Question

What is meant by Wattless current? A capacitor of 15 \(\mu\)F is connected to an AC source of 220 V and 50 Hz. Find out reactance of circuit and rms value of AC current.

Hint:

Remember the mnemonics for phase in AC circuits: "ELI the ICE man". For an inductor (L), EMF (E) leads Current (I). For a capacitor (C), Current (I) leads EMF (E). This helps you remember that for a capacitor, current leads voltage by 90\(^\circ\), which is why the current is wattless in a purely capacitive circuit.

Answer 1

Meaning of Wattless Current:
Wattless current is the component of an alternating current in a circuit that does not contribute to the average power consumed over a full cycle. The average power in an AC circuit is given by \(P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi\), where \(\phi\) is the phase angle between the voltage and current. If the phase angle \(\phi\) is 90\(^\circ\) (as in a purely inductive or purely capacitive circuit), then \(\cos\phi = \cos(90^\circ) = 0\). This makes the average power consumed zero. The current that flows in such a circuit is called wattless current because it flows without any net dissipation of power. It corresponds to the component of current that is in quadrature (90\(^\circ\) out of phase) with the voltage.
Calculations:

Step 1: Understanding the Concept and Formulas:
We have a purely capacitive AC circuit. We need to find the capacitive reactance (\(X_C\)), which is the opposition to the current flow, and then use Ohm's law for AC circuits to find the RMS current (\(I_{\text{rms}}\)). \begin{itemize} \item Capacitive Reactance: \(X_C = \frac{1}{2\pi f C}\) \item RMS Current: \(I_{\text{rms}} = \frac{V_{\text{rms}}}{X_C}\) \end{itemize}

Step 2: Detailed Explanation:
We are given: \begin{itemize} \item Capacitance, \(C = 15 \, \mu\text{F} = 15 \times 10^{-6} \, \text{F}\). \item RMS Voltage, \(V_{\text{rms}} = 220 \, \text{V}\). \item Frequency, \(f = 50 \, \text{Hz}\). \end{itemize} Part 1: Calculate the Reactance (\(X_C\)) of the circuit \[ X_C = \frac{1}{2\pi f C} = \frac{1}{2\pi (50)(15 \times 10^{-6})} \] \[ X_C = \frac{1}{100\pi \times 15 \times 10^{-6}} = \frac{1}{1500\pi \times 10^{-6}} = \frac{10^6}{1500\pi} \] Using \(\pi \approx 3.14159\): \[ X_C \approx \frac{1,000,000}{1500 \times 3.14159} \approx \frac{1,000,000}{4712.385} \approx 212.2 \, \Omega \] Part 2: Calculate the RMS value of AC current (\(I_{\text{rms}}\)) \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{X_C} = \frac{220 \, \text{V}}{212.2 \, \Omega} \] \[ I_{\text{rms}} \approx 1.037 \, \text{A} \]

Step 3: Final Answer:
The reactance of the circuit is approximately 212.2 \(\Omega\), and the rms value of the AC current is approximately 1.04 A.