A 2.2-nF capacitor and one of unknown capacitance are in...

A 2.2-nF capacitor and one of unknown capacitance are in parallel across a $10-\mathrm{V}$ rms sine-wave generator. At $1.0 \mathrm{kHz}$, the generator supplies a total current of $3.4 \mathrm{~mA} \mathrm{rms}$. The generator frequency is then decreased until the rms current drops to $1.2 \mathrm{~mA}$. Find (a) the unknown capacitance and (b) the lower frequency.

Key Concepts

Capacitive Reactance

Capacitive reactance is the opposition that a capacitor offers to the flow of alternating current. It is given by the formula X_C = 1/(2?fC), where f is the frequency and C is the capacitance. This concept is crucial for understanding how the current through a capacitor changes with frequency, making it a key element in AC circuit analysis.

RMS Values in AC Circuits

In AC circuits, RMS (root mean square) values are used to represent the equivalent DC values that would deliver the same power. RMS voltage and current are therefore used to analyze and compare the performance of AC circuits, and they are fundamental to calculating current, voltage, and power in such circuits.

Parallel Capacitor Behavior

When capacitors are connected in parallel, they share the same voltage across them while the total current in the circuit is the sum of the currents through each capacitor. This behavior is important when determining unknown capacitor values based on the total current drawn from a source, as the individual branch currents depend on their respective capacitances and the applied frequency.

Frequency Dependence of Capacitor Behavior

The operation of capacitors in AC circuits is highly dependent on frequency, as the capacitive reactance decreases with increasing frequency. This relationship means that variations in frequency will cause corresponding changes in the current flowing through each capacitor, making frequency a critical parameter when designing and analyzing circuits involving capacitive elements.

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