Question

The emf of self-inductance in the coil at the moments when the electric energy of the capacitor is equal to that of the current in the coil.

Answer 1

The electric energy of the capacitor is $\frac{q^2}{2C}\alpha {\cos }^2{\omega }_{0}t$and of the inductor is
$\frac{1}{2}L{I}^2\alpha {\sin }^2{\omega }_{0}t$
The two are equal when
${\omega }_{0}t=\frac{\pi }{4}$
At that instant the emf of the self-inductance is
$-L\frac{di}{dt}=V_m\cos {\omega }_{0}t=V_m/\sqrt{2}$