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The energy stored in a capacitor is a critical concept in understanding the dynamics of an oscillating LC circuit. A capacitor, a component that holds electric charge, stores energy in the form of an electric field. We measure this energy using the formula \(E_C = \frac{1}{2}CV^2\) where \(E_C\) is the energy in joules, \(C\) is the capacitance in farads, and \(V\) is the voltage across the capacitor in volts.

At time \(t=0\) ms in our circuit, when the capacitor is disconnected from the power supply and connected to the inductor, all of the electrical energy supplied by the power supply is stored within the capacitor. This initial energy plays a key role in the subsequent oscillation of energy between the capacitor and the inductor once the circuit begins to operate.

An inductor, on the other hand, stores energy in a magnetic field when current flows through it. The expression to calculate the energy stored in an inductor is \(E_L = \frac{1}{2}LI^2\), where \(E_L\) is the energy in joules, \(L\) is the inductance in henrys, and \(I\) is the current in amperes.

Since the oscillating LC circuit periodically transfers energy between the capacitor and the inductor, at certain pivotal moments, all of the circuit's energy is housed within the inductor's magnetic field. Knowing when exactly these moments occur functionally depends on the period of the oscillation and can be calculated using the oscillation cycle of the LC circuit.

In an oscillating LC circuit, energy is continuously exchanged between the capacitor and the inductor. In the textbook exercise, the initial energy is wholly stored in the capacitor when connected to the inductor. As the circuit starts oscillating, this energy shuttles back and forth between the electric field of the capacitor and the magnetic field of the inductor.

At time \(t=1.30\) ms, to find the energy in the inductor, we must determine the point in the oscillation cycle we are observing. Ideally, at an exact half-oscillation period, all the energy would be found in the inductor. However, since time does not correspond exactly to a half-period, there could be partial energy transfer between the inductor and capacitor, which must be calculated accordingly.

Angular frequency, denoted by \(\omega\), is fundamental in describing the oscillation of the LC circuit. It is calculated using the formula \(\omega = \frac{1}{\sqrt{LC}}\), where \(L\) is inductance and \(C\) is capacitance. This value is crucial as it determines how fast the energy oscillates between the capacitor and inductor.

Once we have \(\omega\), finding the natural frequency of oscillation \(f\) is straightforward as \(f = \frac{\omega}{2\pi}\). The angular frequency helps us understand the timing of energy shifts within the circuit and allows us to determine the circuit's resonant frequency, where it will naturally oscillate at with no external interference.