Question

A $100\, W$ 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 $ON$.
$\left(\right.$ Let $\left.\alpha_{\text {tungsten }}=4.5 \times 10^{-3} /{ }^{\circ} \, C \right)$

• 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 $
$4.5 \,t_{2}-112.5 =9 \times 10^{3} $
$4.5\, t_{2} =9000+112.5 $
$4.5 \, t_{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 $2025^{\circ} C$.