Question

Rank the following according to the wavelengths of radiant energy each emits, from the shortest wavelengths to the longest: a. A light bulb with a filament glowing at \(4000^{\circ} \mathrm{C}\) b. A rock at room temperature c. A car engine at \(140^{\circ} \mathrm{C}\)

Short answer

1) Light bulb, 2) Car engine, 3) Rock.

Step-by-step solution

Understanding the Planck's Law

Realize that objects emit radiation across a spectrum of wavelengths depending on their temperatures, with most radiation concentrated around a specific wavelength determined by Planck's law. The hotter an object is, the shorter the peak wavelength of the radiation it emits.

Identify the Peak Wavelength Formula

Use Wien's Displacement Law to find the peak wavelength, which states: \( \lambda_{max} = \frac{b}{T} \), where \( \lambda_{max} \) is the peak wavelength, \( b \) is Wien's displacement constant (approximately 2.898 mm·K), and \( T \) is the temperature in Kelvin.

Convert Temperatures to Kelvin

Convert the given Celsius temperatures to Kelvin by using the conversion: \( T_K = T_C + 273.15 \).- Light bulb: \( 4000 + 273.15 = 4273.15 \) K- Car engine: \( 140 + 273.15 = 413.15 \) K- Rock (room temperature): Assuming \( 25^{\circ} \mathrm{C} \), the temperature is \( 25 + 273.15 = 298.15 \) K.

Calculate Peak Wavelengths

Apply Wien's law to compute the peak wavelengths.- Light bulb: \( \lambda_{max} = \frac{2.898}{4273.15} \approx 0.678 \mu m \)- Car engine: \( \lambda_{max} = \frac{2.898}{413.15} \approx 7.01 \mu m \)- Rock: \( \lambda_{max} = \frac{2.898}{298.15} \approx 9.72 \mu m \)

Rank the Objects by Wavelength

Compare the computed peak wavelengths:- Light bulb: \( 0.678 \mu m \)- Car engine: \( 7.01 \mu m \)- Rock: \( 9.72 \mu m \)Therefore, from shortest to longest wavelength: light bulb, car engine, rock.

Key concepts

Radiant Energy

Radiant energy is the energy emitted by objects in the form of electromagnetic radiation. This includes visible light, infrared, ultraviolet, and other types of electromagnetic waves. Every object with a temperature above absolute zero emits some level of radiant energy. - **How it works:** As an object heats up, its atoms and molecules move and vibrate. This motion creates electromagnetic radiation. A warm object, like a heating lamp, emits energy in the form of light and heat that can be felt and seen. - **Key understanding:** The hotter the object, the more radiant energy it emits. The energy is distributed across various wavelengths, with a peak at a specific wavelength where it emits the most energy.

Temperature Conversion

Temperature conversion is a crucial step when dealing with physical laws that require temperature in Kelvin, such as Wien's Displacement Law. The Kelvin scale is the international standard for scientific work because it starts at absolute zero, where all molecular motion stops.- **Conversion formula:** To convert Celsius to Kelvin, use the formula: \( T_K = T_C + 273.15 \). - **Example:** For a light bulb at 4000°C, the temperature in Kelvin is \( 4000 + 273.15 = 4273.15 \) K.- **Relevance:** Converting temperatures ensures that all measurements are scientific and standardized, avoiding inconsistencies when applying formulas that are dependent on temperature in Kelvin.

Planck's Law

Planck's Law describes how objects emit electromagnetic radiation. It states that every object emits radiant energy at a range of wavelengths, but the energy intensity varies with temperature. - **Peak Emission:** According to Planck's Law, at any given temperature, there is a specific wavelength at which the emission of radiant energy is at its highest. This is known as the peak wavelength. - **Application to Everyday Phenomena:** This law explains why hot objects, like stars and light bulbs, emit shorter wavelengths and appear blue, while cooler objects emit longer wavelengths and appear red or infrared. Understanding Planck's Law provides insight into why different objects appear different colors based on their temperatures and helps in calculating the precise wavelengths using Wien’s Law.

Peak Wavelength Computation

The calculation of peak wavelength is an application of Wien's Displacement Law, which allows us to find where an object emits most of its radiant energy.- **Formula:** The formula for this computation is \( \lambda_{max} = \frac{b}{T} \), where \( \lambda_{max} \) is the peak wavelength, \( b \) is the Wien's displacement constant (approximately 2.898 mm·K), and \( T \) is the temperature in Kelvin.- **Step-by-Step Calculation:** - **Example 1:** For a light bulb at 4273.15 K, \( \lambda_{max} = \frac{2.898}{4273.15} \approx 0.678 \text{ micrometers} \). - **Example 2:** For a car engine at 413.15 K, \( \lambda_{max} = \frac{2.898}{413.15} \approx 7.01 \text{ micrometers} \). - **Example 3:** For a rock at 298.15 K, \( \lambda_{max} = \frac{2.898}{298.15} \approx 9.72 \text{ micrometers} \).- **Conclusion on Wavelength Ranking:** When ranking objects, shorter peak wavelengths indicate hotter objects. For example, the light bulb has the shortest wavelength, followed by the car engine, and then the rock. This arrangement correlates directly with their increasing temperatures.