Capacitors of \(0.5 \mu \mathrm{F}\) and \(0.1 \mu \mathrm{F}\) and an inductor of self-inductance \(12 \mathrm{mH}\) and negligible resistance are connected in series. Find the natural angular frequency for discharge of the \(0.5 \mu \mathrm{F}\) capacitor by the methods of \(\mathrm{Sec}\). \(10.2\) without assuming that the capacitors can be replaced by a single equivalent one.

When we talk about connecting capacitors in series, this means we're arranging them one after another in a single line, so the charge stored on each is the same. The total voltage across them is the sum of the voltages across each capacitor.

To understand the effect this has on their capacitance, imagine each capacitor acting as a reservoir for electric charge. Capacitors in series work much like a series of narrow and wide pipes for water – the narrowest pipe restricts the flow, just as the smallest capacitance dictates the total capacity to store charge.

Here's a helpful metaphor: think of a series of buckets connected by holes of different sizes. The amount of water that can flow through all the buckets in a given time is limited by the bucket with the smallest hole. Similarly, in capacitors in series, the available charge storage (capacitance) is limited by the capacitor with the least capacitance.

The concept of equivalent capacitance is akin to reducing a complex network of roads to a single highway that gets you to your destination in the same time. When capacitors are in series, the equivalent capacitance can be thought of as a single capacitor that takes the place of the series without changing the overall electrical behavior.

The formula to find the equivalent capacitance of capacitors in series, \( \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \), might remind you of calculating the total resistance of parallel resistors, and that's no coincidence! It's a way to simplify complex circuits, ensuring we can analyze and understand them without getting lost in the details.

Inductance is the electrical counterpart to inertia in physics. Just as inertia resists changes in motion, inductance resists changes in electric current. It's measured in henries (H), and a higher inductance means a greater opposition to changes in current.

An inductor, usually a coil of wire, stores energy in a magnetic field when current flows through it. Visualize a spring: compress it, and it wants to bounce back. Inductors do the same with current - when the current changes, the inductor 'pushes back' against this change.

Inductance is key in tuning circuits, and it plays a starring role in the LC circuit discussed next. It's all about timing – inductance regulates how fast or slow the electrons dance in the circuit.

An LC circuit is a symphony of energy swapping roles between a capacitor and an inductor. Capacitor stores energy in an electric field, inductor in a magnetic field. Together, they exchange this energy back and forth, which can create oscillations, much like a pendulum swinging through the influence of gravity and momentum.

The natural angular frequency \( \omega \) of this oscillation, much like the frequency of a pendulum's swing, does not depend on the energy itself, but on the properties of the system: the inductance \(L\) and the equivalent capacitance \(C_{eq}\). The formula \( \omega = \sqrt{\frac{1}{LC}} \) unveils the rhythm of this electrical dance – it tells us how fast or slow the circuit oscillates, and it's crucial for designing radios, TVs, and other electronics that need precise control over frequencies.