A 100-W, 120-V incandescent lightbulb has a resistance of 12 Ω when cold (20°C) and 150 Ω when on (hot). Calculate its power consumption (a) at the instant it is turned on, and (b) after a few moments when it is hot.

검증된 단계별 안내

Step 1: Understand the problem. The power consumption of the lightbulb depends on its resistance and the voltage applied. When the bulb is cold, its resistance is 12 Ω, and when it is hot, its resistance is 150 Ω. Use the formula for power: P = V² / R, where P is power, V is voltage, and R is resistance.

Step 2: For part (a), calculate the power consumption at the instant the bulb is turned on (cold resistance). Substitute the given values into the formula: P = (120²) / 12. Simplify the expression to find the power when the bulb is cold.

Step 3: For part (b), calculate the power consumption after the bulb has been on for a while (hot resistance). Substitute the given values into the formula: P = (120²) / 150. Simplify the expression to find the power when the bulb is hot.

Step 4: Compare the results from part (a) and part (b). Notice that the power consumption is higher when the bulb is cold because the resistance is lower, and it decreases as the bulb heats up and its resistance increases.

Step 5: Reflect on the physical meaning. The change in resistance is due to the temperature dependence of the filament material. As the filament heats up, its resistance increases, which reduces the current and power consumption over time.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. It is mathematically expressed as V = I * R. This principle is essential for calculating the current in the lightbulb when it is turned on and when it is hot.

Power in Electrical Circuits

The power (P) consumed by an electrical device is defined as the product of the voltage (V) across it and the current (I) flowing through it, expressed as P = V * I. This concept is crucial for determining the power consumption of the lightbulb at both cold and hot states, as it allows us to calculate how much energy is being used.

Temperature Dependence of Resistance

The resistance of materials, particularly metals, typically increases with temperature. In the case of the incandescent lightbulb, its cold resistance is 12 Ω at 20°C, but it rises to 150 Ω when the bulb is hot. Understanding this temperature dependence is vital for accurately calculating the power consumption at the moment it is turned on versus after it has warmed up.