Question

An oscillating LC circuit has an inductance of 3.00mHand a capacitance of $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{}\mathit{\mu }\mathit{F}$. Calculate the (a) angular frequency and (b) period of the oscillation. (c) At time t =0, the capacitor is charged to$\mathbf{200}\mathbf{}\mathit{\mu }\mathit{F}$and the current is zero. Roughly sketch the charge on the capacitor as a function of time.

Short answer

a). Angular frequency is$5.77\times {10}^{3}rad/sec.$

b). Period oscillation is $1.09\times {10}^{-3}\sec$

c). The graph is plotted for a charge on the capacitor as a function of time.

Step-by-step solution

Given

1. Inductance is 3.00mH.
2. Capacitance is $10.0\mathrm{\mu F}$.
3. Charge is $200\mu C$.

Determining the concept

Use the formula for angular frequency for LC circuit in terms of inductance and capacitor. Then use the formula for angular frequency and find the time.

The formula is as follows:

(a) Determining the angular frequency

Angular frequency:

In an oscillating LC circuit, angular frequency is given as follows:

$\omega =\frac{1}{\sqrt{LC}}$

$\omega =\frac{1}{\sqrt{3\times {10}^{-3}\times 10\times {10}^{-6}}}$

$\mathrm{\omega }=5.77\times {10}^3\mathrm{rad}/\sec$

Hence, angular frequency is $5.77\times {10}^3\mathrm{rad}/\sec .$.

(b) Determining the period oscillation

Period oscillation:

The time period is as follows:

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ 5.77\times {10}^3=\frac{2\mathrm{\pi }}{T} \\ T=1.09\times {10}^{-3}\sec \end{gathered}$$

Hence, period oscillation is $1.09\times {10}^{-3}\sec$.

(c) Determining the sketch charge on the capacitor as a function of time

Sketch charge on the capacitor as a function of time

The charge on the capacitor is given as follows: