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Chapter 20: Problem 19

Two wires have the same cross-sectional area and are joined end to end to form a single wire. One is tungsten, which has a temperature coefficient of resistivity of \(\alpha=0.0045\left(\mathrm{C}^{\circ}\right)^{-1} .\) The other is carbon, for which \(\alpha=-0.0005\left(\mathrm{C}^{\circ}\right)^{-1} .\) The total resistance of the composite wire is the sum of the resistances of the pieces. The total resistance of the composite does not change with temperature. What is the ratio of the lengths of the tungsten and carbon sections? Ignore any changes in length due to thermal expansion.

Short Answer

Expert verified
The ratio of the lengths of tungsten to carbon is \( \frac{1}{9} \).

Step by step solution

01

Understand the problem essentials

We need to find the ratio of the lengths of two wires (made of tungsten and carbon) such that their combined resistance remains constant with temperature changes. This requires that changes in resistance due to temperature cancel each other out.
02

Use the formula for resistance in relation to temperature

The resistance of a material changes with temperature according to the formula: \( R = R_0(1 + \alpha \Delta T) \), where \( R_0 \) is the initial resistance, \( \alpha \) is the temperature coefficient of resistivity, and \( \Delta T \) is the change in temperature.
03

Establish the condition for constant resistance

For the total resistance to remain unchanged with temperature, the increase in resistance due to tungsten must equal the decrease in resistance due to carbon. Thus, \( R_{0t}\alpha_{t}\Delta T = - R_{0c}\alpha_{c}\Delta T \).
04

Cancel out common factors and express in terms of lengths

Since the cross-sectional area is the same for both, the resistance can be expressed proportional to length and resistivity, \( R_0 = \rho \frac{L}{A}\). For equal areas, \( L_{t}\alpha_{t} = - L_{c}\alpha_{c} \). Simplify to get \( \frac{L_{t}}{L_{c}} = -\frac{\alpha_{c}}{\alpha_{t}} \).
05

Substitute the given values

Substitute the given temperature coefficients into the equation: \( \frac{L_{t}}{L_{c}} = -\frac{-0.0005}{0.0045} = \frac{0.0005}{0.0045} \).
06

Simplify the expression to find the length ratio

The value simplifies to \( \frac{L_{t}}{L_{c}} = \frac{1}{9} \), meaning the length of the tungsten wire is 1/9th the length of the carbon wire.

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Key Concepts

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

Temperature Coefficient of Resistivity
The temperature coefficient of resistivity, often symbolized as \(\alpha\), is a measure of how much a material's resistivity changes with temperature. In the context of electrical resistance, it helps us understand how the resistance of a conductor will behave when subjected to temperature changes.
For materials like metals and semiconductors, their resistivity typically increases with temperature. This is because increased thermal energy causes more frequent collisions of charge carriers, which increases resistance.
  • A positive \(\alpha\) means resistivity increases with temperature, characteristic of most metals.
  • A negative \(\alpha\) indicates resistivity decreases with temperature, as seen in some semiconductors like carbon.
This coefficient is crucial for designing circuits that need to operate reliably over a range of temperatures, ensuring the desired electrical properties.
Tungsten
Tungsten is a metal recognized for its exceptional properties and practical applications. It is known for its high melting point, low thermal expansion, and excellent conductivity, making it ideal for high-temperature applications.
In terms of electrical resistivity, tungsten has a positive temperature coefficient. This means its resistance increases as temperature rises, as indicated by its \(\alpha\) value of 0.0045 \(\left(\mathrm{C}^{\circ}\right)^{-1}\).
Some key aspects include:
  • High melting point: useful in making filaments for light bulbs and electronics that endure high temperatures.
  • Strong and durable: contributes to tungsten's popularity in hard materials and alloys.
  • Predictable resistance changes: allows precise control in electrical applications when temperature fluctuations are considered.
Understanding tungsten's thermal properties aids in calculating resistance and designing efficient electrical components.
Carbon
Carbon is a unique element found in numerous forms, including graphite and diamond, but commonly utilized in its amorphous form for electrical components. Unlike metals, carbon has a negative temperature coefficient of resistivity, meaning its resistance decreases as temperature increases.
In this problem, carbon's \(\alpha\) value is -0.0005 \(\left(\mathrm{C}^{\circ}\right)^{-1}\), setting it apart from typical conductors.
Important features of carbon include:
  • Versatile conductive material: often used in resistors and battery electrodes due to its stability over various temperatures.
  • Temperature adaptability: makes carbon suitable for environments with fluctuating temperatures.
  • Improved efficiency: when combined with other materials, carbon can optimize the resistance management of composite wires.
Carbon's properties help balance resistances in mixed-material applications, like the given tungsten-carbon wire example.
Thermal Expansion
Thermal expansion refers to the change in size of a material as it is heated or cooled. Although not directly affecting the resistance in this problem, it is often a crucial consideration in materials science.
Materials expand or contract with temperature changes due to the increase or decrease of kinetic energy between atoms. In this specific case, we ignore thermal expansion to focus entirely on resistivity changes.
However, in practical applications, thermal expansion can be impactful:
  • Structure integrity: different expansion rates can cause mechanical stress or failure when materials are joined.
  • Component design: understanding expansion helps engineers make accurate designs for circuits that must withstand temperature variations.
  • Dimensional changes: expansions in wires can slightly affect resistance, therefore usually a minor correction is considered.
Even if not essential in this problem, thermal expansion remains a significant factor in the design of reliable electrical systems.

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Most popular questions from this chapter

Voltmeter A has an equivalent resistance of \(2.40 \times 10^{5} \Omega\) and a full-scale voltage of \(50.0\) \(V\). Voltmeter \(B\), using the same galvanometer as voltmeter \(A\), has an equivalent resistance of \(1.44 \times 10^{5} \Omega\). What is its full-scale voltage?

A resistor (resistance \(=R\) ) is connected first in parallel and then in series with a \(2.00-\Omega\) resistor. A battery delivers five times as much current to the parallel combination as it does to the series combination. Determine the two possible values for \(R\).

Suppose that the resistance between the walls of a biological cell is \(5.0 \times 10^{9} \Omega\). (a) What is the current when the potential difference between the walls is \(75 \mathrm{mV} ?\) (b) If the current is composed of Na+ ions \((q=+e)\), how many such ions flow in \(0.50 \mathrm{~s}\) ?

The recovery time of a hot water heater is the time required to heat all the water in the unit to the desired temperature. Suppose that a 52-gal \(\left(1.00 \mathrm{gal}=3.79 \times 10^{-3} \mathrm{~m}^{3}\right)\) unit starts with cold water at \(11^{\circ} \mathrm{C}\) and delivers hot water at \(53^{\circ} \mathrm{C}\). The unit is electric and utilizes a resistance heater \((120 \mathrm{~V} \mathrm{ac}, 3.0 \Omega)\) to heat the water. Assuming that no heat is lost to the environment, determine the recovery time (in hours) of the unit.

What resistance must be placed in parallel with a \(155-\Omega\) resistor to make the equivalent resistance \(115 \Omega ?\)
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