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The energy stored in a capacitor is a fundamental concept in electrical circuits. A capacitor stores energy in the electric field between its plates. The formula to calculate this energy is \[ E_C = \frac{1}{2} C V^2 \] where:

• \( E_C \) is the energy stored in the capacitor in joules (J),
• \( C \) is the capacitance in farads (F), and
• \( V \) is the voltage across the capacitor in volts (V).

This equation shows that the energy stored is directly proportional to both the capacitance and the square of the voltage. In practical terms, this means that larger capacitors or higher voltages will store more energy. For the given problem involving a 3.0-µF capacitor at 35 V, substituting these values gives us the energy stored as 1.8375 mJ.

Inductors store energy in a magnetic field created by the current flowing through them. The energy formula for an inductor is \[ E_L = \frac{1}{2} L I^2 \] where:

• \( E_L \) is the energy stored in the inductor in joules (J),
• \( L \) is the inductance in henries (H), and
• \( I \) is the current through the inductor in amperes (A).

Just like with capacitors, the energy stored in an inductor is directly proportional to its inductance and the square of the current passing through it. This means a higher inductance or a larger current will result in more energy stored. To find the correct current that equals the energy stored in the previously mentioned capacitor, we need to use the given inductance value of 5.0 mH and solve for the current.

The relationship between the energy stored in capacitors and inductors forms the basis of resonant circuits and many other electrical applications. Though capacitors and inductors store energy in different ways, they can exchange this energy back and forth in oscillatory systems. Understanding how to equate and measure these energies is crucial for designing circuits, especially in AC systems.
In this exercise, we want the energy stored in a 3.0-µF capacitor to match the energy in a 5.0-mH inductor. By setting \( E_C = E_L \), we can derive a formula that lets us find the necessary current in the inductor to achieve this balance. Step by step, substituting known values and solving the equations allows for determining the respective energy equivalence and the current required.

Capacitor-inductor equivalence is a key concept in AC circuits, especially in LC circuits where these elements can interchange energy. When a capacitor discharges, it can cause current through an inductor, and the inductor, in turn, stores this energy in its magnetic field.
In this scenario, for both components to hold the same amount of energy, we adjust the current through the inductor based on the energy stored with given capacitance. This derives from equating their respective energy formulas.
It illustrates how energy can be balanced between these components. Discovering that the current needed is about 0.857 A offers insights into how these two elements can be harmonized in practical circuit designs. By mastering these foundational principles, students can better understand and predict the behavior of more complex electronic circuits.