Question

In an oscillating $LC$ circuit in which $C=4\mu \text{F}$, the maximum potential difference across the capacitor during the oscillation is $1.5\text{V}$ and the maximum current through the inductor is $50\text{mA}$. How much time is required the charge on the capacitor to rise from zero to its maximum value ?

• A. $100\mu \text{s}$
• B. $200\mu \text{s}$
• C. $300\mu \text{s}$
• D. $400\mu \text{s}$

Answer 1

The correct option is B $200\mu \text{s}$

Given:-

Capacitance, $C=4\mu \text{F}=4\times {10}^{-6}\text{F}$

Maximum potential difference across the capacitor,
$V_0=1.5\text{V}$

Maximum current through the inductor, $i_0=50\text{mA}$
$=50\times {10}^{-3}\text{A}$

We know, $i_0=q_0\omega$

$\Rightarrow i_0=C{V}_0\omega$

$\Rightarrow \omega =\frac{i_0}{C{V}_0}$

$=\frac{50\times {10}^{-3}}{4\times {10}^{-6}\times 1.5}=8.3\times {10}^3\text{rad/s}$

Further, $q=q_0\sin \omega t$

As, at $t=0,q=0$ and

at $t=t,q=q_0$

$\Rightarrow q_0=q_0\sin \omega t$

$\Rightarrow \omega t=\frac{\pi }{2}$

$\Rightarrow t=\frac{\pi }{2\omega }=\frac{3.14}{2\times 8.3\times {10}^3}$

$\therefore t\approx 2\times {10}^{-4}\text{s}=200\mu \text{s}$

Hence, option $\left(\text{B}\right)$ is correct.