Chapter 8: Problem 31

A light bulb manufacturer makes bulbs with different "color temperatures" meaning that the spectrum of light they emit is similar to a blackbody with that temperature. What would be the difference in appearance of the light from bulbs with color temperatures of 2,700 $\mathrm{K}$ and $4,000 \mathrm{~K} ?$

Short Answer

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Answer: The light from the 2,700 K bulb will have a warmer, reddish-orange hue, while the light from the 4,000 K bulb will have a cooler, yellowish-red hue, due to the differences in their peak wavelengths.

Step by step solution

Understanding Wien's Displacement Law

Wien's Displacement Law states that the peak wavelength of the light emitted by a black body is inversely proportional to its temperature. Mathematically, the law is given by: $$\lambda_{\text{max}} = \frac{b}{T},$$ where $\lambda_{\text{max}}$ is the peak wavelength of the emitted light, $T$ is the temperature of the black body, and $b$ is Wien's displacement constant, which has a value of $2.898 \times 10^{-3}\mathrm{m\cdot K}$.

Calculate the peak wavelength for each color temperature

Using Wien's Displacement Law, we can calculate the peak wavelength for the two color temperatures: For $T_1 = 2700\mathrm{~K}$: $$\lambda_{\text{max1}} = \frac{2.898 \times 10^{-3}\mathrm{m\cdot K}}{2700\mathrm{~K}} = 1.07 \times 10^{-6}\mathrm{~m} = 1070\mathrm{~nm}$$ For $T_2 = 4000\mathrm{~K}$: $$\lambda_{\text{max2}} = \frac{2.898 \times 10^{-3}\mathrm{m\cdot K}}{4000\mathrm{~K}} = 7.25 \times 10^{-7}\mathrm{~m} = 725\mathrm{~nm}$$

Determine the color of the light based on the peak wavelength

The color of the light can be determined based on the peak wavelength $\lambda_{\text{max}}$. Here are some approximate wavelength ranges for different colors within the visible light spectrum: - Red: 620-750 nm - Orange: 590-620 nm - Yellow: 570-590 nm - Green: 495-570 nm - Blue: 450-495 nm - Violet: 380-450 nm For $T_1 = 2700\mathrm{~K}$, $\lambda_{\text{max1}} = 1070\mathrm{~nm}$ falls within the red light range. For $T_2 = 4000\mathrm{~K}$, $\lambda_{\text{max2}} = 725\mathrm{~nm}$ also falls within the red light range.

Compare the appearances of the light from the bulbs

Although both bulbs emit light primarily within the red range of the visible light spectrum, the bulb with a color temperature of $2700\mathrm{~K}$ will have a more reddish-orange hue due to its longer peak wavelength, while the bulb with a color temperature of $4000\mathrm{~K}$ will have a more yellowish-red hue due to its shorter peak wavelength. Overall, the light from the $2700\mathrm{~K}$ bulb will appear warmer, and the light from the $4000\mathrm{~K}$ bulb will appear cooler in comparison.

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Short Answer

Step by step solution

Understanding Wien's Displacement Law

Calculate the peak wavelength for each color temperature

Determine the color of the light based on the peak wavelength

Compare the appearances of the light from the bulbs

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