Question

The a.c circuit exhibiting the phenomenon of resonance has/have the circuit element(s)

• A. inductor and resistor
• B. capacitor and resistor
• C. inductor only
• D. capacitor only
• E. inductor and capacitor

Hint:

Resonance frequency: \(f_r = \frac{1}{2\pi\sqrt{LC}}\). At this frequency, the circuit behaves like a pure resistor.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Electrical resonance is a phenomenon that occurs in an AC circuit when the inductive reactance and capacitive reactance are equal in magnitude. This causes the circuit's impedance to be at a minimum (in a series circuit) or maximum (in a parallel circuit), leading to a significant increase in current or voltage at a specific frequency known as the resonant frequency.
Step 2: Detailed Explanation:
Let's analyze the roles of the circuit elements in an AC circuit:
- Resistor (R): Its impedance (\(Z_R = R\)) is independent of frequency. It only dissipates energy.
- Inductor (L): Its impedance, called inductive reactance (\(X_L\)), is directly proportional to frequency (\(X_L = \omega L = 2\pi f L\)). It stores energy in a magnetic field.
- Capacitor (C): Its impedance, called capacitive reactance (\(X_C\)), is inversely proportional to frequency (\(X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}\)). It stores energy in an electric field.
Resonance occurs when the opposing effects of the inductor and capacitor cancel each other out. This happens when their reactances are equal:
\[ X_L = X_C \] \[ \omega L = \frac{1}{\omega C} \] For this condition to be met, the circuit must contain both an inductor (\(L\)) and a capacitor (\(C\)).
- An RL circuit or an RC circuit cannot resonate because they only have one type of reactance.
- Circuits with only an inductor or only a capacitor cannot resonate.
- A resistor is usually present in practical resonant circuits (like an RLC circuit), but it is not essential for the phenomenon of resonance itself. The core requirement is the presence of both L and C to allow for the exchange of energy between the magnetic and electric fields.
Step 3: Final Answer:
The essential circuit elements required for exhibiting resonance are an inductor and a capacitor. This corresponds to option (E).