, the single bar in part \((a)\) or the two bars in part \((b)\) of the drawing? Why? (c) Which situation, if either, absorbs more power from the room, the single bar in part (a) or the two bars in part ( \(b\) ) of the drawing? Justify your reasoning.

Step-by-Step Solution

Answer

In both cases (single or cut bar), the net radiant power is zero as they are in thermal equilibrium with the room.

Step 1: Understanding Radiant Power Emission

Radiant power emitted by an object is given by \[ P = \sigma A e(T^4) \] where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, \( e \) is the emissivity, and \( T \) is the temperature of the object in Kelvin.If the object and room are at the same temperature, the net power emitted is zero, since emitted power equals absorbed power, maintaining thermal equilibrium.

Step 2: Net Radiant Power in Original State

Part (a) of the exercise asks whether the net power emitted is positive, negative, or zero for the original bar -- Since the temperature difference between the bar and the room is zero (same temperature), the net radiant power emitted is zero, maintaining thermal equilibrium.

Step 3: Modification by Cutting the Bar

When the bar is cut into two equal parts, as in part (b), the surface area exposed to the room increases, while the temperature remains the same. Increased surface area affects both emission and absorption rate proportionally, but the net power exchange due to having no temperature difference still remains zero.

Step 4: Comparing Radiant Power Emission

Part (b) asks about emission differences. By cutting the bar, the total surface area exposed increases, suggesting that each bar individually might have a slightly altered radiative exchange. However, as temperature is constant and they emit and absorb at the same rate, net power emission remains zero in both scenarios.

Step 5: Comparing Power Absorption from the Room

The same principle applies for absorption. Each shape in a shared thermal environment absorbs radiant energy equally. Because the bars are at the same temperature as the surrounding room, absorption equals emission, maintaining equilibrium in both scenarios, so net absorption remains zero in both situations.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle that describes the power radiated from a black body in terms of its temperature. This law states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of the black body's absolute temperature. The equation is given by:\[ P = \sigma A eT^4 \]Where:- \( P \) is the radiant power emitted- \( \sigma \) is the Stefan-Boltzmann constant - \( A \) is the surface area- \( e \) is the emissivity of the material- \( T \) is the absolute temperature in KelvinUnderstandably, this means that even a small increase in temperature results in a significantly larger increase in the radiated energy, given the fourth power relationship. The law is crucial in studying stars, planets, and any objects exchanging thermal radiation.

Radiant Power Emission

Radiant power emission refers to the energy emitted by a body in the form of thermal radiation. Every object emits some level of radiation depending on its temperature, surface area, and emissivity. In the context of the Stefan-Boltzmann Law, radiant power emission is a core aspect because it quantifies how much energy an object loses as radiation. For systems in thermal equilibrium, such as the objects in this exercise scenario, the radiant power emitted equals the power absorbed from the surroundings. This means the net power emission is zero when temperature differences are absent. This principle is why the bars, regardless of how they are divided, do not emit a net radiant power when they are at the same temperature as their environment.

Surface Area and Emissivity

The surface area and emissivity of an object are critical in determining its radiant power emission. Surface area defines the extent of the object's exposure to its surroundings. Greater surface area implies a greater ability to absorb and emit thermal radiation. Emissivity is a measure of how effectively a surface emits thermal radiation relative to a perfect black body. It ranges from 0 to 1, with 1 indicating a perfect emitter. Surfaces with high emissivity are more efficient at radiating heat. Although the bars in the exercise are cut, which increases the surface area, the basic principle here is that increased surface area does not change the net emission unless there is a temperature gradient. This is because both emission and absorption scale with surface area but are matched at thermal equilibrium.

Thermal Radiation

Thermal radiation is a form of energy emitted by all objects based on their temperature. It is an electromagnetic phenomenon that spans a range of wavelengths, primarily in the infrared for common temperatures on Earth. Thermal radiation is omnidirectional and does not require a medium, distinguishing it from conduction and convection. All objects continually exchange thermal radiation, and the balance of this exchange determines if an object gains or loses heat. In thermal equilibrium, an object absorbs and emits radiation at equal rates. This is critical in the exercise context where the bar and the room are at the same temperature. Hence, no net power is lost or gained through thermal radiation in either scenario — whether the bar remains whole or is split, illustrating the self-regulating nature of thermal systems in equilibrium.