Question

The resistance of the wire made of silver at \(27^{\circ} \mathrm{C}\) temperature is equal to \(2.1 \Omega\) while at \(100^{\circ} \mathrm{C}\) it is \(2.7 \Omega\) calculate the temperature coefficient of the resistivity of silver. Take the reference temperature equal to \(20^{\circ} \mathrm{C}\) (A) \(4.02 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\) (B) \(0.402 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\) (C) \(40.2 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\) (D) \(4.02 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\)

Short answer

The temperature coefficient of resistivity of silver is approximately \(4.02 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\).

Step-by-step solution

List the given information

We are provided with the following values: - Resistance at \(27^{\circ}\mathrm{C}\): \(R_{27} = 2.1 \Omega\) - Resistance at \(100^{\circ}\mathrm{C}\): \(R_{100} = 2.7 \Omega\) - Reference temperature: \(t_0 = 20^{\circ}\mathrm{C}\)

Set up two equations using the resistance formula

We can set up two equations using the temperature coefficient of resistivity formula: - Equation 1: \(R_{27} = R_0[1 + \alpha (27 - 20)]\) - Equation 2: \(R_{100} = R_0[1 + \alpha (100 - 20)]\)

Solve for R_0 using Equation 1

Using Equation 1, we can solve for the reference resistance \(R_0\): \(2.1 = R_0[1 + \alpha (7)]\)

Substitute R_0 from Equation 1 into Equation 2

Now we substitute the expression for \(R_0\) from Step 3 into Equation 2: \(2.7 = \frac{2.1}{1 + 7\alpha}[1 + 80\alpha]\)

Solve for alpha

Now, we need to solve the equation for \(\alpha\): \(\frac{2.7}{1 + 80\alpha} = \frac{2.1}{1 + 7\alpha}\) Cross-multiplying, we get: \(2.7 (1 + 7\alpha) = 2.1 (1 + 80\alpha)\) Expanding both sides: \(2.7 + 18.9\alpha = 2.1 + 168\alpha\) Now we can isolate \(\alpha\): \(150.9\alpha = 0.6\) \(\alpha = \frac{0.6}{150.9}\)

Calculate the value of alpha and choose the correct option

Now, we can calculate the value of \(\alpha\): \(\alpha \approx 0.00402^{\circ}\mathrm{C}^{-1}\) Now, we can see that the correct option is: (A) \(4.02 \times 10^{-3}{ }^{\circ} \mathrm{C}^{-1}\)

Key concepts

silver

Silver, a precious metal, is highly regarded for its excellent conductivity. It is widely used in electrical circuits due to its ability to transport electrical current with minimal resistance.
This metal is preferred in applications where high conductivity is crucial, such as in high-frequency engineering and specialized cables.
One of the factors influencing silver's conductivity is its purity, with commercial-grade silver being nearly 100% pure.
Despite its efficiency, silver is less commonly used than copper in large scale applications because of its higher cost. However, in technology requiring top performance, silver's superior conductivity can outweigh the expense.
In our discussed exercise, a silver wire's resistance was evaluated to understand how its resistivity is affected by temperature changes, showcasing one of the thermal properties of this metal.

resistance

Resistance is a fundamental concept in electronics, describing the opposition to the flow of electric current within a conductor.
It is typically measured in ohms (denoted by the symbol \(\Omega\)).
Resistance can vary greatly among different materials and is influenced by factors such as material type, length, cross-sectional area, and temperature.In a conductor like silver, resistance affects how efficiently electricity can flow through the material.
Lower resistance means better conductivity and more efficient transmission of electrical current.
The resistance observed in a wire at any temperature can be calculated using Ohm’s Law: \[ R = \rho \frac{L}{A} \]where \( R \) is the resistance, \( \rho \) is the resistivity, \( L \) is the length of the conductor, and \( A \) is its cross-sectional area.
Understanding resistance and its effect is critical in designing circuits that perform reliably under varying conditions.

temperature effect on resistance

The effect of temperature on resistance is a key consideration in electrical engineering. As temperature changes, the resistance of a material can significantly fluctuate.
Many conductive materials, including silver, exhibit increased resistance as their temperature rises.
This happens because higher temperatures cause more vibrations in the lattice structure of the material, making it harder for free electrons to flow through the lattice.
The temperature coefficient of resistivity provides a quantifiable measure of how resistance changes with temperature: \[ R = R_0 [1 + \alpha (T - T_0)] \]where \( R \) is the resistance at temperature \( T \), \( R_0 \) is the reference resistance at reference temperature \( T_0 \), and \( \alpha \) is the temperature coefficient of resistivity.
In the given exercise, this concept is applied to calculate how the resistance of silver wire changes with temperature.
Engineers use this information to anticipate and manage temperature-related changes in electronic circuits.

thermal properties of materials

The thermal properties of materials are crucial in determining their applications in various industries, including electronics and construction. These properties dictate how materials respond to temperature changes in terms of expansion, conductivity, and resistance.
In the case of conductive materials like silver, the thermal property of interest is how temperature affects electrical resistivity. This affects how efficiently the material can conduct electricity under varying temperatures.
Other important thermal properties include heat capacity, thermal expansion, and thermal conductivity.
For instance, materials with high thermal conductivity can efficiently transfer heat, which can be desirable for heat dissipation in electronic components.
Understanding the thermal properties is essential, especially when designing components expected to operate under temperature fluctuations. This knowledge helps manage potential risks such as overheating or structural failure due to thermal expansion.