Problem 11 - Physics

Question

An 8.2-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 supplied by the generator is approximately 0.442 A.

Step 1: Calculate Inductive Reactance

The inductive reactance \(X_L\) is calculated using the formula: \[ X_L = 2 \pi f L \] where \(f\) is the frequency and \(L\) is the inductance. Substituting the given values, \(f = 620 \text{ Hz}\) and \(L = 8.2 \text{ mH} = 8.2 \times 10^{-3} \text{ H}\), we find: \[ X_L = 2 \pi \times 620 \times 8.2 \times 10^{-3} \approx 31.947 \text{ ohms} \].

Step 2: Calculate RMS Current

The RMS current \(I_{rms}\) can be determined using Ohm's Law for AC circuits: \[ I_{rms} = \frac{V_{rms}}{X_L} \] where \(V_{rms} = 10.0 \text{ V} \). Substituting the values, we get: \[ I_{rms} = \frac{10.0}{31.947} \approx 0.313 \text{ A} \].

Step 3: Calculate Peak Current

The peak current \(I_{peak}\) is related to the RMS current by the relation: \[ I_{peak} = \sqrt{2} \times I_{rms} \]. Using the \(I_{rms} = 0.313 \text{ A}\) calculated earlier, we find: \[ I_{peak} = \sqrt{2} \times 0.313 \approx 0.442 \text{ A} \].

Key Concepts

AC Circuits

An AC (Alternating Current) circuit is a type of electrical circuit where the current changes direction periodically. In AC circuits, voltages and currents oscillate in a sinusoidal manner, typically due to an ac generator. Frequencies in AC circuits are measured in hertz (Hz), which indicates how many cycles per second the current completes.

AC circuits have various components such as resistors, inductors, and capacitors. Each component affects the circuit differently:

• Inductor: Opposes changes in current and introduces inductive reactance.
• Resistor: Opposes current and introduces resistance to the circuit.
• Capacitor: Stores energy, affecting the voltage and introducing capacitive reactance.

In AC circuits, we often refer to values like RMS (Root Mean Square) voltages and currents, as these give a meaningful representation of what the circuit experiences on average, despite the constant oscillation.

Ohm's Law

Ohm's Law is a fundamental principle used to understand the relationship between voltage, current, and resistance or reactance. In AC circuits, Ohm's Law is slightly adapted due to the presence of reactance, such as inductive reactance, instead of simple resistance.

For AC circuits, the equation becomes:

• \( I_{rms} = \frac{V_{rms}}{X_L} \)

Here, \( I_{rms} \) is the root mean square current, \( V_{rms} \) is the root mean square voltage, and \( X_L \) is the inductive reactance. Inductive reactance, \( X_L \), measures how much an inductor in a circuit resists the flow of AC. It increases with higher frequency and higher inductance due to the formula:

\[ X_L = 2 \pi f L \]This formula represents how the inductance \( L \), measured in henrys (H), and frequency \( f \) affect the circuit's reactance and, consequently, the current that can flow.

Peak Current Calculation

Calculating the peak current is an important step in understanding the maximum potential flow of electricity in an AC circuit. The peak current \( I_{peak} \) is related to the RMS (Root Mean Square) current, which is a type of electrical current measurement used for AC circuits.

The peak current can be found using the formula:

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

This formula tells us that the peak current is \( \sqrt{2} \approx 1.414 \) times greater than the RMS current value. Here, the RMS current is already smoothened to reflect the average current of the AC, but knowing the peak gives insight into the circuit's maximum load.To calculate the peak current from the given problem, we use the RMS current \( I_{rms} \) of 0.313 A, previously determined with Ohm's Law for AC circuits. The peak current, therefore, is approximately 0.442 A, which is the maximum amplitude of the current that the generator can deliver to the circuit.