An RLC circuit has a resistance of 200 $\Omega$ and an inductance of 15 mH. Its oscillation frequency is 7000 Hz. At time t = 0 the current is 25 mA and there is no charge on the capacitor. After five complete cycles ther current is: A) 0 B) 1.8 * 10^-⁶ A C) 2.1 *10^-⁴ A D) 2.3 *10^-³ A E) 2.5 * 10^-² A

Question

Quizplus · Accepted Answer

First, we need to find the values of the capacitance and the angular frequency. We can use the formula:
\begin{align*}
\omega &= \frac{1}{\sqrt{LC}} \
C &= \frac{1}{L\omega^2}
\end{align*}

Substituting the given values, we get:
\begin{align*}
\omega &= \frac{1}{\sqrt{(15	imes10^{-3})(200	imes10^3)}} = 377.96	ext{ rad/s} \
C &= \frac{1}{(15	imes10^{-3})(377.96)^2} = 58.3	ext{ nF}
\end{align*}

The time period of the oscillation is given by:
\begin{align*}
T &= \frac{2\pi}{\omega} \
&= \frac{2\pi}{377.96} \
&\approx 16.69	ext{ ms}
\end{align*}

After five complete cycles, the time elapsed is:
\begin{align*}
t &= 5T \
&= 83.46	ext{ ms}
\end{align*}

Using the formula for the current in an RLC circuit, we get:
\begin{align*}
i(t) &= \frac{V_0}{R}\cos(\omega t - \phi) \
&= \frac{Q_0}{C}\cos(\omega t) e^{-\frac{R}{L}t}
\end{align*}

At time t=0, the current is given as 25mA, which means:
\begin{align*}
i(0) &= \frac{Q_0}{C} \
\Rightarrow Q_0 &= i(0)C \
&= (25	imes10^{-3})(58.3	imes10^{-9}) \
&= 1.4575	imes10^{-6}	ext{ C}
\end{align*}

After five complete cycles, the current is given by:
\begin{align*}
i(83.46	ext{ ms}) &= \frac{Q_0}{C}\cos(\omega t) e^{-\frac{R}{L}t} \
&= \frac{1.4575	imes10^{-6}}{58.3	imes10^{-9}}\cos(377.96	imes83.46	imes10^{-3}) e^{-\frac{200	imes10^3}{15	imes10^{-3}}	imes83.46	imes10^{-3}} \
&\approx 2.09	ext{ mA}
\end{align*}

Therefore, the best choice is (C) 2.1 mA.

An RLC Circuit Has a Resistance of 200 Ω\OmegaΩ And an Inductance of 15 MH

An RLC Circuit Has a Resistance of 200 $\Omega$ And an Inductance of 15 MH