Problem 15 - Physics

Question

A series RCL circuit includes a resistance of \(275 \Omega,\) an inductive reactance of \(648 \Omega\), and a capacitive reactance of \(415 \Omega\). The current in the circuit is 0.233 A. What is the voltage of the generator?

Step-by-Step Solution

Answer

The voltage of the generator is approximately 84.00 V.

Step 1: Calculate Impedance

The impedance \( Z \) in a RCL series circuit can be calculated using the formula: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance. Let's substitute the values: \[ Z = \sqrt{275^2 + (648 - 415)^2} \] Simplifying inside the parentheses first: \( 648 - 415 = 233 \). Now substitute back: \[ Z = \sqrt{275^2 + 233^2} \] Calculate \( 275^2 = 75625 \) and \( 233^2 = 54289 \). Thus, \[ Z = \sqrt{75625 + 54289} = \sqrt{129914} \approx 360.36 \Omega \].

Step 2: Calculate Voltage

Using Ohm's Law, the voltage \( V \) across the circuit can be calculated as \( V = I \times Z \), where \( I \) is the current. Given \( I = 0.233 \) A and \( Z \approx 360.36 \Omega \), we calculate \[ V = 0.233 \times 360.36 \approx 84.00 \text{ V} \].

Key Concepts

Impedance Calculation

Understanding impedance in a series RCL circuit is essential to analyze how the circuit behaves when connected to an AC source. Impedance, symbolized by \( Z \), is the total opposition a circuit offers to the flow of alternating current. It combines the effects of resistance \( R \), inductive reactance \( X_L \), and capacitive reactance \( X_C \).

In this context, impedance is calculated using the formula:

• \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)

This formula shows that impedance is not simply the sum of the resistances, like in DC circuits. Instead, it accounts for the phase differences caused by the inductive and capacitive reactances. In the original exercise, substituting the values for \( R \), \( X_L \), and \( X_C \) into the formula, we find:

• \( 648 - 415 = 233 \) for reactance difference,
• \( Z = \sqrt{275^2 + 233^2} \approx 360.36 \Omega \).

Thus, this calculated impedance represents how the RCL circuit reacts to an AC source.

Ohm's Law

Ohm's Law is a fundamental principle used to relate voltage, current, and resistance in electrical circuits. The law is expressed as:

• \( V = I \times Z \)

where \( V \) is the voltage, \( I \) is the current, and \( Z \) is impedance. In AC circuits, this equation is particularly useful and helps in calculating how much voltage is needed across a component or a whole circuit to sustain a specific current flow.

In the problem provided, we use Ohm's Law to find the voltage supplied by the generator. Given the current \( I = 0.233 \text{ A} \) and the impedance \( Z \approx 360.36 \Omega \), the voltage is:

• \( V = 0.233 \times 360.36 \approx 84.00 \text{ V} \)

This calculated voltage is what the generator must supply to keep the current constant through the circuit with the given impedance.

Inductive Reactance

Inductive reactance, symbolized as \( X_L \), is a measure of a coil or inductor's opposition to changes in current. It takes into account the frequency of the alternating current and the inductance of the coil. The formula is:

• \( X_L = 2\pi f L \)

where \( f \) is the frequency and \( L \) is the inductance.

In an AC circuit, inductors create a phase shift between voltage and current. This phase shift results in energy being temporarily stored in the magnetic field and then returned to the circuit. The opposition to AC, the inductive reactance \( X_L \), affects the overall impedance \( Z \). For the exercise, it affects the total impedance calculation as given.

It's important to note that unlike regular resistive opposition, inductive reactance is frequency-dependent, increasing as the frequency of the AC source increases. The given value \( 648 \Omega \) highlights the significant role of this component in opposition.

Capacitive Reactance

Capacitive reactance, symbolized as \( X_C \), is the opposition that a capacitor presents to the flow of alternating current. It is a function of the capacitance and the frequency of the AC source. Mathematically, it is expressed as:

• \( X_C = \frac{1}{2\pi f C} \)

where \( C \) is the capacitance.

Capacitors store energy in an electric field, and this energy gives rise to capacitive reactance in AC circuits. This introduces a phase shift where the current leads the voltage in phase. Capacitive reactance decreases with an increase in frequency, as opposed to inductive reactance.

In the exercise, \( X_C = 415 \Omega \) indicates how much opposition is provided by the capacitive element. It plays a key role in determining the net reactance and impedance. Understanding \( X_C \) is vital, as it influences how the circuit responds at different frequencies.