Problem 5 - College Physics

Question

\- A hair dryer is rated at \(1200 \mathrm{~W}\) when plugged into a \(120-\mathrm{V}\) outlet. Find \((\mathrm{a})\) its rms current, \((\mathrm{b})\) its peak current, and (c) its resistance.

Step-by-Step Solution

Answer

(a) rms current: 10 A, (b) peak current: 14.14 A, (c) resistance: 12 Ω.

Step 1: Calculate rms current

The rms (root mean square) current can be calculated using the power formula for alternating current: \( P = V_{rms} \times I_{rms} \). Rearrange the formula to solve for \( I_{rms} \): \( I_{rms} = \frac{P}{V_{rms}} \). Substitute \( P = 1200 \text{ W} \) and \( V_{rms} = 120 \text{ V} \) to find \( I_{rms} \): \( I_{rms} = \frac{1200}{120} = 10 \, \text{A} \).

Step 2: Calculate peak current

The peak current \( I_{peak} \) is related to the rms current by the formula \( I_{peak} = I_{rms} \times \sqrt{2} \). Use \( I_{rms} = 10 \, \text{A} \) from Step 1: \( I_{peak} = 10 \times \sqrt{2} \approx 14.14 \text{ A} \).

Step 3: Calculate resistance

Using Ohm's law, the resistance \( R \) can be calculated with \( V = I \times R \). Rearrange to solve for \( R \): \( R = \frac{V}{I} \). Use \( V = 120 \text{ V} \) and \( I_{rms} = 10 \text{ A} \): \( R = \frac{120}{10} = 12 \Omega \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in physics, particularly when understanding electrical circuits. It explains the relationship between voltage (V), current (I), and resistance (R) in a circuit. Ohm's Law is expressed as:\[ V = I \times R \]This means the voltage across a conductor is equal to the current flowing through it, multiplied by the resistance. This formula is crucial for calculating the electrical resistance or the current in a circuit if the other values are known. In the context of the exercise, finding the hair dryer's resistance required using Ohm’s Law. Given the voltage and the current, the resistance was determined by rearranging the formula to \( R = \frac{V}{I} \). This calculation is simple yet vital in understanding how efficiently a device like a hair dryer operates. It demonstrates how the electric potential difference influences the current, moderated by the resistance.

AC Power Calculation

AC Power Calculation is essential when dealing with alternating current (AC) circuits, as it helps determine the power consumed by electrical devices. For AC circuits, the power rating of a device is calculated using the formula:\[ P = V_{rms} \times I_{rms} \]In this equation, \( P \) represents power in watts, \( V_{rms} \) is the root mean square voltage, and \( I_{rms} \) is the root mean square current. This formula calculates the actual power consumption because, in AC circuits, the voltage and the current vary sinusoidally, not uniformly, like in DC circuits. In our context, the hair dryer's power rating of \(1200 \text{ W} \) and an \(120 \text{ V} \) outlet was used to find the rms current. Thus, understanding AC Power Calculation allows one to accurately assess electrical devices' demands in real-world AC setups.

Electrical Resistance

Electrical Resistance is a measure of the opposition that a circuit offers to the flow of electric current. The SI unit for resistance is the ohm (\( \Omega \)), and it can be understood intuitively as the "friction" that electrical current experiences as it moves through a conductor. It is a fundamental concept that affects how electrical systems and components behave.In the given exercise, by calculating the resistance of the hair dryer, we learn how much the device resists the electrical current supplied by a household outlet. Factors influencing resistance include the material type, length, and cross-sectional area of the conductor. Calculating resistance using Ohm's Law helps in designing electrical circuits and selecting the correct electrical components to ensure device safety and efficiency. Being able to compute resistance relationships makes it possible to troubleshoot and optimize electrical and electronic systems effectively.