Problem 158 - A Complete Resource Book in Physics for JEE Main

Question

The resistance of a bulb filament is \(100 \Omega\) at a temperature \(100^{\circ} \mathrm{C}\). If its temperature coefficient of resistance be \(0.005 /{ }^{\circ} \mathrm{C}\), its resistance will become \(200 \Omega\) at a temperature of (A) \(300^{\circ} \mathrm{C}\) (B) \(400^{\circ} \mathrm{C}\) (C) \(500^{\circ} \mathrm{C}\) (D) \(200^{\circ} \mathrm{C}\)c

Step-by-Step Solution

Answer

The temperature at which the resistance becomes \(200 \Omega\) is \(400^{\circ} \mathrm{C}\).

Step 1: Identify given quantities

From the problem we can identify the initial resistance \(R_{0} = 100 \Omega\), the temperature coefficient of resistance \(\alpha = 0.005 /{ }^{\circ} \mathrm{C}\), and the resistance we want to reach\( R_{T} = 200 \Omega\).

Step 2: Write down the equation for resistance as function of temperature

The general formula to calculate the resistance at a certain temperature is \(R_{T} = R_{0} * (1 + \alpha * T)\).

Step 3: Solve the equation for temperature T

We want to find the temperature at which the resistance becomes \(200 \Omega\). So we solve the formula from step 2 for T: \(T = (R_{T} / R_{0} - 1) / \alpha \).

Step 4: Substitute the given values and find the final temperature

Substitute the given values into the equation \(T = (200 / 100 - 1) / 0.005\). Solve the calculation to find the value of T.

Key Concepts

Temperature Coefficient of Resistance

The temperature coefficient of resistance is an essential concept when studying how materials change with temperature. Every material has a specific reaction to temperature changes, and this is quantified by the temperature coefficient of resistance, often represented as \( \alpha \). This coefficient tells us how much the resistance of a material will change with a change in temperature.

• High \( \alpha \): Resistance changes a lot with temperature.
• Low \( \alpha \): Resistance is more stable with temperature changes.

For example, metals typically have positive temperature coefficients, meaning their resistance increases as the temperature goes up. In the given exercise, the coefficient \( \alpha = 0.005 /{ }^{\circ} \mathrm{C} \) means that for every increase of 1 degree Celsius in temperature, the resistance of the filament increases by 0.5% of its original value.

Ohm's Law

Ohm's Law is one of the fundamental principles in electronics and physics. It describes the relationship between voltage, current, and resistance in an electrical circuit. The law is usually expressed with the formula:\[ V = I \times R \]where:

• \( V \) is the voltage across the conductor.
• \( I \) is the current flowing through the conductor.
• \( R \) is the resistance of the conductor.

Ohm's Law helps us understand how changing one element affects the others in a circuit. For instance, if you increase the resistance \( R \) while keeping the voltage \( V \) constant, the current \( I \) will decrease. This principle is universal and crucial for solving problems relating to resistance, like the one in the exercise.

Filament Bulb Resistance

The resistance of a filament bulb is an interesting case because it doesn't remain constant with use. This is largely due to the temperature coefficient of resistance, which affects metals like those used in a filament. As the filament heats up, its temperature increases significantly, resulting in a corresponding rise in its resistance. This is why bulbs consume more power as they warm up to their operating temperature. Over time, this change can cause wear and tear on the filament, eventually leading to the bulb burning out. Understanding filament bulb resistance is not just academic; it has practical implications in designing circuits that incorporate these components. Knowing the starting and operating resistances allows engineers to ensure that bulbs work efficiently and safely in various environments.