# How do we understand L C Oscillations (Inductor - Capacitor Circuit Oscillation) Intuitively?

How do we understand L C Oscillations (Inductor - Capacitor Circuit Oscillation) Intuitively? How to intuitively understand the Phase Lag and Imagine What is happening in the circuit exactly without the analogy of SHM? Currently my understanding of L C Oscillation is only mathematical. I am convinced with the mathematics, but I am not able to explain it intuitively to other person that why it is happening so. Let's start with a capacitor charged to $V_0$, with energy $U_E=\frac {CV_0^2}2$.

When we connect an inductor, the voltage on the capacitor is applied to the inductor and the current through the incductor starts to grow according to the known formula: $\frac {dI(t)}{dt}=\frac {V(t)} L$.

Looking at this formula, we can see that, as the capacitor is discharged and its voltage decreases, the growth of the current slows down. At the point, when the voltage on the capacitor becomes zero, the current stops growing, which means that it has reached its maximum.

Since the voltage on the capacitor crosses zero, when the current through the inductor is at its maximum, we can say that there is a $90^\circ$ phase difference between the voltage and the current.

So, as the capacitor (the analog of a mechanical spring) has just reached its neutral point and ran out of its potential energy, the inductor (the analog of a mass) has reached the maximum of its magnetic energy, $U_B=\frac {LI_0^2}2$, and, as the moving mass continues to move, the flowing current in the inductor continues to flow. This inertia is the reason the oscillations do not stop.

The next quarter cycle is symmetric to the first, both mathematically and physically, and the second half of the cycle is just a mirror image of the first half.

The $90^\circ$ phase shift between the voltage and the current signifies the fact that two quarters of each cycle, when the direction of the voltage coincides with the direction of the current, the energy is passed from the capacitor to the inductor, while the other two quarters of the cycle, when the current flows against the voltage, the energy is passed from the inductor to the capacitor.

• you say that the inertia is the reason it does not stop, and this is all well and good for mass but what about charges, how are you defining inertia of charges? – Prakhar Nagpal Jul 31 at 14:07

Basically what is happening is the capacitor and inductor are passing energy back and forth.

The energy stored in a capacitor's electric field when its voltage is maximum (your first, fifth and ninth figures) is given by

$$E_C = \frac{1}{2}CV^2$$

When the capacitor losses all its charge it no longer has any stored energy (your third and seventh figures). It becomes stored in the magnetic field of the inductor when the current is maximum, which is given by

$$E_L = \frac{1}{2}LI^2$$

Realistically there is alway some resistance in the circuit dissipating energy so the oscillations are damped until all the energy is dissipated in resistance and the current goes to zero.

Hope this helps.