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Impedance in an RLC circuit is the total opposition that the circuit presents to alternating current (AC). It is symbolized as \( Z \), and uniquely combines resistive and reactive elements. The formula to calculate impedance is:

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

Here:

• \( R \) is the resistance in ohms (Ω),
• \( X_L \) is the inductive reactance,
• \( X_C \) is the capacitive reactance.

The RLC circuit combines these components, where the impedance depends on the balance between inductive and capacitive reactance. This relationship means even if resistance is constant, impedance can vary significantly with frequency changes in the circuit. To find the impedance in this AC circuit, we use Ohm's Law \( V = IZ \). So, \( Z \) can be rearranged and calculated as \( Z = \frac{V}{I} \), where voltage \( V \) and current \( I \) are known quantities. With given values, solving gives \( Z \) as \( 184.40 \Omega \).

Inductive reactance \( X_L \) is a measure of an inductor's opposition to a changing current. It's directly dependent on both the frequency of the AC source and the inductance of the coil. The equation for calculating inductive reactance is:

• \[ X_L = 2\pi f L \]

Where:

• \( f \) is the frequency in hertz (Hz),
• \( L \) is the inductance in henrys (H).

Inductive reactance increases with frequency. This means as the frequency rises, the inductor's opposition to current flow also increases. In this exercise, you must find values for two unknown frequencies using the known inductance \( L = 5.42 \times 10^{-3} \) H. By substituting this into the formula, you'll discover how \( X_L \) behaves in relation to different frequencies.

Capacitive reactance \( X_C \) describes how a capacitor resists the change in voltage across its plates in an AC circuit. Unlike inductive reactance, capacitive reactance decreases with an increase in frequency. It is calculated using the formula:

• \[ X_C = \frac{1}{2\pi f C} \]

Where:

• \( f \) is the frequency in hertz (Hz),
• \( C \) is the capacitance in farads (F).

In simpler terms, a higher frequency means the capacitor "reacts" less to the voltage changes, acting like a short circuit at very high frequencies. For this circuit, the capacitance is \( C = 0.200 \times 10^{-6} \) F. Plug this into the formula to find how different frequencies impact \( X_C \).'s role in the overall circuit impedance.

Ohm's Law is fundamental and widely used to analyze circuits, including AC circuits. Unlike DC circuits where resistance is key, in AC circuits, impedance takes a center role. Ohm’s Law for AC circuits can be expressed as:

• \( V = IZ \)

Where:

• \( V \) is the voltage across the circuit,
• \( I \) is the current through the circuit,
• \( Z \) is the impedance as a function of resistance and reactance.

It showcases the relationship between voltage, current, and impedance. Importantly, impedance affects how the current behaves with an applied alternating voltage.
Using Ohm’s Law, given a voltage \( V = 26.0 \) V and a current \( I = 0.141 \) A, you can directly calculate the impedance of the circuit \( Z \). This calculation is crucial for determining factors like frequency that influence current characteristics in an AC circuit, ultimately solving for different current pathways seen in the exercise.