Problem 163 - Master Resource Book in JEE Main Physics

Question

Statement I The temperature dependence of resistance is usually given as \(R=R_{0}(1+\Delta t)\). The resistance of a wire changes from \(100 \Omega\) to \(150 \Omega\) when its temperature is increased from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\). This implies that \(\alpha=2.5 \times 10^{-3} \mathrm{C}^{-1}\). Statement II \(R=R_{0}(1+\alpha \Delta t)\) is valid only when the change in the temperature is small and \(\Delta R=\left(R-R_{0}\right) \ll

Step-by-Step Solution

Answer

The correct answer is (b).

Step 1: Understanding the Formula

The formula given is \( R = R_0(1 + \alpha \Delta t) \) where \( R_0 \) is the original resistance, \( \alpha \) is the temperature coefficient of resistance, and \( \Delta t \) is the change in temperature.

Step 2: Calculating \( \alpha \) from Statement I

Given that \( R_0 = 100 \ \Omega \) and the final resistance \( R = 150 \ \Omega \), with \( \Delta t = 227\degree C - 27\degree C = 200\degree C \). Using the formula: \( 150 = 100(1 + \alpha \times 200) \). Solving for \( \alpha \), we get \( \alpha = \frac{0.5}{200} = 2.5 \times 10^{-3} \ \degree C^{-1} \). The calculation matches with the given statement, making Statement I true.

Step 3: Analyzing Statement II

Statement II suggests that the formula \( R = R_0(1 + \alpha \Delta t) \) is valid only for small changes in temperature where \( \Delta R \ll R_0 \). Here, \( \Delta R = 150 - 100 = 50 \ \Omega \), which is larger than \( R_0 \), indicating \( \Delta R less \!\! < R_0 \). Thus, Statement II is valid, but not applicable fully for the conditions given in Statement I.

Step 4: Conclusion

Both Statement I and Statement II are true in isolation. However, Statement II doesn't serve as a valid explanation for the truth of Statement I since the temperature change is large and \( \Delta R \gg R_0 \). Therefore, the correct answer is (b).

Key Concepts

Resistance Change

When we talk about resistance change, we're referring to how the electrical resistance of a material varies due to changes within its environment, primarily temperature.

• Definition: Resistance is the opposition faced by electrical current as it travels through a conductor.
• Factors affecting resistance: It's mostly affected by temperature, length, and cross-sectional area of the conductor.

Let's understand this with an example. If you have a wire with an initial resistance of 100 Ω and you observe its resistance increasing to 150 Ω, there has been a resistance change. This change could be instigated by a number of factors, with temperature being a major influencer as in our given problem. Generally, resistance increases with an increase in temperature for most conductive materials. This is because atoms inside the metal vibrate more at higher temperatures, making it harder for the electrons to pass through, thus increasing resistance.

Temperature Dependence of Resistance

The temperature dependence of resistance is a fundamental principle, indicating that the resistance of almost all conductors increases with an increase in temperature.

• When temperature rises, atoms vibrate more vigorously.
• This increased vibration creates more obstacles for the flowing electrons.

The mathematical expression representing this relationship is given by:\[ R = R_0(1 + \alpha \Delta t) \]Here, \( R \) is the final resistance, \( R_0 \) is the original resistance, \( \alpha \) is the temperature coefficient of resistance, and \( \Delta t \) is the change in temperature. The value of \( \alpha \) is characteristic of the material and indicates how much resistance changes per degree of temperature change.This equation holds accurately for small temperature changes. However, for large temperature changes where resistance alters significantly, the linear approximation used may not fully apply.

Electrical Resistance

Electrical resistance is a measure of how much a material opposes the flow of electric current. It's measured in ohms (Ω) and varies with - material type,- temperature, and- physical dimensions of the conductor.The higher the resistance, the more energy is required to push electric current through the material.Ohm's Law is essential to understanding electrical resistance:\[ V = IR \]Here, \( V \) stands for voltage, \( I \) for current, and \( R \) for resistance. This formula shows that voltage across a conductor is directly proportional to the current flowing through it for a fixed resistance.In summary, resistance is vital because it helps control and manage the flow of electrical energy, ensuring safety and functionality in electrical appliances and systems. When working on electronics, understanding resistance and how it's influenced by environmental factors is key to troubleshooting and improving circuit performance.