Question

A Tungsten filament for a light bulb is found to have a resistance of 0.25 Ω when
measured at a temperature of 20◦ C. When the light bulb has been on for a substantial
period of time, the filament has heated up and the resistance is found to be 3.50 Ω.
What is the temperature of the heated filament? Tungsten’s temperature coefficient
of resistivity is 4.5 × 10−3 per degree C, and do not consider thermal expansion of the
filament.

Answer 1

Answer Question #65353 – Physics – Electric Circuit

A Tungsten filament for a light bulb is found to have a resistance of $0.25\ \Omega$ when measured at a temperature of $20{}^{\circ}\mathrm{C}$. When the light bulb has been on for a substantial period of time, the filament has heated up and the resistance is found to be $3.50\ \Omega$. What is the temperature of the heated filament? Tungsten's temperature coefficient of resistivity is $4.5\cdot 10^{-3}$ per degree C, and do not consider thermal expansion of the filament.

Solution. Temperature dependence of electrical resistance of tungsten filament bulbs excluding thermal expansion, it is possible to find a formula

$$R = R_{0} \left(1 + \alpha (T - T_{0})\right)$$

where $R = 3.5\ \Omega$ – resistance at temperature $T$, $R_{0} = 0.25\ \Omega$ – resistance at temperature $T_{0} = 20^{0}\ \mathrm{C}$, $\alpha = 4.5 \cdot 10^{-3}$ per degree C – temperature coefficient of resistance.

Therefore

$$\frac{R}{R_{0}} = 1 + \alpha (T - T_{0}) \rightarrow T = T_{0} + \frac{1}{\alpha} \left(\frac{R}{R_{0}} - 1\right) \approx 2909^{0}\ \mathrm{C}.$$

Answer. $2909^{0}\ \mathrm{C}$.