Problem 23 - Applied Physics

Question

A maximum voltage of \(34.0 \mathrm{~V}\) is developed by an ac generator that delivers a maximum current of \(0.170 \mathrm{~A}\) to a circuit. (a) What is the effective voltage of the generator? (b) Find the effective current delivered to the circuit by the generator. (c) Find the resistance of the circuit.

Step-by-Step Solution

Answer

(a) 24.05 V, (b) 0.120 A, (c) 200.42 ohms

Step 1: Understanding the Problem

We are given the maximum voltage (peak voltage) of the AC generator as \( V_{max} = 34.0 \text{ V} \) and the maximum current (peak current) as \( I_{max} = 0.170 \text{ A} \). We need to find (a) the effective voltage (RMS voltage), (b) the effective current (RMS current), and (c) the resistance of the circuit.

Step 1: Calculate RMS Voltage

The effective voltage (RMS voltage) for an AC circuit can be found using the relation \( V_{rms} = \frac{V_{max}}{\sqrt{2}} \). Substituting the given values, we get:\[V_{rms} = \frac{34.0}{\sqrt{2}} = \frac{34.0}{1.414} \approx 24.05 \text{ V}\]

Step 2: Calculate RMS Current

The effective current (RMS current) is calculated in a similar manner using: \( I_{rms} = \frac{I_{max}}{\sqrt{2}} \). Plugging the numbers in, we have:\[I_{rms} = \frac{0.170}{\sqrt{2}} = \frac{0.170}{1.414} \approx 0.120 \text{ A}\]

Step 3: Find the Resistance of the Circuit

Using Ohm's law for resistances in AC circuits, the resistance \( R \) can be found using the RMS values:\[R = \frac{V_{rms}}{I_{rms}}\]Substituting the calculated effective voltage and current, we get:\[R = \frac{24.05}{0.120} \approx 200.42 \text{ ohms}\]

Key Concepts

RMS Voltage

RMS Voltage, or root mean square voltage, is a critical concept in AC circuits. It provides a way to express the effective value of an AC voltage, allowing for meaningful comparisons with DC voltages. In essence, RMS Voltage represents the equivalent DC voltage that would deliver the same amount of power to a load as the AC voltage. This is vital for ensuring circuits function efficiently and safely.

The formula for finding RMS voltage, given the peak (or maximum) voltage, is:

• \[ V_{rms} = \frac{V_{max}}{\sqrt{2}} \]

This formula stems from statistical and mathematical analyses of AC signals, capturing the voltage's true power capability. In our example, the generator's peak voltage is 34.0 volts; applying the RMS formula results in an effective voltage of approximately 24.05 volts.

Understanding RMS voltage is essential for tasks involving power calculations and matching AC devices with proper power sources.

RMS Current

Like RMS voltage, RMS current, or root mean square current, is used in AC circuits to provide a standard, measurable, and comparable value. This concept helps us determine the real effective current flowing in an AC system. In many ways, RMS current parallels RMS voltage. It shows the amount of current that delivers the same power to a load as would a DC current of the same RMS value.

To calculate RMS current from peak current, we use:

• \[ I_{rms} = \frac{I_{max}}{\sqrt{2}} \]

This formula is derived from the same principles used for RMS voltage calculations. In the case study, the given peak current is 0.170 A, which converts to an RMS value of about 0.120 A.

RMS current is crucial in ensuring electronic devices, and circuits operate within their safe current ratings to avoid overheating and potential damage.

Ohm's Law

Ohm's Law is a fundamental principle for understanding AC circuits, relating voltage, current, and resistance. It states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them.

For AC circuits, we typically use the RMS values to apply Ohm's Law:

• \[ R = \frac{V_{rms}}{I_{rms}} \]

This relationship helps determine circuit elements such as resistance from known voltage and current values. For instance, using our calculated RMS voltage and current, the resistance in the example circuit is determined to be approximately 200.42 ohms.

Ohm's Law is valuable for designing circuits and diagnosing issues, making it one of the cornerstones of electrical engineering.