•• At high temperatures (T ≳ 500 K) the resistivity of tungsten is approximately proportional to the absolute temperature (ρ ∞ T). A hot tungsten light-bulb filament maintains a stable temperature because the electrical power into the bulb (P = I2R = V2/R)is exactly balanced by the power out due to cooling by radiation and conduction. At high temperatures the cooling is dominated by radiation, and the power radiated is given by the formula, P = A ε σ T4, where A is the surface area of the body and both ε and σ are constants, called the emissivity and the Stephan–Boltzmann constant,* respectively. (a) Show that the steady-state temperature of a hot lightbulb filament with voltage V is proportional to V2/5. (b) Show that the temperature of a hot filament carrying a current I is proportional to I2/3.
*Beware: The Greek letter σ (sigma) is used to denote both the conductivity and the Stephan–Boltzmann constant. Don’t confuse them.
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