(II) A capacitor is placed in parallel with some device, B, as in Fig. 30–18b, to filter out stray high-frequency signals, but to allow ordinary 60.0-Hz ac to pass through with little loss. Suppose that circuit B in Fig. 30–18b is a resistance R = 530 Ω connected to ground, and that C = 0.35 μF. Calculate the ratio of the capacitor’s current amplitude to the incoming current’s amplitude if the incoming current has a frequency of (a) 60.0 Hz; (b) 60.0 kHz.

Verified step by step guidance

1. Step 1: Understand the problem setup. The capacitor is in parallel with a resistor, and the goal is to calculate the ratio of the capacitor's current amplitude to the incoming current amplitude for two different frequencies: 60.0 Hz and 60.0 kHz. This involves analyzing the impedance of the capacitor and resistor at these frequencies.
2. Step 2: Recall the formula for the capacitive reactance, which is given by \( X_C = \frac{1}{2 \pi f C} \), where \( f \) is the frequency and \( C \) is the capacitance. This reactance determines how the capacitor responds to alternating current at different frequencies.
3. Step 3: Calculate the impedance of the capacitor \( Z_C \) using \( Z_C = X_C \), and the impedance of the resistor \( Z_R \) is simply \( R \). Since the capacitor and resistor are in parallel, the total impedance \( Z_{total} \) can be calculated using the formula for parallel impedances: \( \frac{1}{Z_{total}} = \frac{1}{Z_C} + \frac{1}{Z_R} \).
4. Step 4: Determine the current amplitude through the capacitor \( I_C \) using Ohm's Law: \( I_C = \frac{V}{Z_C} \), where \( V \) is the voltage across the circuit. Similarly, the incoming current amplitude \( I_{total} \) is given by \( I_{total} = \frac{V}{Z_{total}} \). The ratio of the capacitor's current amplitude to the incoming current amplitude is \( \frac{I_C}{I_{total}} \).
5. Step 5: Substitute the given values (\( R = 530 \ \Omega \), \( C = 0.35 \ \mu F \), \( f = 60.0 \ Hz \) and \( f = 60.0 \ kHz \)) into the formulas derived in the previous steps. Perform the calculations for each frequency to find the ratio \( \frac{I_C}{I_{total}} \) for both cases.

Key Concepts

Capacitive Reactance

Capacitive reactance (Xc) is the opposition that a capacitor offers to alternating current (AC) due to its capacitance. It is frequency-dependent and is calculated using the formula Xc = 1/(2πfC), where f is the frequency and C is the capacitance. At lower frequencies, capacitive reactance is higher, which means the capacitor allows less current to pass through, while at higher frequencies, it decreases, allowing more current.

Impedance in AC Circuits

Impedance (Z) in AC circuits is the total opposition to current flow, combining both resistance (R) and reactance (X). For a circuit with a resistor and a capacitor in parallel, the impedance can be calculated using the formula Z = R || Xc, where '||' denotes the parallel combination. Understanding impedance is crucial for analyzing how much current flows through each component at different frequencies.

Current Amplitude Ratio

The current amplitude ratio compares the current flowing through the capacitor to the total incoming current. This ratio is influenced by the frequency of the incoming signal and the capacitive reactance. By calculating the individual currents at different frequencies, one can determine how effectively the capacitor filters out high-frequency signals while allowing lower frequencies, like 60 Hz, to pass with minimal loss.