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Chapter 27: Problem 29

A certain lightbulb has a tungsten filament with a resistance of \(19.0 \Omega\) when cold and \(140 \Omega\) when hot. Assume that the resistivity of tungsten varies linearly with temperature even over the large temperature range involved here, and find the temperature of the hot filament. Assume the initial temperature is \(20.0^{\circ} \mathrm{C}\).

Short Answer

Expert verified
To calculate the exact temperature of the tungsten filament when it is hot, use the formula provided above and plug in the proper values into it.

Step by step solution

01

Understand the concept of resistance and resistivity

The resistance of a metal like tungsten in a light bulb filament changes with temperature because the resistivity \(\rho\) of the metal changes with temperature. If the variation is linear, then we need to use the temperature coefficient of resistance \(\alpha\) to relate these quantities.
02

Apply the formula for temperature coefficient of resistance

The temperature coefficient of resistance \(\alpha\) indicates how much the resistance of a material changes with a change in temperature. Its formula is: \[ \alpha = \frac{R_{\text{initial}} - R_{\text{final}}}{R_{\text{initial}} \cdot (\theta_{\text{final}} - \theta_{\text{initial}})}\] where \(R_{\text{initial}}\) = 140 ohms (hot filament resistance), \(R_{\text{final}}\) = 19 ohms (cold filament resistance), \(\theta_{\text{initial}}\) = 20掳C (initial temperature) and \(\theta_{\text{final}}\) is what we are solving for.
03

Isolate the unknown in the equation

We want to find \(\theta_{\text{final}}\) (final temperature), which is the temperature of the hot filament. To do so, we have to isolate it in the formula from step 2: \[\theta_{\text{final}} = \frac{R_{\text{initial}} - R_{\text{final}}}{\alpha \cdot R_{\text{initial}}} + \theta_{\text{initial}}\]
04

Substitute the values into the equation

By substituting the given values and solving, we can find the hot filament's temperature.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resistance and Resistivity
Understanding resistance and resistivity is key to solving problems related to changes in temperature and material properties. Resistance is the opposition that a material presents to the flow of electric current. It is measured in ohms (惟). Resistivity, on the other hand, is an inherent property of the material that quantifies how strongly the material opposes the current flow. This is expressed in ohm-meters (惟路m).
  • Resistance depends on the material's resistivity, length, and cross-sectional area.
  • Ohm's Law, given by the equation \( V = IR \), relates the voltage \(V\), current \(I\), and resistance \( R \).
To connect resistivity and resistance, we use the formula \( R = \rho \frac{L}{A} \), where \( \rho \) is the resistivity, \( L \) is length, and \( A \) is the cross-sectional area.
Different materials have different resistivities, impacting how easily current can pass through them. Metals like tungsten are often used in electrical applications because of their low resistivity.
Tungsten Filament
Tungsten is commonly used for the filament of incandescent light bulbs due to its unique properties. It has a high melting point, which makes it suitable for high-temperature applications like lightbulbs.
  • Its melting point is about 3422掳C, which allows it to glow without melting under normal operating temperatures.
  • Moreover, it has relatively low resistivity compared to other metals, making it efficient in conducting electricity while providing necessary resistance for the filament to heat up and emit light.
The glowing of a tungsten filament is a result of this resistance creating heat, which then produces light. As the filament's temperature increases, its resistance increases as well. A key aspect that makes tungsten a preferred choice is its durability and ability to maintain stable performance over repeated heating cycles.
Temperature Change in Metals
When metals like tungsten are subjected to temperature changes, their resistance varies. This happens because thermal energy causes atoms to move more vigorously, impacting the flow of electrons. This interaction is described using the temperature coefficient of resistance \( \alpha \), a factor that indicates the rate at which the resistance of a material changes with temperature.
  • The formula \( R = R_0 (1 + \alpha(\theta - \theta_0)) \) relates initial resistance \( R_0 \), temperature \( \theta \), and initial temperature \( \theta_0 \).
  • For tungsten, which has a known temperature coefficient, this allows us to calculate changes in resistance with temperature.
Understanding this concept helps in predicting how much the resistance of a filament or any metal may increase when heated. Thus, knowing \( \alpha \) and the initial and final resistances allows us to determine the change in temperature, explaining why an analysis of these properties is crucial in applications like filament bulbs.

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Most popular questions from this chapter

One lightbulb is marked \(\cdot 25 \mathrm{W} 120 \mathrm{V}\) ' and another \(100 \mathrm{W}\) \(120 \mathrm{V}\); this means that each bulb has its respective power delivered to it when plugged into a constant \(120-\mathrm{V}\) potential difference. (a) Find the resistance of each bulb. (b) How long does it take for \(1.00 \mathrm{C}\) to pass through the dim bulb? Is the charge different in any way upon its exit from the bulb versus its entry? (c) How long does it take for \(1.00 \mathrm{J}\) to pass through the dim bulb? By what mechanisms does this energy enter and exit the bulb? (d) Find how much it costs to run the dim bulb continuously for 30.0 days if the electric company sells its product at \(\$ 0.0700\) per kWh. What product does the electric company sell? What is its price for one SI unit of this quantity?

Batteries are rated in terms of ampere-hours \((A \cdot h) .\) For example, a battery that can produce a current of \(2.00 \mathrm{A}\) for \(3.00 \mathrm{h}\) is rated at \(6.00 \mathrm{A} \cdot \mathrm{h} .\) (a) What is the total energy, in kilowatt-hours, stored in a \(12.0-\mathrm{V}\) battery rated at \(55.0 \mathrm{A} \cdot \mathrm{h} ?\) (b) At \(\$ 0.0600\) per kilowatt-hour, what is the value of the electricity produced by this battery?

The resistance of a platinum wire is to be calibrated for low-temperature measurements. A platinum wire with resistance \(1.00 \Omega\) at \(20.0^{\circ} \mathrm{C}\) is immersed in liquid nitrogen at \(77 \mathrm{K}\left(-196^{\circ} \mathrm{C}\right) .\) If the temperature response of the platinum wire is linear, what is the expected resistance of the platinum wire at \(-196^{\circ} \mathrm{C} ? \quad\left(\alpha_{\text {platinum }}=3.92 \times 10^{-3} /^{\circ} \mathrm{C}\right)\)

Suppose that the current through a conductor decreases exponentially with time according to the equation \(I(t)=I_{0} e^{-t / \tau}\) where \(I_{0}\) is the initial current (at \(t=0),\) and \(\tau\) is a constant having dimensions of time. Consider a fixed observation point within the conductor. (a) How much charge passes this point between \(t=0\) and \(i=\pi^{2}\) (b) How much charge passes this point between \(t=0\) and \(t=10 \tau ?\) (c) What If? How much charge passes this point between \(t=0\) and \(t=\infty ?\)

If the current carried by a conductor is doubled, what happens to the (a) charge carrier density? (b) current density? \((c)\) electron drift velocity? (d) average time interval between collisions?
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