From mechanics, you may recall that when the acceleration of an object is proportional to its coordinate, d2xdt2=−kmx=−ω2x , such motion is called simple harmonic motion, and the coordinate depends on time as x(t)=Acos(ωt+ϕ), where ϕ, the argument of the harmonic function at t=0, is called the phase constant. Find a similar expression for the charge q(t) on the capacitor in this circuit. Do not forget to determine the correct value of ϕ based on the initial conditions described in the problem. Express your answer in terms of q0 , L, and C. Use the cosine function in your answer.

1) The circuit of this problem is a system formed by an Ac voltage source and a capacitor, in this case all the voltage of the source is equal to the voltage at the terminals of the capacitor

ΔV = Δ[tex]V_{C}[/tex]

we assume that the source has a voltage of the form

ΔV = ΔV₀o sin wt

The capacitance of a capacitor is

C = q / ΔV

q = C ΔV sin wt

the current in the circuit is

i = dq / dt

i = c ΔV₀ w cos wt

if we use

cos wt = sin (wt + π / 2)

we make this change by being a resonant oscillation

we substitute

i = w C ΔV₀ sin (wt + π/2)

With this answer we see that the current in capacitor has a phase factor of π/2 with respect to the current

2) Another possible circuit is an LC circuit.

In this case the voltage alternates between the inductor and the capacitor

V_{L} + V_{C} = 0

L di / dt + q / C = 0

the current is

i = dq / dt

they ask us for a solution so that

L d²q / dt² + 1 / C q = 0

d²q / dt² + 1 / LC q = 0

this is a quadratic differential equation with solution of the form

q = A sin (wt + Ф)

to find the constant we derive the proposed solution and enter it into the equation

di / dt = Aw cos (wt + Ф)

d²i / dt²= - A w² sin (wt + Ф)

- A w² + 1 /LC A = 0

w = √ (1 / LC)

To find the phase factor, for this we use the initial conditions for t = 0

in the case of condensate for t = or the charge is zero

0 = A sin Ф

Ф = 0

q = q₀ sin (wt)