Problem 180 - A Complete Resource Book in Physics for JEE Main

Question

The temperature dependence of resistances of \(\mathrm{Cu}\) and Si (not doped) in the temperature range \(300-400 \mathrm{~K}\), is best described by (A) linear increase for Cu, exponential increase for Si. (B) linear increase for Cu, exponential decrease for Si. (C) linear decrease for Cu, linear decrease for Si. (D) linear increase for Cu, linear increase of Si.

Step-by-Step Solution

Answer

The correct description of the temperature dependence of resistances of Copper (Cu) and Silicon (Si) in the temperature range \(300-400 \mathrm{~K}\) is (B) linear increase for Cu, exponential decrease for Si. This is because Copper is a metal, and its resistance follows a linear increase with temperature, whereas Silicon is a semiconductor, and its resistance follows an exponential decrease in temperature.

Step 1: Identify the resistance behavior of Copper

Copper is a metal, and its resistance is found to linearly increase with temperature.

Step 2: Identify the resistance behavior of Si (not doped)

Silicon (Si) is a semiconductor, and the resistance of a semiconductor is found to exponentially decrease with temperature.

Step 3: Match the correct option

Now, we need to find the option that correctly matches the temperature dependence of Cu and Si. The correct description of the temperature dependence of resistances of Cu and Si should be: - Linear increase for Cu - Exponential decrease for Si Out of the given options, the description that matches the behavior of Copper and Silicon is Option (B). #Answer#: (B) linear increase for Cu, exponential decrease for Si.

Key Concepts

Electrical Resistance

When it comes to understanding electric circuits, a fundamental concept is electrical resistance, symbolized by R. It is essentially the opposition to the flow of electric current through a material. The resistance of a material can be influenced by several factors, such as its temperature, geometric dimensions, and the material's intrinsic properties.

For instance, when you increase the temperature of a conductor, the vibrations of its atoms intensify, which in turn scatters the electrons more. This scattering causes greater resistance to the flow of electricity. This concept helps unravel why devices that heat up during operation can become less efficient over time or require cooling systems to maintain performance.

Properties of Copper

Copper (Cu) is known for its excellent conductivity and is widely used in electrical wiring and electronics. One fundamental property of copper pertinent to electric circuits is its relatively low resistivity. However, this doesn't mean that copper's resistance remains constant; rather, it increases linearly with temperature — a behavior typical for most metals.

This temperature coefficient of resistance is taken into consideration when designing electrical systems, such as in choosing the size of wiring in building construction or in circuit design, to ensure safety and efficiency even under varying temperatures.

Semiconductor Properties

A semiconductor like silicon (Si) behaves differently from a conductor like copper. At absolute zero temperature, the semiconductor behaves like an insulator with almost no free charge carriers. However, as the temperature increases, it gains more charge carriers - electrons and holes. This increase in charge carriers reduces the resistance of silicon, thus showing an exponential decrease in resistance with temperature.

Such temperature-dependent behavior is crucial in the field of electronics, as it forms the basis for the functioning of various devices, such as diodes, transistors, and solar cells. Even in temperature sensors, the semiconductor's predictable change in resistance with temperature is exploited for accurate measurements.

Thermal Effects in Conductors and Semiconductors

The thermal effects on materials like conductors and semiconductors have significant technical implications. In conductors, such as copper, the increase in temperature leads to an increase in atomic vibrations. This impedes the free flow of electrons, causing the resistance to increase linearly with temperature. However, semiconductors reveal a different pattern. As these materials are heated, more electron-hole pairs are created, which enhances conductivity, hence the resistance decreases exponentially.

Understanding these differences between conductors and semiconductors is essential when designing electronic components. It ensures that circuits will function reliably under varying environmental conditions and is a cornerstone for materials science and electronic engineering education.