Question

The frequency of the output signal of an LC oscillator circuit is $F\ \text{Hz}$ with a capacitance of $0.1\ \mu\text{F}$. If the value of the capacitor is increased to $0.2\ \mu\text{F}$, then the frequency of the output signal will be

• A. $\frac{F}{\sqrt{2}}\ \text{Hz}$
• B. $\frac{F}{3}\ \text{Hz}$
• C. $\frac{F}{2}\ \text{Hz}$
• D. $2F\ \text{Hz}$

Hint:

Whenever capacitance increases, the resonant frequency must drop because the system takes longer to charge and discharge. Since the capacitance doubled ($0.1 \rightarrow 0.2$), the frequency must decrease by a factor of $\sqrt{2}$. This visual checklist immediately eliminates option (D) since it predicts a frequency increase!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
An $LC$ oscillator circuit generates an AC signal at a specific resonant frequency governed by an inductor and a capacitor. Given an initial frequency $F$ at a capacitance of $0.1\ \mu\text{F}$, we need to find the new output frequency when the capacitance is increased to $0.2\ \mu\text{F}$.

Step 2: Key Formula or Approach:
The resonant frequency ($F$) of a standard ideal $LC$ oscillator circuit is given by the formula: $$F = \frac{1}{2\pi\sqrt{LC}}$$ Since the inductor value $L$ remains completely constant, the frequency is inversely proportional to the square root of the capacitance: $$F \propto \frac{1}{\sqrt{C}}$$

Step 3: Detailed Explanation:
Let's list our initial and final states: Initial capacitance, $C_1 = 0.1\ \mu\text{F}$ Initial frequency, $F_1 = F$ Final capacitance, $C_2 = 0.2\ \mu\text{F}$ Final frequency, $F_2$ Setting up our inverse square root ratio equation: $$\frac{F_2}{F_1} = \sqrt{\frac{C_1}{C_2}}$$ Substitute our known values into the ratio: $$\frac{F_2}{F} = \sqrt{\frac{0.1\ \mu\text{F}}{0.2\ \mu\text{F}}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$$ Isolating the new frequency parameter $F_2$: $$F_2 = \frac{F}{\sqrt{2}}\ \text{Hz}$$

Step 4: Final Answer:
The new frequency of the output signal is $\frac{F}{\sqrt{2}}\ \text{Hz}$, which matches option (A).