Problem 11 - Physics

Question

An \(8.2-\mathrm{mH}\) inductor is connected to an ac generator \((10.0 \mathrm{~V} \mathrm{rms}, 620 \mathrm{~Hz})\) Determine the peak value of the current supplied by the generator.

Step-by-Step Solution

Answer

The peak current is approximately 0.444 A.

Step 1: Calculate Inductive Reactance

First, find the inductive reactance \( X_L \) using the formula \( X_L = 2 \pi f L \), where \( f = 620 \, \text{Hz} \) is the frequency and \( L = 8.2 \, \text{mH} = 8.2 \times 10^{-3} \, \text{H} \) is the inductance. Plug in the values: \[ X_L = 2 \pi \times 620 \times 8.2 \times 10^{-3} \approx 31.86 \, \Omega. \]

Step 2: Calculate RMS Current

Use the formula for RMS current: \( I_{\text{rms}} = \frac{V_{\text{rms}}}{X_L} \), where \( V_{\text{rms}} = 10.0 \, \text{V} \). Substitute the values to find \( I_{\text{rms}} \): \[ I_{\text{rms}} = \frac{10.0}{31.86} \approx 0.314 \, \text{A}. \]

Step 3: Calculate Peak Current

Find the peak current \( I_0 \) using the relation \( I_0 = \sqrt{2} \times I_{\text{rms}} \). Use the value of \( I_{\text{rms}} \) obtained in the previous step: \[ I_0 = \sqrt{2} \times 0.314 \approx 0.444 \, \text{A}. \]

Key Concepts

Understanding AC Generators

An alternating current (AC) generator is a device that converts mechanical energy into electrical energy. It works on the principle of electromagnetic induction. The generator consists of a loop of wire rotating in a magnetic field, which induces an electric current in the wire.

Key points about AC generators include:

• They produce an alternating current which periodically reverses direction.
• The voltage generated follows a sinusoidal wave pattern.
• An AC generator is characterized by its frequency, measured in hertz (Hz), which indicates the number of cycles per second.
• In our exercise, the generator has a frequency of 620 Hz and provides an RMS voltage of 10.0 V.

This type of generator is commonly used in power stations to supply electricity to homes and businesses.

RMS Current Demystified

RMS, or root mean square current, is a crucial concept in AC circuits. It provides a measure of the equivalent direct current (DC) value that would provide the same power in a resistive load.

In practical terms, the RMS current value allows us to understand and calculate the effective power being delivered by an AC generator.

RMS current is given by the formula:

• \( I_{\text{rms}} = \frac{V_{\text{rms}}}{X_L} \)

where:

• \( V_{\text{rms}} \) is the root mean square voltage.
• \( X_L \) is the inductive reactance, found using \( X_L = 2\pi f L \).

For example, in our problem, we calculated \( I_{\text{rms}} \) to be approximately 0.314 A, indicating how much current is effectively flowing through the circuit.

The Significance of Peak Current

Peak current, often represented as \( I_0 \), is the maximum current magnitude that flows through the circuit during a sinusoidal AC cycle. It's crucial to understand it as it represents the highest value of current that an electrical device will encounter.

In a sinusoidal waveform, the peak value is related to the RMS value by the following relationship:

• \( I_0 = \sqrt{2} \times I_{\text{rms}} \)

This equation tells us that the peak current is the RMS current multiplied by the square root of 2, providing us with the crest factor of a sine wave.

For instance, the calculated peak current in our example is approximately 0.444 A. Knowing the peak current is essential in designing electrical components, ensuring they can handle the maximum current without damage.