Which graph shows the correct variation of r.m.s. current $I$ with frequency $f$ of a.c. in case of (LCR) parallel resonance circuit?
Show Hint
A parallel LCR resonant loop is often called a rejector circuit because it actively blocks or rejects current at its resonant frequency, creating a sharp valley in the current graph. A series LCR loop is an acceptor circuit and forms the exact opposite shape—a sharp current peak.
Step 1: Understanding the Question:
The question asks us to identify the correct graphical curve that properly displays the relationship between the root-mean-square current ($I$) and the source frequency ($f$) inside a parallel LCR resonant circuit network.
Step 2: Key Formula or Approach:
1. In a series resonant circuit, total impedance reaches a minimum at resonance, causing current to peak.
2. Conversely, in a parallel LCR circuit, the combination of inductor and capacitor branches exhibits a unique behavior: at the exact resonant frequency ($f_r$), the net admittance drops to a minimum because the individual branch currents cancel each other out. This causes the total network impedance ($Z$) to surge to a maximum:
$$Z_{max} = \frac{L}{CR}$$
3. Since total current is inversely proportional to circuit impedance ($I = \frac{V}{Z}$), a maximum in impedance forces the line current to drop to a minimum at resonance.
Step 3: Detailed Explanation:
Let's analyze how the current behaves across different frequency domains for a parallel LCR filter loop:
* At very low frequencies ($f \rightarrow 0$), the inductive reactance drops ($X_L = \omega L \rightarrow 0$), which allows a large current to pass easily through the inductor branch.
* At very high frequencies ($f \rightarrow \infty$), the capacitive reactance drops ($X_C = \frac{1}{\omega C} \rightarrow 0$), which allows a large current to pass easily through the capacitor branch.
* At the exact resonant frequency ($f = f_r$), the parallel assembly reaches maximum impedance, creating an anti-resonant block that minimizes the net current drawn from the source.
Therefore, the graph must form a dip or a valley shape, starting high, dipping to a minimum at $f_r$, and rising high again. Graph curve Q represents this anti-resonance dip shape perfectly.
Step 4: Final Answer:
The graph showing the correct variation is curve Q, corresponding to option (A).
