In a series circuit with a resistor (R), inductor (L), and capacitor (C), the voltages across each component add up vectorially (because of the phase differences between them). To calculate the supply voltage and the resonance frequency, we need to use the following steps.

Step 1: Supply Voltage Calculation

The total supply voltage $V_s$ in a series RLC circuit is the vector sum of the voltages across R, L, and C:

$V_s = \sqrt{V_R^2 + (V_L - V_C)^2}$

Where:

• $V_R = 5 \, \text{V}$ is the voltage across the resistor.
• $V_L = 8 \, \text{V}$ is the voltage across the inductor.
• $V_C = 10 \, \text{V}$ is the voltage across the capacitor.

Substitute the values:

$V_s = \sqrt{5^2 + (8 - 10)^2}$ $V_s = \sqrt{25 + (-2)^2}$ $V_s = \sqrt{25 + 4} = \sqrt{29}$ $V_s \approx 5.39 \, \text{V}$

So, the supply voltage $V_s$ is approximately 5.39 V.

Step 2: Resonance Frequency Calculation

The resonance frequency $f_0$ of an RLC circuit is given by the formula:

$f_0 = \frac{1}{2 \pi \sqrt{LC}}$

However, we don't have the values of $L$ and $C$ directly. But we can find them using the relationship between the voltages across L and C at resonance. At resonance, the reactances of the inductor and the capacitor are equal in magnitude, but opposite in sign, so the voltage across them cancels out.

The voltage across the inductor $V_L$ and the capacitor $V_C$ are related to the total voltage in the series circuit:

$\frac{V_L}{V_s} = \frac{X_L}{Z}$ $\frac{V_C}{V_s} = \frac{X_C}{Z}$

At resonance, the inductive reactance $X_L$ and the capacitive reactance $X_C$ are equal:

$X_L = X_C$

Given that the values of $V_L$ and $V_C$ are 8V and 10V respectively, the resonance frequency would occur when the total reactance cancels out, and we would use a condition like:

$f_0 = \frac{1}{2 \pi \sqrt{LC}}$

But to fully calculate this, we would need further details about either the inductance or capacitance in the system or specific resonance conditions.

Summary

1. The supply voltage $V_s$ is approximately 5.39 V.
2. The resonance frequency requires additional values of inductance or capacitance, which are not given directly in the question. Further calculations or data are needed.

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Would you like more details on any specific part of this calculation? Or do you have further questions?

Here are some follow-up questions to explore:

1. What is the impedance of an RLC circuit at resonance?
2. How does the phase difference between voltage and current behave at resonance?
3. What is the relationship between impedance and resonance in an RLC circuit?
4. How would you calculate the quality factor $Q$ of this circuit?
5. Can you derive the expression for the resonant frequency in terms of $R$, $L$, and $C$?

Tip: In RLC circuits, resonance occurs when the inductive and capacitive reactances cancel each other out, which leads to minimal impedance and maximum current.