·Size of a lightbulb filament. The operating temperature of...

$\cdot$ Size of a lightbulb filament. The operating temperature of a tungsten filament in an incandescent lightbulb is $2450 \mathrm{K},$ and its emissivity is $0.35 .$ Find the surface area of the filament of a 150 $\mathrm{W}$ bulb if all the electrical energy consumed by the bulb is radiated by the filament as light. (In reality, only a small fraction of the radiation appears as visible light.)

Key Concepts

Emissivity

Emissivity is a measure of a material's ability to emit thermal radiation compared to that of an ideal blackbody. It is a dimensionless factor with values ranging from 0 to 1, where 1 corresponds to a perfect emitter. In practical applications, the actual radiated power is adjusted by multiplying the blackbody value by the emissivity, making it a crucial parameter in determining how efficiently a real object radiates heat.

Blackbody Radiation Concept

The concept of blackbody radiation provides the theoretical framework for understanding the emission of electromagnetic radiation by a hot object. While no real object perfectly behaves as a blackbody, many objects can be approximated as one. This approximation simplifies the analysis of thermal radiation by allowing the application of idealized formulas like the Stefan-Boltzmann Law.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in thermal radiation which states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature. This law, expressed as P = ?AT^4 (where ? is the Stefan-Boltzmann constant and T is temperature in Kelvin), is essential for relating temperature to the radiative power output of an object.

Energy Conversion and Radiative Surface Area

In systems like incandescent bulbs, electrical energy is converted into thermal energy, which is then radiated from the filament. By equating the electrical power input to the radiative power (considering the filament’s emissivity), one can solve for the required surface area. This approach links the physical dimensions of the radiating body directly to its thermal emission characteristics.

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