The tungsten filament of a certain 100- W lightbulb radiates...

The tungsten filament of a certain $100-$ W lightbulb radiates $2.00 \mathrm{W}$ of light. (The other $98 \mathrm{W}$ is carried away by convection and conduction.) The filament has a surface area of $0.250 \mathrm{mm}^{2}$ and an emissivity of $0.950 .$ Find the filament's temperature. (The melting point of tungsten is $3683 \mathrm{K} .$ )

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Key Concepts

Stefan-Boltzmann Law

This law defines the power radiated per unit area of a body as proportional to the fourth power of its absolute temperature. It is central to calculating temperatures of radiating bodies by relating the emitted power to their surface area, emissivity, and temperature.

Emissivity

Emissivity is a factor that quantifies how efficiently a real object radiates energy compared with an ideal blackbody, which has an emissivity of 1. It adjusts the Stefan-Boltzmann law to account for materials that do not emit radiation perfectly.

Radiative Heat Transfer

This concept involves the energy emitted from a body as electromagnetic radiation. In contexts like heated filaments, radiative heat transfer is a key mechanism of energy loss, described mathematically by the Stefan-Boltzmann law and modified by the emissivity of the material.

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