In FIGURE P32.54, what is the current supplied by the emf when (a) the frequency is very small and (b) the frequency is very large?

Verified step by step guidance

1. Analyze the circuit in FIGURE P32.54. It likely contains an emf source, a resistor (R), an inductor (L), and/or a capacitor (C). The behavior of the circuit depends on the frequency of the emf source.
2. For part (a), when the frequency is very small (approaching 0 Hz), the inductor behaves like a short circuit (its reactance, \(X_L = \omega L\), where \(\omega = 2\pi f\), approaches 0), and the capacitor behaves like an open circuit (its reactance, \(X_C = \frac{1}{\omega C}\), approaches infinity). Analyze the circuit under these conditions to determine the current.
3. For part (b), when the frequency is very large (approaching infinity), the inductor behaves like an open circuit (its reactance, \(X_L = \omega L\), becomes very large), and the capacitor behaves like a short circuit (its reactance, \(X_C = \frac{1}{\omega C}\), approaches 0). Analyze the circuit under these conditions to determine the current.
4. Use Ohm's Law, \(I = \frac{V}{Z}\), where \(Z\) is the total impedance of the circuit, to calculate the current in each case. For very small frequency, \(Z\) will primarily depend on the resistor and capacitor, while for very large frequency, \(Z\) will primarily depend on the resistor and inductor.
5. Substitute the appropriate values for \(Z\) in each case to express the current as a function of the emf and the circuit components. Simplify the expressions to find the current for both very small and very large frequencies.

Key Concepts

Electromotive Force (emf)

Electromotive force (emf) is the voltage generated by a source, such as a battery or generator, when no current is flowing. It represents the energy per unit charge provided by the source to move charges through a circuit. Understanding emf is crucial for analyzing how it influences current flow in different frequency scenarios.

Impedance

Impedance is the total opposition that a circuit presents to the flow of alternating current (AC) at a given frequency. It combines resistance and reactance, which varies with frequency. At low frequencies, impedance is primarily resistive, while at high frequencies, reactance becomes significant, affecting the current supplied by the emf.

AC Circuit Behavior

The behavior of AC circuits changes with frequency due to the presence of inductors and capacitors, which introduce reactance. At very low frequencies, capacitive reactance is high, limiting current flow, while at very high frequencies, inductive reactance dominates. Understanding these behaviors is essential for predicting how current will respond to varying frequencies in the circuit.