91Ó°ÊÓ

In an AC circuit, capacitive reactance (X_C) is an important concept to understand. It describes the opposition that a capacitor gives to the flow of alternating current (AC). Capacitive reactance is defined by the formula:\[X_C = \frac{1}{2\pi f C}\]where:

• \( f \) is the frequency of the AC supply, measured in Hertz (Hz).
• \( C \) is the capacitance, measured in Farads (F).

The higher the frequency or the smaller the capacitance, the smaller the capacitive reactance. This means that X_C is inversely proportional to both the frequency and the capacitance. In the given exercise, after adding a second capacitor in series, the total capacitance is halved, which causes the capacitive reactance to increase. This increase in X_C signifies that the circuit now opposes AC flow more strongly than before due to the altered capacitance.

Impedance (Z) is the overall resistance that a circuit presents to the flow of alternating current. It takes into account both the resistance and the reactance elements present in the circuit. For an RLC circuit, which consists of resistors, inductors, and capacitors in series, impedance is calculated as:\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]Where:

• \( R \) is the resistance.
• \( X_L \) is the inductive reactance.
• \( X_C \) is the capacitive reactance.

In the discussed case, when the capacitive reactance is increased by adding another capacitor, it influences the impedance. A higher capacitive reactance increases the impedance since it contributes more significantly to the difference (X_L - X_C). Thus, the overall opposition that the circuit presents to the AC source increases.

The current amplitude (I) in an AC circuit indicates the maximum instantaneous value of the current passing through the circuit. It is determined by the impedance and the peak voltage (\mathscr{E}_m) of the AC source via Ohm's Law for AC systems:\[I = \frac{\mathscr{E}_m}{Z}\]The current amplitude depends inversely on the impedance; the higher the impedance, the lower the current amplitude. Hence, in our situation, as impedance increases due to the addition of the second capacitor, we observe a decrease in current amplitude. The circuit thus delivers less peak current compared to before because the increased opposition (higher impedance) reduces the flow of AC through the circuit.

An RLC circuit is a circuit consisting of a Resistor ( R ), an Inductor ( L ), and a Capacitor ( C ) connected in series or parallel. In this type of circuit, each component exhibits distinctive behaviors:

• The resistor impedes both AC and DC equally and converts electrical energy into heat.
• The inductor provides inductive reactance, hindering the change in current and storing energy as a magnetic field.
• The capacitor yields capacitive reactance, limiting the change in voltage and holding electrical energy in an electric field.

For a series RLC circuit as referenced in the exercise, the interplay of these components determines the net impedance experienced by the circuit. When a new capacitor is added, the characteristics of interacting elements change significantly, thus impacting both the capacitive and total impedance of the circuit.

Inductive reactance (X_L) expresses how much an inductor opposes the flow of alternating current. It is directly proportional to both the frequency and the inductance. Inductive reactance is calculated with the formula:\[X_L = 2\pi f L\]where:

• \( f \) is the frequency.
• \( L \) is the inductance, measured in Henrys (H).

In the original problem, the inductive reactance was calculated as 377 \, \Omega for a frequency of 400 \, \rm{Hz}. Because X_L is not affected by capacitance alterations, it remains the same even after modifications are made to the capacitors. Inductive reactance plays a crucial role together with capacitive reactance to determine the overall impedance (Z) of the RLC circuit and thus the behavior of current flow.