A 10-µF capacitor is plugged into a 110 V-rms 60-Hz voltage source, with an ammeter in series. What is the rms value of the current through the capacitor?

1. Break Down the Problem

To find the rms value of the current through the capacitor, we need to:

1. Calculate the capacitive reactance $X_C$ of the capacitor.
2. Use Ohm's Law for AC circuits to find the current $I_{rms}$ flowing through the capacitor.

2. Relevant Concepts

The formulas we will use are:

1. Capacitive reactance: $X_C = \frac{1}{2 \pi f C}$ where:

   • $f$ is the frequency (60 Hz)
   • $C$ is the capacitance (10 µF = $10 \times 10^{-6}$ F)

2. Current through the capacitor in an AC circuit: $I_{rms} = \frac{V_{rms}}{X_C}$

3. Analysis and Detail

1. Calculate the Capacitive Reactance $X_C$: $X_C = \frac{1}{2 \pi (60 \, \text{Hz})(10 \times 10^{-6} \, \text{F})}$ $X_C = \frac{1}{2 \pi (60)(10 \times 10^{-6})} = \frac{1}{0.000037699} \approx 2652.58 \, \Omega$

2. Calculate the rms Current $I_{rms}$: $I_{rms} = \frac{V_{rms}}{X_C} = \frac{110 \, \text{V}}{2652.58 \, \Omega} \approx 0.0414 \, \text{A}$

4. Verify and Summarize

• Capacitive reactance was calculated correctly using the relevant formula.
• The rms current was found by applying Ohm's Law to the AC voltage and capacitive reactance.

Final Answer

The rms value of the current through the capacitor is approximately $0.0414 \, \text{A}$.