Question

What values of phase constant $\mathit{\varphi }$in Eq. 31-12 allow situations (a), (c), (e), and (g) of Fig. 31-1 to occur at $\mathit{t}\mathbf{=}\mathbf{0}$?

Short answer

1. The value of phase constant at time$t=0$, at whichsituation a occurs is$0\pm 2n\pi$ .
2. The value of phase constant at time$t=0$, at whichsituation c occurs is$\frac{\pi }{2}\pm n\pi$.
3. The value of phase constant at time$t=0$, at whichsituation e occurs is$\pi \pm 2n\pi$.
4. The value of phase constant at time $t=0$, at which situation g occurs is $\frac{3\pi }{2}\pm 2n\pi$.

Step-by-step solution

The given data

The charge of the capacitor varies as $q=Q\text{cos}\left(\omega t+\varphi \right)$.

Understanding the concept of LC circuit mechanism

Figure 31-1 describes the oscillatory variation of energy in an LC circuit. We can determine various phases of the situations described in the question by considering the time t = 0.

Formula:

The charge of the capacitor varies as $\mathit{q}\mathbf{=}\mathit{Q}\mathbf{}\mathbf{\text{cos}}\left(\omega t+\varphi \right)$ (i)

a) Calculation of phase constant at which situation a occurs

For $t=0$, we can say using equation (i) that the charge will vary as

$q=Q\text{cos}\left(\varphi \right)............................\left(i\right)$

Situation a indicates that the charge on the capacitor is maximum. Thus, using equation (i), we get the maximum charge condition as follows:

$q=Q$

This implies that

$$\begin{gathered} \cos \varphi =1 \\ \varphi =0\pm 2n\pi \end{gathered}$$

Hence, the value of the phase constant for situation a is $0\pm 2n\pi$.

b) Calculation of phase constant at which situation c occurs

Situation c indicates that the charge on the capacitor is zero. Thus, using equation (i), we can get the phase constant as follows:

$$\begin{gathered} dq=0 \\ \cos \varphi =0 \\ \varphi =\frac{\pi }{2}\pm n\pi \end{gathered}$$

Hence, the value of the phase constant is $\frac{\pi }{2}\pm n\pi$.

c) Calculation of phase constant at which situation e occurs

Situation e indicates that the charge on the capacitor is maximum.

But capacitor is fully charged withtheopposite polarity. Thus, using equation (i), we can get the phase constant as follows:

$$\begin{gathered} dq=-Q \\ \cos \varphi =-1 \\ \varphi =\pi \pm 2n\pi \end{gathered}$$

But the charge on the capacitor is with opposite polarity.

Hence, the value of the phase constant is $\pi \pm 2n\pi$.

d) Calculation of phase constant at which situation g occurs

Situation g indicates that the charge on the capacitor is zero and the current is flowing in the opposite direction. Thus, using equation (i), we can get the value of the phase constant as follows:

$$\begin{gathered} dq=0 \\ \cos \varphi =0 \\ \varphi =\frac{3\pi }{2}\pm 2n\pi \end{gathered}$$

Hence, the value of the phase constant is $\frac{3\pi }{2}\pm 2n\pi$.