Differential Equation. How can I get the values of L and C? I don´t know how to

Question

consider an LC system

with a source of potential difference E(t) = E₀cos(ωt)

The equation for the voltage V = Vc across the capacitor is

LC (d²V ⁄ dt² )+ V = E₀cos(ωt)

Given the frequency ω, find the values of L and C so that there can be resonance in the system.

Answer 1

This is a differential equation describing a driven harmonic oscillator.

Its solution is of the form

V = Acos(ωt) (1)

for some amplitude A.

If this exercise is about deriving (1) from first principles, then

I am sorry, I cannot help you.

If this exercise is about finding the value of LC that maximizes A,

i.e., produces resonance, then read further.

Plug (1) into the differential equation

-LCω²Acos(ωt) + Acos(ωt) = E₀cos(ωt)

Express A in terms of the other quantities

A = E₀/(1 - LCω²) (2)

A becomes infinite when

1 - LCω² = 0

LC = 1/ω²

This is the value of LC than maximizes A.

In practice A does not become infinity because there is resistance not

modeled by your differential equation.

But that is the value of LC that will tune the circuit to ω.

The wording of the question sounds as if you were supposed to find separate values of L and C. In reality only their product matters.

When tuning an old fashioned radio to some frequency, you are changing the capacitance only.