At the resonant frequency f0=1 MHz, a series resonant circuit with R=50 Ωhas |𝐕R|=2 V and |𝐕L|=2

step 1 In this problem I can write the value of mod of i is equal to mod of VR by r which is equal to 2

At the resonant frequency f0=1 MHz, a series resonant...

At the resonant frequency $f_{0}=1 \mathrm{MHz}$, a series resonant circuit with $R=50 \Omega$ has $\left|\mathbf{V}_{R}\right|=2 \mathrm{V}$ and $\left|\mathbf{V}_{L}\right|=20 \mathrm{V}$. Determine the
values of $L$ and $C$. What is the value of $\left|\mathbf{V}_{C}\right| ?$

Key Concepts

Series Resonance

In a series RLC circuit at resonance, the inductive and capacitive reactances exactly cancel each other out, meaning that the total impedance of the circuit is purely resistive. This cancellation is the core principle behind resonance, ensuring that the circuit draws maximum current for a given voltage and that the individual reactive voltages may be much higher than the applied voltage despite their cancelation in the overall voltage drop.

Resonant Frequency

The resonant frequency is the frequency at which the inductive reactance equals the capacitive reactance. It is given by the formula f? = 1/(2??(LC)). This formula connects the values of the inductance and capacitance to the frequency at which resonance occurs, making it a fundamental tool in designing and analyzing RLC circuits.

Inductive and Capacitive Reactance

Inductive reactance (X_L) is given by 2?fL and quantifies how an inductor opposes changes in current, while capacitive reactance (X_C) is given by 1/(2?fC) and represents how a capacitor resists changes in voltage. At resonance, these two reactances are equal in magnitude, ensuring the net reactive component of the circuit’s impedance is zero.

Quality Factor (Q)

The quality factor Q is a measure of how 'sharp' or selective the resonance is in an RLC circuit. In a series resonant circuit, Q is defined as the ratio of the reactive voltage (either inductive or capacitive) to the resistive voltage, that is, Q = X/R. It indicates the factor by which the voltages across the reactive components are amplified compared to the applied voltage.

Voltage Magnification in Resonant Circuits

In a resonant circuit, even though the applied voltage is dropped entirely across the resistor at resonance, the voltages across the inductor and capacitor can be much higher. These magnified voltages are proportional to the quality factor (Q) of the circuit. The phenomenon of voltage magnification arises because the energy oscillates between the magnetic field of the inductor and the electric field of the capacitor.

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