Question

A $100W$ tungsten light bulb has a resistance of $250\Omega$ when it as turned $ON$ and $25\Omega$ when lurned $OFF$. The ambient room temperature is $25^{\circ }C$. Find the temperature of the filament when the bulb is turned $ON$.
$($ Let ${\alpha }_{\text{tungsten }}=4.5\times 10^{-3}/{}^{\circ }C)$

• A. $2600^{\circ }C$
• B. $2025^{\circ }C$
• C. $2500^{\circ }C$
• D. $2625^{\circ }C$

Answer 1

Answer: (B)

If $t_2$ and $t_1$ be the temperature of filament, when light bulb is turned ON and turned OFF, respectively then,
$R_{t_2}=250\Omega$
$R_{t_1}=25\Omega$
${\alpha }_{\text{tungsten }}=4.5\times 10^{-3}/{}^{\circ }C$
We know that, $\alpha =\frac{R_{t_2}-R_{t_1}}{R_{t_1}\left(t_2-t_1\right)}$
Putting the given values, we get
$4.5\times 10^{-3}=\frac{250-25}{25\left(t_2-25\right)}$
$4.5\times 10^{-3}=\frac{9}{t_2-25}$
$4.5\times 10^{-3}{t}_2-112.5\times 10^{-3}=9$
$4.5t_2-112.5=9\times 10^3$
$4.5t_2=9000+112.5$
$4.5t_2=9112.5$
$t_2=\frac{9112.5}{4.5}=2025^{\circ }C$
Hence, the temperature of the filament when the bulb is turned on $2025^{\circ }C$.