## Related questions with answers

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 $2900 \mathrm{~K}$, whereas the filament in a halogen bulb may operate at $3400 \mathrm{~K}$. The human eye is most sensitive to a frequency around $5.5 \times 10^{14} \mathrm{~Hz}$. Which bulb produces a peak frequency closer to this value?

Solution

VerifiedGiven that the temperature of filament of halogen bulb, $T_h= 3400\ \text{K}$ and the temperature of filament of standard bulb, $T_{s} = 2900\ \text{K}$.

We know from the Wein's Displacement law

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

First, peak frequency $f_{std}$ for the standard bulb

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

Similarly, peak frequency $f_{hal}$ for the halogen bulb

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

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

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

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