When I was reading the LC circuit in my textbook I came across the derivation of equations of instantaneous charge and current.
7.8 LC Oscillations
We know that a capacitor and an inductor can store electrical and magnetic energy, respectively. When a capacitor (initially charged) is connected to an inductor, the charge on the capacitor and the current in the circuit exhibit the phenomenon of electrical oscillations similar to oscillations in mechanical systems (Chapter 14, Class XI).
Let a capacitor be charged $q_m$ (at $t=0$) and connected [to] an inductor as shown in Fig. 7.18.
The moment the circuit is completed, the charge on the capacitor starts decreasing, giving rise to current in the circuit. Let $q$ and $i$ be the charge and current in the circuit at time $t$. Since $\mathrm{d} i/\mathrm{d}t$ is positive, the induced emf in $L$ will have polarity as shown, i.e. $v_b<v_a$. According to Kirchhoff's loop rule,
$$
\frac{q}{C} – L \frac{\mathrm{d}i}{\mathrm{d}t} = 0
\tag{7.39}
$$
$i = -(\mathrm{d}q/\mathrm{d}t)$ in the present case (as $q$ decreases, $i$ increases). Therefore, Eq. (7.39) becomes:
$$
\frac{\mathrm{d}^2 q}{\mathrm{d}t^2}- \frac{1}{LC}q = 0
\tag{7.40}
$$
This equation has the form $\frac{\mathrm{d}^2 x}{\mathrm{d}t^2}- \omega_0^2 x = 0$ for a simple harmonic oscillator. The charge, therefore, oscillates with a natural frequency
$$
\omega_0 = \frac{1}{\sqrt{LC}}
\tag{7.41}
$$
and varies sinusoidally with time as
$$
q = q_m \cos(\omega_0 t + \phi)
\tag{7.42}
$$
where $q_m$ is the maximum value of $q$ and $\phi$ is a phase constant. Since $q=q_m$ at $t=0$, whe have $\cos(\phi)=1$ or $\phi=0$.
Which is no problem, but when I got to the derivation to current, it said something like
The current $i\left(= -\frac{\mathrm{d}q}{\mathrm{d}t}\right)$ is given by
$$
i = i_m \sin(\omega_0t)
$$
where $i_m = \omega_0 q_m$
Which I don't quite understand.
It says in the third line,
$$
i_m = \omega_0 q_m,
$$
where $\omega_0$ is the natural frequency of the LC circuit, $i_m$ is the maximum current and $q_m$ is the maximum charge.
So, in other words, it's trying to say that $\omega_0 =1/T$, which is certainly not correct, as $\omega$ is an angular frequency and not just a frequency, i.e. it has the unit of radians per second and not just $\mathrm{s}^{-1}$.
However, we know that
$$
\omega_0 = \frac{1}{\sqrt{LC}},
$$
so this could help us, as by substituting the units of $L$ (coefficient of self-inductance) and $C$ (coefficient of capacitance), we can get the unit of $\omega_0$. But, surprisingly, we would get
$$
[\omega_0] = 1/(\mathrm{s}^2)^{1/2}
$$
which ultimately is
$$
[\omega_0] = \mathrm{s}^{-1}.
$$
So, basically, it's saying that angular frequency is equal to just "frequency", which contradicts the fact their units are different. How can this be true?
Then again, one might ask, how current can oscillate "angularly"?
So my question is, are both natural frequency and angular frequency the same thing? And if they are, then can someone please explain the relationship
$$
f_0 = \frac{\omega_0}{2\pi} = \frac{1}{2\pi\sqrt{LC}}
$$
to me?
And also of course, the Wikipedia page on LC circuits.
To summarize, I just wanted to know few things:
-
How is angular frequency (radians per second) (has kind-of "physical meaning") be equal to just frequency (cycles per second) [has all meanings {physical and electrical (oscillation of electrons)} but here, In our case, just focusing to electrical]?
-
If they are both equal, then, how can electrons oscillate angularly?
-
If angular frequency is not equal to frequency, then, is my textbook wrong?
Also, I think I'm missing something, probably my basics are not so clear.
Best Answer
No. A resonant system has a natural frequency which is characteristic of the system. From the Wikipedia article Natural frequency:
The natural frequency of system may be expressed in units of ordinary frequency $(\mathrm{Hz})$ or angular frequency $(\mathrm{\frac{rad}{s}})$
Both ordinary frequency and angular frequency have the same dimension of inverse time $[\mathrm{T^{-1}}]$ but ordinary frequency express the number of cycles per second while angular frequency expresses the number of radians per second. Since there are $2\pi$ radians in one cycle, it follows that angular frequency is just the ordinary frequency multiplied by $2\pi\mathrm{\frac{rad}{cycle}}$.
Typically, the symbol for ordinary frequency is $f$ or $\nu$ while the symbol angular frequency is $\omega$ thus
$$\omega = 2\pi\nu$$
You can see this clearly in the expression for the energy of a photon of frequency $\nu$ which can be expressed as
$$E_\gamma = h\nu = \frac{h}{2\pi}2\pi\nu = \hbar\omega$$