91Ó°ÊÓ

An oscillating circuit, often referred to as an LC circuit, is a fundamental circuit in electronics consisting of an inductor (L) and a capacitor (C). These components are connected in series or parallel and result in oscillatory behavior. The oscillations occur because energy alternates between the electric field of the capacitor and the magnetic field of the inductor.
When the capacitor releases its stored charge, it creates a current through the inductor, storing energy in its magnetic field. This current decreases, causing the capacitor to charge again with reversed polarity. Consequently, the circuit oscillates at a specific resonant frequency, dictated by its inductance and capacitance.
Key characteristics of oscillating circuits include:

• The resonant frequency, which determines the rate of oscillation, is given by \( f = \frac{1}{2\pi\sqrt{LC}} \).
• No external energy source is needed after initial charge, as energy is conserved and oscillates between the components.
• Damping can occur due to resistance, causing oscillations to decrease over time.

In an LC circuit, the current is not constant but varies sinusoidally with time. This behavior can be described mathematically. For the given exercise, the current is expressed as \( i(t) = (1.60) \sin(2500t + 0.680) \).
Key points about current in LC circuits include:

• The amplitude of the current is determined by the initial charge and resistance (though this example assumes ideal conditions).
• The phase of the current is influenced by the initial conditions of the circuit.
• The maximum value of the current occurs when the sine function reaches its peak value of 1. At this point, the argument of the sine function fulfills \( 2500t + 0.680 = \frac{\pi}{2} + 2\pi n \) for integer \( n \).
• Solving for \( t \) gives the precise moment when current first reaches maximum after \( t=0 \).

Inductance is a measure of how much a conductor opposes a change in current. In an LC circuit, inductance \( L \) plays a critical role in defining its resonant frequency alongside capacitance \( C \).
For the exercise, the angular frequency \( \omega \) is given as 2500 rad/s. The relationship between inductance and angular frequency can be expressed as \( \omega = \frac{1}{\sqrt{LC}} \).
Steps to calculate the inductance include:

• Rearranging the formula to solve for \( L \), we get \( L = \frac{1}{(\omega)^2 C} \).
• Substitute \( C = 64.0 \times 10^{-6} F \) and \( \omega = 2500 \) rad/s into the rearranged formula.
• Solving gives \( L = 2.5 \times 10^{-3} H \) or 2.5 mH (millihenrys).

Understanding the role of inductance helps in designing circuits with specific oscillatory behaviors depending on the application.

The energy stored within an LC circuit is an important property as it influences the circuit's performance and behavior. In the context of the exercise, the total energy can be determined using the formula:
\[ E = \frac{1}{2} L I^2 \]
where \( I \) is the peak current and \( L \) is the inductance.
Steps for energy calculation in the provided exercise include:

• Calculate the maximum current \( I = 1.60 A \), obtained from the current formula.
• Substitute the calculated inductance \( L = 2.5 \times 10^{-3} H \) and \( I = 1.60 A \) into the energy formula.
• Perform the calculation to find \( E = 3.2 \times 10^{-3} \) joules or 3.2 millijoules.

Understanding energy in LC circuits is crucial for applications requiring efficient energy transfer and minimal power loss.