If the capacitance C in a parallel RLC circuit decreases, what happens to resonant frequency f₀? 

Explanation:

Resonant Frequency in a Parallel RLC Circuit

Definition: Resonant frequency (f₀) in a parallel RLC circuit is the frequency at which the inductive reactance and capacitive reactance become equal in magnitude, causing the circuit to exhibit purely resistive behavior. At this frequency, the impedance of the circuit is maximized, and the circuit resonates.

Formula: The resonant frequency (f₀) for a parallel RLC circuit is given by the following equation:

f₀ = 1 / (2 × π × √(L × C))

Where:

• L: Inductance of the inductor (in henries, H)
• C: Capacitance of the capacitor (in farads, F)

Working Principle: Resonance occurs in a parallel RLC circuit when the energy alternates between the inductor and capacitor, creating oscillations at the resonant frequency. At this point, the inductive reactance (X_L) and capacitive reactance (X_C) cancel each other out, leading to the circuit behaving as though only the resistance is present.

Impact of Decreasing Capacitance (C):

From the formula for resonant frequency (f₀):

f₀ = 1 / (2 × π × √(L × C))

It is evident that the resonant frequency is inversely proportional to the square root of the product of inductance (L) and capacitance (C). Specifically:

• If C decreases, the denominator (√(L × C)) becomes smaller.
• This causes the value of f₀ to increase because the resonant frequency is inversely proportional to the square root of C.

Therefore, when the capacitance (C) in a parallel RLC circuit decreases, the resonant frequency (f₀) increases.

Correct Option Analysis:

The correct option is:

Option 2: Increases

This option correctly identifies the effect of decreasing capacitance on the resonant frequency in a parallel RLC circuit. As explained above, a reduction in capacitance leads to an increase in the resonant frequency due to their inverse relationship in the resonant frequency formula.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Decreases

This option is incorrect because decreasing capacitance does not cause the resonant frequency to decrease. As shown in the formula, resonant frequency is inversely proportional to the square root of capacitance (C); hence, a reduction in C increases the value of f₀.

Option 3: Remains constant

This option is incorrect because the resonant frequency depends on the values of both inductance (L) and capacitance (C). Changes in either L or C will affect the resonant frequency. If capacitance decreases, f₀ will increase, as explained earlier.

Option 4: Becomes zero

This option is incorrect because the resonant frequency cannot become zero simply by decreasing capacitance. Even with very small values of C, the resonant frequency remains finite and increases. Resonant frequency becoming zero would imply the absence of oscillation, which is not the case here.

Conclusion:

Understanding the relationship between capacitance and resonant frequency is crucial for analyzing parallel RLC circuits. The resonant frequency is determined by the values of inductance and capacitance. When capacitance decreases, the resonant frequency increases due to their inverse proportionality. This principle is fundamental in electronics and telecommunications, where resonant circuits are used for signal filtering and frequency selection.