Problem 9 - College Physics

Question

A sinusoidal ac voltage source in a circuit produces a maximum voltage of 12.0 \(\mathrm{V}\) and an rms current of 7.50 \(\mathrm{mA}\) . Find (a) the voltage and current amplitudes and (b) the rms voltage of this source.

Step-by-Step Solution

Answer

(a) Voltage amplitude is 12.0 V, current amplitude is 10.60 mA. (b) RMS voltage is 8.49 V.

Step 1: Understanding Sinusoidal Signals

In alternating current (AC) circuits, voltages and currents vary sinusoidally with time. The maximum value of a sinusoidal waveform is called the amplitude, and the root mean square (RMS) value is a measure of the effective level of a waveform.

Step 2: Determine Voltage Amplitude

The voltage amplitude is the maximum voltage provided by the source. Given in the problem, the maximum voltage or amplitude is 12.0 V.

Step 3: Calculate Current Amplitude

Given the RMS current is 7.50 mA, you can calculate the current amplitude using the formula: \( I_0 = I_{rms} \cdot \sqrt{2} \).Substitute the given values to find the current amplitude: \( I_0 = 7.50 \, \text{mA} \times \sqrt{2} \approx 10.60 \, \text{mA} \).

Step 4: Calculate RMS Voltage

The RMS voltage can be determined from the amplitude using the formula: \( V_{rms} = \frac{V_0}{\sqrt{2}} \).Substitute the known amplitude: \( V_{rms} = \frac{12.0 \, \text{V}}{\sqrt{2}} \approx 8.49 \, \text{V} \).

Key Concepts

Sinusoidal Waveform

AC circuits are characterized by voltages and currents that vary with time in a sinusoidal manner. Imagine a smooth, wavy motion like a gentle ocean wave. This wave-like form can be described mathematically by sine functions. The sinusoidal waveform is the most fundamental type of oscillating signal used in AC circuits. It has many important properties:

• Amplitude: This is the peak value of the wave, representing the highest point on the waveform.
• Frequency: This tells us how many cycles occur in one second, usually measured in hertz (Hz).
• Phase: This shows the waveform's alignment concerning a reference point.

Understanding these aspects allows us to make sense of how energy flows through AC circuits. We use them to analyze different signals we encounter in electronics.

Voltage Amplitude

Voltage amplitude is a crucial parameter of sinusoidal waveforms. It represents the maximum voltage level that the waveform reaches. Think of it like the tallest peak of a mountain range. In the problem, the voltage amplitude is directly given as 12.0 V. This is the maximum voltage output the source can deliver at any point in time. Knowing the voltage amplitude helps in designing circuits to ensure the components can handle the power they receive without damage.

Current Amplitude

Just as voltage has amplitude, so does current. This is the peak or maximum value of current in an AC circuit. To find the current amplitude from the RMS current, we use the formula: \[ I_0 = I_{rms} \times \sqrt{2} \]where \( I_{rms} \) is the root mean square (RMS) current value. The RMS value offers a measure of AC current that signifies its ability to deliver power. It takes the form of a continuous, direct current value emotionally equivalent to the same power. In this exercise, substituting the known RMS current of 7.50 mA leads to an amplitude of approximately 10.60 mA. This step is essential in analyzing how much current can flow through a system.

RMS Voltage

The RMS voltage gives insight into the effective value of a fluctuating AC waveform. It helps us understand how much work it does compared to a DC voltage level. We use the formula:\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]where \( V_0 \) is the voltage amplitude. Plugging in the amplitude of 12.0 V, we find \( V_{rms} \approx 8.49 \) V. This means that an AC system using an amplitude of 12.0 V delivers the same power as a DC system of approximately 8.49 V. Using RMS values makes it easier to calculate and predict real-world circuit performance without complicated computations.