Question

Daylight and incandescent light may be approximated as a blackbody at the effective surface temperatures of \(5800 \mathrm{~K}\) and \(2800 \mathrm{~K}\), respectively. Determine the wavelength at maximum emission of radiation for each of the lighting sources.

Short answer

Answer: The wavelength at maximum emission for daylight is 500 nm, and for incandescent light, it is 1040 nm.

Step-by-step solution

Understanding Wien's Displacement Law

Wien's Displacement Law relates the wavelength at maximum emission to the temperature of the blackbody, which is given by: \(\lambda_{max} = \dfrac{b}{T}\) Where \(\lambda_{max}\) is the wavelength at maximum emission, \(b\) is Wien's displacement constant (\(b \approx 2.9 \times 10^{-3} \mathrm{~m K}\)), and \(T\) is the temperature of the blackbody in kelvins. We will apply this formula to both lighting sources to find the wavelengths at maximum emission.

Calculate wavelength for daylight source

Given the temperature of the daylight source as \(T = 5800 \mathrm{~K}\). Then using Wien's displacement law: \(\lambda_{max_{daylight}} = \dfrac{b}{T_{daylight}}\) \(\lambda_{max_{daylight}} = \dfrac{2.9 \times 10^{-3} \mathrm{~m K}}{5800 \mathrm{~K}}\) \(\lambda_{max_{daylight}} = 5\times10^{-7} \mathrm{~m} = 500 \mathrm{~nm}\) The wavelength at maximum emission for daylight source is \(500 \mathrm{~nm}\).

Calculate wavelength for incandescent light source

Given the temperature of the incandescent light source as \(T = 2800 \mathrm{~K}\). Then using Wien's displacement law: \(\lambda_{max_{incandescent}} = \dfrac{b}{T_{incandescent}}\) \(\lambda_{max_{incandescent}} = \dfrac{2.9 \times 10^{-3} \mathrm{~m K}}{2800 \mathrm{~K}}\) \(\lambda_{max_{incandescent}} = 1.04 \times 10^{-6} \mathrm{~m} = 1040 \mathrm{~nm}\) The wavelength at maximum emission for the incandescent light source is \(1040 \mathrm{~nm}\).

Conclusion

In conclusion, the wavelength at maximum emission for daylight is \(500 \mathrm{~nm}\), and for incandescent light, it is \(1040 \mathrm{~nm}\).

Key concepts

Blackbody Radiation

Blackbody radiation is a concept in physics that describes how objects emit radiation or light based on their temperature. Imagine a perfect object that absorbs all incoming light and reflects none. This theoretical object is a 'blackbody'. In reality, no perfect blackbody exists, but many real-world objects approximate this behavior.

When a blackbody is heated, it emits light, visible or invisible, depending on its temperature. This emission covers a range of wavelengths. The intensity of the emitted radiation changes with wavelength, having a specific distribution. This distribution is called the blackbody radiation spectrum. The spectrum peaks at particular wavelengths, which shift as temperature changes.

Key characteristics of blackbody radiation include:

• The radiation emitted is continuous and covers a broad range of wavelengths.
• The peak wavelength is inversely proportional to the temperature of the blackbody. This means as temperature increases, the peak wavelength decreases.
• Blackbody radiation plays a vital role in many fields, from understanding stellar objects to designing energy-efficient lighting.

Understanding blackbody radiation is crucial for developing an intuition about how temperature affects light emission.

Wavelength at Maximum Emission

The wavelength at maximum emission is a critical concept in understanding blackbody radiation. It defines the wavelength at which the intensity of light is highest for a given temperature. Wien's Displacement Law gives us a clear mathematical framework to determine this wavelength based on temperature.

Wien's Displacement Law: This law states that the product of the absolute temperature of a blackbody and the wavelength at maximum emission is constant. The formula is given by:
\[\lambda_{max} = \dfrac{b}{T}\]
Here, \(\lambda_{max}\) denotes the wavelength at which the emission is maximum, \(b\) is Wien's displacement constant \( \approx 2.9 \times 10^{-3} \mathrm{~m \cdot K} \), and \(T\) is the temperature in Kelvin.

This relationship tells us:

• As the temperature of the blackbody increases, the peak wavelength shifts toward shorter wavelengths (e.g., from infrared to visible range).
• For cooler objects, the peak is in the infrared range, explaining why some objects glow less visibly as they cool down.

Applying this understanding allows us to calculate light characteristics for various temperature-based sources, just like we see in the sunlight and incandescent lighting examples.

Daylight and Incandescent Light

Daylight and incandescent light are two common sources of artificial and natural lighting in our everyday environment. Both can be understood as emitters of blackbody radiation, albeit at different temperatures.

Daylight: Daylight can be considered equivalent to a blackbody emitting at a temperature of about \(5800 \mathrm{~K}\). The wavelength at maximum emission for sunlight is calculated using Wien's law and found to be approximately \(500 \mathrm{~nm}\), which falls in the visible spectrum, contributing to the bright and vibrant light we perceive.

Incandescent Light: Traditional incandescent bulbs operate at a lower temperature, around \(2800 \mathrm{~K}\). When applying Wien’s displacement law here, the maximum emission wavelength is about \(1040 \mathrm{~nm}\), which lies in the infrared region. This calculation explains why incandescent lights are warm and not as energy-efficient as other light types, as much of the energy is lost as heat.

These differences highlight why daylight appears more balanced in color versus the warm tint of incandescent lighting.

• Daylight covers a broad spectrum, shining most intensely at visible wavelengths.
• The warmth of incandescent light is due to its peak emission in the infrared.

Understanding these principles aids not only in designing various lighting but also in appreciating the natural and artificial light interactions.