A 100-W incandescent light bulb has a cylindrical tungsten...

$\mathrm{A} 100-\mathrm{W}$ incandescent light bulb has a cylindrical tungsten filament 30.0 $\mathrm{cm}$ long, 0.40 $\mathrm{mm}$ in diameter, and with an emissivity of $0.26 .$ (a) What is the temperature of the filament? (b) For what wavelength does the spectral emittance of the bulb peak? (c) Incandescent light bulbs are not very efficient sources of visible light. Explain why this is so.

$\mathrm{A} 100-\mathrm{W}$ incandescent light bulb has a cylindrical tungsten filament 30.0 $\mathrm{cm}$ long, 0.40 $\mathrm{mm}$ in diameter, and with an emissivity of $0.26 .$ (a) What is the temperature of the filament? (b) For what wavelength does the spectral emittance of the bulb
peak? (c) Incandescent light bulbs are not very efficient sources of visible light. Explain why this is so.

Key Concepts

Stefan-Boltzmann Law

This fundamental law of thermal radiation states that the power radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature. In practical applications, it is modified by the emissivity of the material, which accounts for deviations from ideal blackbody behavior. This law is used to relate the filament’s temperature to the power it radiates.

Emissivity

Emissivity is a measure of a material's ability to emit energy as thermal radiation compared to an ideal blackbody. It ranges from 0 to 1, where 1 represents a perfect emitter. In practical problems, such as the tungsten filament, the given emissivity factor is used to adjust the ideal blackbody formulas to more accurately predict the filament's radiative properties.

Wien’s Displacement Law

Wien’s displacement law establishes the relationship between the temperature of a blackbody and the wavelength at which it emits radiation most intensely. As the temperature increases, the peak wavelength shifts to shorter wavelengths. This law is essential for determining the spectral peak of the filament's emission.

Blackbody Radiation

Blackbody radiation refers to the theoretical spectrum of light emitted by an object that absorbs all incoming radiation, regardless of wavelength. Although real objects, like tungsten filaments, are not perfect blackbodies, they are often approximated as such to simplify the analysis of their thermal radiation and spectral properties.

Efficiency of Incandescent Light Bulbs

The efficiency of incandescent bulbs is related to the fact that a significant portion of the energy they radiate is in the infrared region rather than the visible spectrum. This inefficiency arises because, at the temperatures at which filaments operate, most of the thermal radiation falls outside the range of wavelengths detectable by the human eye, leading to wasted energy in the form of heat.

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