91Ó°ÊÓ

Energy in the electric field is $50.0\%$of that in the magnetic field.

Using the relation between the total energy, electrical energy in the capacitor, and magnetic energy in the inductor, find the charge when the energy in the electric field is $50.0\%$of that in the magnetic field. By using the equation of time-dependent charge on the capacitor, find the fraction of the period that must elapse following the time when the capacitor is fully charged.

The formula is as follows:

$U=U_E+U_B$

$U=\frac{Q^2}{\text{2}C}$

$q=Q\cos \mathrm{Ó\neg }t$

Where, U is potential energy, Q, q is charges.

If $U_E$is the electrical energy in the capacitor and $U_{B_{}}$is the magnetic energy in the inductor, then the total energy in the LCcircuit is given by$U=U_E+U_B$

According to the given condition in the problem,

At time $U_E=\frac{\text{50}}{\text{100}}{U}_B=\frac{\text{1}}{\text{2}}{U}_B$, i.e.$U_B=\text{2}{U}_E$

Therefore, the above equation becomes,

$U=U_E+\text{2}{U}_E=\text{3.00}{U}_E$

To charge the capacitor at the time ,

$U_E=\frac{q^2}{\text{2}C}$

And the total energy is,

$U=\frac{Q^2}{\text{2}C}$

As$U=\text{3.00}{U}_E$

Substituting the values of $U$and $U_E$,

$\begin{array}{rcl}\frac{Q^2}{\text{2}C}& =& \text{3.00}\frac{q^2}{\text{2}C}\\ q^2& =& \frac{Q^2}{\text{2}C}\frac{\text{2}C}{\text{3.00}}\\ q& =& \frac{Q}{\sqrt{\text{3.00}}}\\ q& =& \text{0.577}Q\end{array}$

Therefore, the charge when the energy in the electric field is $50.0\%$of that in the magnetic field $0.577Q$.

The time-dependent charge on the capacitor is given by,

$q=Q\text{cos}\mathrm{Ó\neg }t$

But for a fully charged capacitor,

$q=\text{0.577}Q$

Therefore,

role="math" localid="1663159650227" $\begin{array}{rcl}Q\cos \mathrm{Ó\neg }t& =& 0.577Q\\ \cos \mathrm{Ó\neg }t& =& 0.577\\ \mathrm{Ó\neg }t& =& 0.955rad\end{array}$

But it is known,

$\mathrm{Ó\neg }=\frac{\text{2}\mathrm{Ï€}}{T}$

Therefore,

$\begin{array}{rcl}\frac{\text{2}\mathrm{Ï€}t}{T}& =& \text{0.955}\\ \frac{t}{T}& =& \frac{\text{0.955}}{\text{2}\mathrm{Ï€}}\\ \frac{t}{T}& =& \text{0.152}\end{array}$

Therefore, the fraction of the period elapsed following the time when the capacitor is fully charged is $0.152$.

Find the charge when the energy in the electric field is of that in the magnetic field using the relation between the total energy, electrical energy in the capacitor, and magnetic energy in the inductor. Also, find the fraction of the period elapsed following the time when the capacitor is fully charged using the equation of time-dependent charge on the capacitor.