A capacitor is connected to a 15 kHz oscillator. The peak current is 65 mA when the rms voltage is 6.0 V. What is the value of the capacitance C?

Verified step by step guidance

1. Step 1: Recall the formula for capacitive reactance, which is given by \( X_C = \frac{1}{2 \pi f C} \), where \( X_C \) is the capacitive reactance, \( f \) is the frequency, and \( C \) is the capacitance. This formula relates the frequency and capacitance to the opposition to current flow in a capacitor.
2. Step 2: Use Ohm's Law for AC circuits, which states \( I = \frac{V}{X_C} \), where \( I \) is the current, \( V \) is the voltage, and \( X_C \) is the capacitive reactance. Rearrange this formula to solve for \( X_C \): \( X_C = \frac{V}{I} \).
3. Step 3: Substitute the given values for the rms voltage (\( V = 6.0 \; \text{V} \)) and the peak current (\( I_{\text{peak}} = 65 \; \text{mA} \)). Note that the rms current \( I_{\text{rms}} \) is related to the peak current by \( I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}} \). Calculate \( I_{\text{rms}} \) first, then use it to find \( X_C \).
4. Step 4: Once \( X_C \) is determined, use the formula \( X_C = \frac{1}{2 \pi f C} \) to solve for the capacitance \( C \). Rearrange the formula to \( C = \frac{1}{2 \pi f X_C} \). Substitute the frequency \( f = 15 \; \text{kHz} = 15,000 \; \text{Hz} \) and the calculated \( X_C \) into this equation.
5. Step 5: Perform the calculations to find the value of \( C \). Ensure that the units are consistent throughout the calculation (e.g., convert mA to A and kHz to Hz). The final result will give the capacitance in Farads (F).

Key Concepts

Capacitance

Capacitance is the ability of a capacitor to store electrical energy in an electric field, measured in farads (F). It is defined as the ratio of the electric charge stored on one plate of the capacitor to the voltage across the plates. In AC circuits, capacitance affects how the capacitor reacts to changing voltages and currents, influencing the overall impedance of the circuit.

RMS Voltage

RMS (Root Mean Square) voltage is a statistical measure of the magnitude of a varying voltage. It represents the equivalent DC voltage that would deliver the same power to a load. In AC circuits, RMS voltage is crucial for calculating power and current, as it provides a consistent value for analysis, especially when dealing with sinusoidal waveforms.

Peak Current

Peak current refers to the maximum instantaneous current flowing through a circuit during one cycle of an AC waveform. It is important for understanding the behavior of components like capacitors in AC circuits, as it helps determine how much current the capacitor can handle at its maximum charge. The relationship between peak current, RMS voltage, and capacitance is essential for calculating the value of the capacitor.