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Modern halogen lightbulbs allow their filaments to operate at a higher temperature than the filaments in standard incandescent bulbs. For comparison, the filament in a standard lightbulb operates at about 2900 K2900 \mathrm{~K}, whereas the filament in a halogen bulb may operate at 3400 K3400 \mathrm{~K}. The human eye is most sensitive to a frequency around 5.5×1014 Hz5.5 \times 10^{14} \mathrm{~Hz}. Which bulb produces a peak frequency closer to this value?


Answered 2 years ago
Answered 2 years ago
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Given that the temperature of filament of halogen bulb, Th=3400 KT_h= 3400\ \text{K} and the temperature of filament of standard bulb, Ts=2900 KT_{s} = 2900\ \text{K}.

We know from the Wein's Displacement law

fpeak=(5.88×1010 s1K1)T\begin{equation} f_{peak} = (5.88\times 10^{10}\ \text{s}^{-1}\cdot \text{K}^{-1})T \end{equation}

First, peak frequency fstdf_{std} for the standard bulb

fstd=(5.88×1010 s1K1)Ts=(5.88×1010 s1K1)(2900 K)=1.7×1014 Hz\begin{align*} f_{std} & = (5.88\times 10^{10}\ \text{s}^{-1}\cdot \text{K}^{-1})T_{s}\\ & = (5.88\times 10^{10}\ \text{s}^{-1}\cdot \text{K}^{-1})(2900\ \text{K})\\ & = 1.7\times 10^{14}\ \text{Hz} \end{align*}

Similarly, peak frequency fhalf_{hal} for the halogen bulb

fhal=(5.88×1010 s1K1)Th=(5.88×1010 s1K1)(3400 K)=1.99×1014 Hz\begin{align*} f_{hal} &= (5.88\times 10^{10}\ \text{s}^{-1}\cdot \text{K}^{-1})T_{h}\\ & = (5.88\times 10^{10}\ \text{s}^{-1}\cdot \text{K}^{-1})(3400\ \text{K})\\ & = 1.99\times10^{14}\ \text{Hz} \end{align*}

Given that human eye is sensitive to a frequency around 5.5×1014 Hz5.5\times10^{14}\ \text{Hz}.

Therefore, Halogen Bulb\boxed{\textbf{Halogen Bulb}} produces peak frequency closer to 5.5×1014 Hz5.5\times 10^{14}\ \text{Hz} than the standard bulb.

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