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In an AC circuit, inductive reactance describes how an inductor opposes the change in current. This is due to the inductor's property of storing energy in a magnetic field as current changes. Physically, it's described by the formula \( X_L = 2\pi f L \), where \( X_L \) is inductive reactance, \( f \) is the frequency, and \( L \) is the inductance. Reactance has units of ohms (\( \Omega \)).
For our exercise, the reactance of the inductor is given as \( 900 \ \Omega \).

• When the current flows through the inductor, it lags the voltage by 90 degrees due to this reactance.
• This reactance is crucial for determining the voltage across the inductor when analyzing AC circuits.

In the context of the problem, using the given inductive reactance allows us to find that the current through the circuit is 0.5 A when the voltage across the inductor is 450 V. This mirrors the inductor's natural opposition to current changes.

Capacitive reactance in an AC circuit refers to how a capacitor resists the change in voltage across it. It happens because the capacitor stores energy in an electric field. The formula for capacitive reactance is \( X_C = \frac{1}{2\pi f C} \), where \( X_C \) is the capacitive reactance, \( f \) is frequency, and \( C \) is capacitance. Like inductive reactance, capacitive reactance is measured in ohms (\( \Omega \)).
In the given exercise, the capacitor has a reactance of \( 500 \ \Omega \).

• This causes the current in a capacitor to lead the voltage by 90 degrees.
• Knowing this reactance helps to determine the voltage across the capacitor, especially in our problems involving series circuits.

For the circuit involved in the problem, with a current of 0.5 A determined from the inductor's reactance, the voltage across the capacitor is calculated to be 250 V. It's important to note the phase relationship between different components to solve such circuits accurately.

Ohm's Law is a fundamental principle used to relate voltage, current, and resistance in electrical circuits. It is expressed simply as \( V = I \times R \), where \( V \) represents voltage, \( I \) current, and \( R \) resistance.
In our scenario, Ohm's Law allows us to find the voltage across the resistor. The current, calculated as 0.5 A using the inductor's law, is multiplied by the resistor's value of \( 300 \ \Omega\), yielding a voltage of 150 V.

• This law is essential as it applies to each component in a circuit, not just the whole system.
• It supports our understanding of how individual components affect overall circuit behavior.

By using Ohm's Law, we can also calculate the power dissipated in each resistance as shown in the solution, aiding in a complete analysis of AC circuits.

Phasor addition is crucial in analyzing AC circuits because it allows the combination of different voltage or current vectors, considering both magnitude and phase.
In an AC circuit with sinusoidal sources, voltages and currents are often out of phase due to the reactive components - inductors and capacitors.

• Phasors transform time-varying signals into a format that simplifies the mathematics of adding them.
• Instead of directly adding voltages across a capacitor, resistor, and inductor, we consider their phase differences.

The exercise includes calculating the source voltage using phasor addition, which results in a total impedance calculation \( Z = \sqrt{R^2 + (X_L - X_C)^2} \).
This allows us to determine the source voltage amplitude, given the current. Understanding phasor addition is vital for combining AC signals correctly and predicting the behavior of AC circuits comprehensively.