1. Natural Resonant Frequency

In a series RLC circuit, the resonance occurs when the inductive reactance equals the capacitive reactance ($X_L = X_C$). The resonant frequency $f_0$ is given by: $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$

2. Frequency for Maximum Voltage across Capacitor ($f_C$)

The voltage across the capacitor $V_C$ is given by the product of the current $I$ and the capacitive reactance $X_C = \frac{1}{\omega C}$. $$V_C = I X_C = \frac{V}{\sqrt{R^2 + (\omega L – \frac{1}{\omega C})^2}} \cdot \frac{1}{\omega C}$$ $$V_C = \frac{V}{\omega C \sqrt{R^2 + (\omega L – \frac{1}{\omega C})^2}} = \frac{V}{\sqrt{\omega^2 C^2 R^2 + (\omega^2 LC – 1)^2}}$$

To find the maximum $V_C$, we must minimize the denominator. Solving $\frac{d}{d\omega}(\text{denominator}) = 0$ yields the frequency $f_C$: $$f_C = f_0 \sqrt{1 – \frac{R^2 C}{2L}}$$ Since the term inside the square root is less than 1, it implies: $$f_C < f_0 \implies f_C < \frac{1}{2\pi\sqrt{LC}}$$

3. Frequency for Maximum Voltage across Inductor ($f_L$)

The voltage across the inductor $V_L$ is given by $I X_L$ where $X_L = \omega L$. $$V_L = I X_L = \frac{V \omega L}{\sqrt{R^2 + (\omega L – \frac{1}{\omega C})^2}}$$ Similar to the capacitor derivation, maximizing this function with respect to frequency yields $f_L$: $$f_L = \frac{f_0}{\sqrt{1 – \frac{R^2 C}{2L}}}$$ Since the denominator is less than 1, the value of $f_L$ will be greater than $f_0$: $$f_L > f_0 \implies f_L > \frac{1}{2\pi\sqrt{LC}}$$