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Chapter 4: Problem 23

The operating temperature of a tungsten filament in an incandescent lamp is \(2460 \mathrm{~K}\), and its total emissivity is \(0.30\). Find the surface area of the filament of a \(100-\mathrm{W}\) lamp.

Short Answer

Expert verified
The surface area of the filament is approximately \( 1.604 \times 10^{-2} \text{ m}^2 \).

Step by step solution

01

Identify the Known Quantities

Given: 1. Operating temperature of the tungsten filament, \( T = 2460 \text{ K} \)2. Total emissivity, \( \text{e} = 0.30 \)3. Power of the lamp, \( P = 100 \text{ W} \)
02

Use Stefan-Boltzmann Law

The Stefan-Boltzmann law relates the power radiated by a black body to its temperature and surface area. The formula, adjusted for emissivity, is:\[ P = e \times \text{Stefan-Boltzmann constant} \times A \times T^4 \]Where \( P \) is the power, \( e \) the emissivity, \( A \) the surface area, and \( T \) the temperature. The Stefan-Boltzmann constant is \( \text{σ} = 5.67 \times 10^{-8} \text{ W} \text{m}^{-2} \text{K}^{-4} \).
03

Rearrange the Formula to Solve for Surface Area

Rearrange the formula to isolate \( A \) (surface area):\[ A = \frac{P}{e \times \text{σ} \times T^4} \]
04

Plug in the Known Values

Substitute the given values into the rearranged formula:\[ A = \frac{100 \text{ W}}{0.30 \times 5.67 \times 10^{-8} \text{ W} \text{m}^{-2} \text{K}^{-4} \times (2460 \text{ K})^4} \]
05

Calculate the Surface Area

First, calculate \( T^4 \): \[ (2460)^4 = 3.66 \times 10^{13} \text{ K}^4 \] Then, calculate the denominator:\[ 0.30 \times 5.67 \times 10^{-8} \times 3.66 \times 10^{13} \text{ W} \text{m}^{-2} = 6.23154 \times 10^3 \text{ W} \]Finally, compute the surface area:\[ A = \frac{100 \text{ W}}{6.23154 \times 10^3 \text{ W}} = 1.604 \times 10^{-2} \text{ m}^2 \]

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Key Concepts

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

Black Body Radiation
Black body radiation is a type of electromagnetic radiation emitted by an object that absorbs all incident radiation, regardless of frequency or angle of incidence. Such objects are known as black bodies. A perfect black body in equilibrium emits radiation in a characteristic, continuous spectrum that depends solely on its temperature. The Stefan-Boltzmann Law is rooted in this concept, stating that a black body's power emission is proportional to the fourth power of its absolute temperature.
Emissivity
Emissivity is a measure of an object's ability to emit infrared energy compared to a perfect black body. It ranges from 0 to 1, with 1 representing a perfect emitter (black body) and 0 representing a perfect reflector. In real-world scenarios, no material is a perfect black body; hence, emissivity is always less than 1. For example, the emissivity of the tungsten filament in this exercise is 0.30, meaning it emits 30% of the radiation a perfect black body would at the same temperature.
Surface Area Calculation
Calculating the surface area of an incandescent lamp's filament involves using the Stefan-Boltzmann Law. Given the power output, temperature, and emissivity, the formula \[ A = \frac{P}{e \times \sigma \times T^4} \] allows us to solve for surface area. This formula takes into account the emitted power, the emissivity factor, the Stefan-Boltzmann constant (σ), and the temperature raised to the fourth power. By plugging in the known values and solving, we find the filament's surface area.
Incandescent Lamp
An incandescent lamp produces light by heating a tungsten filament to a high temperature, causing it to glow brightly. When electric current flows through the filament, it encounters resistance, which generates heat. This heat raises the filament's temperature to over 2000 K, producing light via thermal radiation. However, only a small fraction of electrical energy is converted into visible light, with most being emitted as heat.
Heat Transfer
Heat transfer is the process by which thermal energy flows from an area of higher temperature to one of lower temperature. In the context of black body radiation, heat transfer occurs primarily through radiation. The tungsten filament in an incandescent lamp transfers heat via radiation to its surroundings. The Stefan-Boltzmann Law, which involves emissivity and the surface temperature, helps quantify this heat transfer process. Emissivity affects how effectively an object can radiate heat, which is crucial for understanding the filament's thermal characteristics.

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Most popular questions from this chapter

The molar heat capacity at constant pressure \(C_{P} / n\) of a gas varies with the temperature according to the equation $$ \frac{C_{P}}{n}=a+b T-\frac{c}{T^{2}} $$ where \(a, b\), and \(c\) are constants. How much heat is transferred during an isobaric process in which \(n\) moles of gas experience a temperature rise from \(T_{i}\) to \(T_{f} ?\)

A combustion cxperiment is performed by burning a mixture of fuel and oxygen in a constant-volume container surrounded by a water bath. During the experiment, the temperature of the water rises. If the system is the mixture of fuel and oxygen: (a) Has heat been transferred? (b) Has work been done? (c) What is the sign of \(\Delta U\) ?

A cylinder with rigid well-insulated walls is divided into two parts by a rigid insulating wall with a small hole in it. A frictionless, insulated piston is held against the perforated partition, thus preventing the gas that is on the other side from seeping through the hole. The gas is maintained at a pressure \(P_{i}\) by another frictionless insulated piston. Imagine both pistons to move simultaneously in such a way that, as the gas streams through the hole, the pressure remains at a constant value \(P_{i}\) on one side of the dividing wall and at a constant lower value \(P_{f}\) on the other side, until all the gas is forced through the hole. (Note: this process is called a throttling process.) Prove that $$ U_{i}+P_{i} V_{i}=U_{f}+P_{f} V_{f} $$

The air above the surface of a freshwater lake is at a temperature \(T_{A}\), while the water is at its freezing point \(T_{i}\), where \(T_{A}

A copper wire of length \(1.317 \mathrm{~m}\) and diameter \(3.26 \times 10^{-4} \mathrm{~m}\) is blackened and placed along the axis of an evacuated glass tube. The wire is connected to a battery, a rheostat, an ammeter, and a voltmeter, and the current is increased until, at the moment the wire is about to melt, the ammeter reads \(12.8 \mathrm{~A}\) and the voltmeter reads \(20.2 \mathrm{~V}\). Assuming that all the energy supplied was radiated and that the radiation from the glass tube is negligible, calculate the melting temperature of copper.
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