An LC circuit with a 20 - \(\mu\) F capacitor oscillates with period \(5.0 \mathrm{ms} .\) The peak current is \(25 \mathrm{mA} .\) Find (a) the inductance and (b) the peak voltage.

Inductance in an LC circuit essentially dictates how much the circuit resists changes in current flow. Its unit is the henry (H). In our case with a periodic oscillation, the inductance can be extracted from the following formula which relates the period of an LC circuit:

\[ L = \frac{T^2}{(2\pi)^2C} \]
Here, we rearrange the standard oscillation period formula to solve for the inductance (L). Given that the period (T) is 5.0 ms and the capacitance (C) is 20 \(\mu\) F, we substitute these values into our formula. This allows us to calculate L without necessarily understanding the entire mathematical underpinning of the oscillation but through algebraic manipulation of the formula.

To truly understand this formula, one should know that the LC circuit's natural oscillation period is analogous to the pendulums swing: it is the time it takes to complete one full cycle of electric charge flow to and fro, akin to the pendulum's back and forth. The inductance (L) determines how sluggish the charge flow is, much like the length of a pendulum's arm influences the time it takes to swing.

Understanding the period of an LC circuit is fundamental to analyzing its behavior. The period is the time duration of one complete oscillation of current and is denoted by (T). For a standard LC circuit where 'L' is inductance and 'C' is capacitance, the period is given by this formula:

\[ T = 2\pi\sqrt{LC} \]
This equation, akin to the formula for a spring-mass system's period, reveals that the period of oscillation is directly proportional to the square root of the product of inductance and capacitance. If we tweak either L or C, the period changes accordingly, meaning that the LC circuit's oscillation can be fine-tuned for various applications in electronics, such as resonators or filters.

For ease of understanding, think of it as a heartbeat, where the LC circuit 'beats' electric charge back and forth through its system - the period is the time between these heartbeats. With a fixed capacitance in our problem, the inductance will determine the frequency of these oscillations, demonstrating the intimate relationship between these two components.

The peak voltage in an LC circuit represents the maximum potential difference between the plates of the circuit's capacitor during an oscillation cycle. Determining the peak voltage can be directly related to the energy stored in the LC circuit. With the peak current (I) and inductance (L) known, one can calculate the total magnetic field energy in the circuit using this energy formula:

\[ U = \frac{1}{2}LI^2 \]
Using the energy (U) and the capacitance (C), we can then find the peak voltage (V) through the formula:

\[ V = \sqrt{\frac{2U}{C}} \]
This approach involves conservation of energy principles where electrical energy is converted to magnetic energy and vice versa, without any loss, in a perfect LC circuit. V can be visualized as the height of a swing at its highest point—it represents the maximum energy the capacitors electric field can store in the form of electrostatic potential before it starts swinging back, converting that stored energy back into a current in the circuit.