91Ó°ÊÓ

In an LC circuit, which consists of an inductor (L) and a capacitor (C), understanding the concept of maximum charge is essential. The maximum charge, denoted by \( Q_{max} \), is the peak amount of electrical charge that the capacitor holds during its oscillation cycle.
An easy way to remember the formula for maximum charge is by using this relationship:

• \( Q_{max} = I_{max} \times \sqrt{L/C} \)

Here, \( I_{max} \) refers to the maximum current that flows through the circuit. From the example, we know \( I_{max} = 0.750 \) A.
Given:

• \( L = 0.0800 \) H (inductance of the inductor)
• \( C = 1.25 \times 10^{-9} \) F (capacitance of the capacitor)

Replacing each value into the equation, \( Q_{max} = 0.750 \times \sqrt{\frac{0.0800}{1.25 \times 10^{-9}}} \). After calculating, \( Q_{max} \) is approximately 190.49 \( \mu \)C. This means that when the energy is fully stored in the capacitor, it holds this maximum charge amount.

The oscillation frequency in an LC circuit refers to how often the current changes direction per second, measured in hertz (Hz). In this circuit, the frequency is determined by both the inductor and the capacitor.
The formula to find the oscillation frequency \( f \) is:

• \( f = \frac{1}{2\pi} \sqrt{\frac{1}{LC}} \)

Using this, we start by substituting known values:

• \( L = 0.0800 \) H
• \( C = 1.25 \times 10^{-9} \) F

Subsequently, the frequency is calculated as \( f = \frac{1}{2\pi} \times \sqrt{\frac{1}{0.0800 \times 1.25 \times 10^{-9}}} \). Through computation, the frequency comes out approximately 503.29 kHz. A higher oscillation frequency means the circuit energy shifts quickly between the capacitor and inductor, enabling swift charge and discharge cycles.

At various times during oscillation, energy alternates between the capacitor and the inductor. When examining the energy stored within an inductor, we use a specific formula:

• \( E_L = \frac{1}{2} L I^2(t) \)

Where \( I(t) \) is the current at the specific time \( t \), influenced by \( \omega \), the angular frequency.
With \( \omega = 2\pi f \), and from our previous section, \( f \approx 503.29 \text{ kHz} \). Thus, \( \omega \approx 3162389 \text{ rad/s} \).
Let's calculate the current at \( t = 2.5 \) ms.

• First, find \( \omega t \): \( 3162389 \times 0.0025 \approx 7905.97 \)
• Next, use the sine function: \( I(t) = 0.750 \times \sin(7905.97) \approx 0.6495 \text{ A} \)

The final step finds energy at \( t = 2.5 \) ms: \( E_L = \frac{1}{2} \times 0.0800 \times (0.6495)^2 \). This sums up as \( E_L \approx 0.01688 \text{ J} \). This process demonstrates how energy shifts even during an oscillation cycle, representing the dynamics in the circuit.