Answer
$0.152~~$ of a period must elapse.
Work Step by Step
If the energy stored in the electric field is 50.0% of the energy stored in the magnetic field, then the electric field stores $\frac{1}{3}$ of the total energy and the magnetic field stores $\frac{2}{3}$ of the total energy at that moment.
In part (a), we found that the charge on the capacitor is $~~0.577~Q$
We can use an equation for the charge on the capacitor to find an expression for $t$:
$q = Q~cos~\omega t = 0.577~Q$
$cos~\omega t = 0.577$
$\omega t = cos^{-1}~(0.577)$
$\omega t = 0.955~rad$
$t = \frac{0.955~rad}{\omega}$
We can express $t$ as a fraction of the period $T$:
$\frac{t}{T} = \frac{0.955/\omega}{2\pi/\omega} = \frac{0.955}{2\pi} = 0.152$
$0.152~~$ of a period must elapse.
