Maximum Charge

Let the capacitor voltage be $V_C$ , and let the inductor current be $I_L$ . The governing equations for the circuit are:

$2E - V_C = L \dot{I}_L \\ I_L = C \dot{V}_C$

Substituting the second equation in to the first yields:

$2E - V_C = L C \ddot{V}_C$

The homogeneous equation is:

$\ddot{V}_C = - \frac{1}{L C} V_C = - \omega^2 V_C \\ \omega = \frac{1}{\sqrt{LC}}$

The homogeneous solution is sinusoidal, and then we must add a constant term as well.

$V_C = A \sin (\omega t ) + B \cos (\omega t) + D \\ \dot{V}_C = A \omega \cos (\omega t ) - B \omega \sin (\omega t) \\ \ddot{V}_C = -A \omega^2 \sin (\omega t ) - B \omega^2 \cos (\omega t)$

Initial conditions:

$V_C (0) = E = B + D\\ \dot{V}_C (0) = 0 = A \omega \\ \ddot{V}_C (0) = \frac{E}{L C} = -B \omega^2 = -\frac{B}{L C}$

Solving for the constants yields:

$A = 0 \\ B = -E \\ D = 2 E$

Finally:

$V_C = -E \cos (\omega t) + 2 E$

The maximum value of $V_C$ is therefore $3 E$ , and the maximum charge is $3 C E$ .

To solve numerically, we only need to consider the basic equations.

$2E - V_C = L \dot{I}_L \\ I_L = C \dot{V}_C$

Isolate the derivative terms and numerically integrate.

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import math

dt = 10.0**(-5.0)

L = 1.0
C = 1.0
E = 1.0

##########################

t = 0.0
count = 0

I = 0.0
VC = E

Id = (2.0*E - VC)/L
VCd = I/C

VCmax = 0.0

while t <= 10.0:

    I = I + Id*dt
    VC = VC + VCd*dt

    Id = (2.0*E - VC)/L
    VCd = I/C

    if VC > VCmax:
        VCmax = VC

    t = t + dt
    count = count + 1

    #if count % 1000 == 0:
        #print t,VC

print ""
print ""

print dt
print VCmax

#1e-05
#3.00004712501

Let $q(0)= CE, q'(0) = 0$ for this oscillator circuit. By Kirchhoff's Voltage Law, we have the differential equation:

$Lq''(t) + \frac{1}{C} q(t) = 2E$ (i)

and taking the Laplace Transform of (i) gives:

$L[s^{2}Q(s) - (CE)s - 0] + \frac{1}{C} \cdot Q(s) = \frac{2E}{s} \Rightarrow Q(s) = \frac{2E + (LCE)s^2}{s(Ls^2 + 1/C)} = \frac{2CE}{s} - \frac{(LCE)s}{Ls^2 + 1/C}$ (ii)

and the inverse Laplace Transform of (ii) yields:

$q(t) = (CE) \cdot [2 - \cos(\frac{t}{\sqrt{LC}})]$

which has a maximum capacitor charge of $q_{MAX} = \boxed{3CE}.$