User:Larryv/sandbox

Math
Charge on capacitor at time $$t$$:

$$Q(t) = Q_0 \cos \frac{t}{\sqrt{LC}}$$

Differentiating this gives us the current:

$$ I(t) = - \frac{dQ}{dt} = \frac{Q_0}{\sqrt{LC}} \sin \frac{t}{\sqrt{LC}} $$

Differentiating this again lets us determine the inductor's induced EMF:

$$ \mathcal{E} = -L \frac{dI}{dt} = -L \left( \frac{Q_0}{LC} \cos \frac{t}{\sqrt{LC}} \right)= - \frac{Q_0}{C} \cos \frac{t}{\sqrt{LC}} $$

From Faraday's Law:

$$\mathcal{E} = -N \frac{d\phi_\mathrm{B,S}}{dt}$$

$$ -N \frac{d\phi_\mathrm{B,S}}{dt} = - \frac{Q_0}{C} \cos \frac{t}{\sqrt{LC}}$$

Solving for $$d \phi_\mathrm{B,S} / dt$$ and integrating gives us

$$\phi_\mathrm{B,S} = \frac{Q_0 \sqrt{L}}{N \sqrt{C}} \sin \frac{t}{\sqrt{LC}}$$