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Alternating current, or AC, is a type of electrical current where the flow of electric charge periodically reverses direction. Unlike direct current (DC), which flows only in a single direction, AC continuously changes its polarity. Electricity in homes and businesses is primarily AC because it can be easily transformed to different voltages. This makes it highly efficient for transmission over long distances. The waveform of AC is often sinusoidal which means it smoothly alternates over time in a smooth and continuous curve. In the given exercise, the function \(E = 200 \sqrt{2} \sin(100t)\ \text{V}\) represents an AC voltage with a peak value of \(200\sqrt{2}\) volts and an angular frequency of \(100\ \text{rad/s}\). Understanding AC is fundamental to analyzing behaviors in AC circuits.

Capacitive reactance, denoted as \(X_C\), is a measure of a capacitor's resistance to changes in voltage in an AC circuit. Distinct from resistance, which applies to DC circuits, reactance applies to AC and varies with frequency. Key aspects to remember about capacitive reactance include:

• It is inversely proportional to the frequency of the AC signal and the capacitance of the capacitor (\(X_C = \frac{1}{\omega C}\)).
• Higher frequencies result in lower reactance, allowing more current to pass through the capacitor.
• In our exercise, using a capacitance \(C = 1 \mathrm{mF}\) and frequency \(\omega = 100\ \text{rad/s}\), \(X_C\) is calculated to be \(10 \ \Omega\).

A solid grasp of capacitive reactance allows proper analysis and prediction of current flow in AC circuits with capacitors.

Ohm's Law, well-known in DC circuits, also applies to AC circuits but is adapted to account for AC-specific elements like reactance and impedances. In AC, instead of just using resistance \(R\), we consider impedance \(Z\), which includes both resistance and reactance (either inductive or capacitive).In capacitive AC circuits, Ohm's Law is expressed as \(I_{rms} = \frac{V_{rms}}{X_C}\), where:

• \(I_{rms}\) is the root mean square (RMS) current.
• \(V_{rms}\) is the RMS voltage.
• \(X_C\) is the capacitive reactance.

Applying Ohm's Law for AC circuits to our capacitor example, and given our calculated \(X_C\) and \(V_{rms}=200 \text{ V}\), the current \(I_{rms}\) is \(20 \text{ mA}\). This highlights how AC circuit analysis utilizes analogous relationships to DC, adjusted for alternating current characteristics.

Root Mean Square (RMS) voltage is a way of expressing the effective voltage of an AC system. While the peak voltage may indicate the maximum potential difference, RMS voltage allows us to understand how much power an AC source will deliver; this is important because power depends on both the square of voltage and current. The important features of RMS voltage are:

• It is calculated as \(V_{rms} = \frac{V_{peak}}{\sqrt{2}}\) for sinusoidal waves.
• Gives the equivalent DC value which provides the same power to a load.
• In the exercise, converting the supplied peak voltage \(200\sqrt{2}\) volts to RMS results in \(200\ \text{V}\).

Understanding RMS voltage is key to adequately analyzing AC circuits, enabling you to predict how much actual work or power the AC system can provide.