Question

An LC circuit contains a \(20 \mathrm{mH}\) inductor and a \(50 \mu \mathrm{F}\) capacitor with an initial charge of \(10 \mathrm{mc}\). The resistance of the circuit is negligible. At the instant the circuit is closed be \(t=0 .\) At what time is the energy stored completely magnetic. (a) \(\mathrm{t}=0 \mathrm{~ms}\) (b) \(\mathrm{t}=1.54 \mathrm{~ms}\) (c) \(\mathrm{t}=3.14 \mathrm{~ms}\) (d) \(\mathrm{t}=6.28 \mathrm{~ms}\)

Short answer

The time when the energy stored in the LC circuit is completely magnetic is approximately \(t = 1.54\,ms\). The correct answer is (b).

Step-by-step solution

Identify Given Values

In this problem, we are given: - Inductor (L) = \(20mH\) or \(0.02H\) - Capacitor (C) = \(50\mu F\) or \(50\times10^{-6}F\) - Initial charge on the capacitor (Q) = \(10mc\) or \(10 \times 10^{-3}C\) Our goal is to find the time (t) when the energy stored in the LC circuit is completely magnetic.

Recall the Energy Functions and the Value of Angular Frequency

To solve this problem, we need to recall the following formulas: 1. Energy stored in a capacitor: \(E_C = \frac{1}{2} C V_C^2\) 2. Energy stored in an inductor: \(E_L = \frac{1}{2} L I_L^2\) 3. Angular frequency in an LC circuit: \(\omega = \frac{1}{\sqrt{LC}}\) In this problem, we are interested in finding the time when \(E_C = 0\), which means that all the energy is stored in the inductor.

Calculate the Angular Frequency

Using the formula \(\omega = \frac{1}{\sqrt{LC}}\), we can find the angular frequency: \(\omega = \frac{1}{\sqrt{(0.02)(50\times10^{-6})}} \approx 1000\,rad/s\)

Find the Voltage and Current Functions

Now that we have the angular frequency, we can find the voltage and current functions across the capacitor and inductor, respectively: 1. Voltage across the capacitor: \(V_C(t) = Qcos(\omega t)\) 2. Current through the inductor: \(I_L(t) = -\frac{Q\omega}{L}sin(\omega t)\) Here, Q is the initial charge on the capacitor.

Determine the Time When the Energy Stored in the Capacitor is Zero

To find the time when the energy stored in the capacitor is zero, we need to find the time when the voltage across the capacitor is zero. From the voltage function, it is clear that this happens when \(cos(\omega t) = 0\). The cosine function has its first zero crossing at \(t = \frac{\pi}{2\omega}\). We can plug in the calculated angular frequency to find the time: \(t = \frac{\pi}{2(1000)} \approx 1.57 \times 10^{-3}\,s \)

Round Off the Answer and Compare with Options

We can round our answer to two decimal places: \(t \approx 1.57\,ms\) Comparing our answer with the available options, the closest one is: (b) \(t = 1.54\,ms\) So the answer is (b). At approximately \(t = 1.54\,ms\), the energy stored in the LC circuit is completely magnetic.