91Ó°ÊÓ

An LC circuit consists of an inductor (L) and a capacitor (C). In this problem, you're given the current function as a function of time, which is: \(i(t) = 1.60 \text{ A} \times \text{sin}(2500 \text{ rad/s} \times t + 0.680 \text{ rad})\). The current oscillates because of the energies exchanged between the inductor’s magnetic field and the capacitor’s electric field.

The current function tells us how the current varies with time. To find the time when the current is maximum, look for when the sine function equals 1. Mathematically, this happens when the argument of the sine function is \(\frac{\pi}{2}\).

So, set the argument \(2500t + 0.680\) equal to \( \frac{\pi}{2} \). Then, solve for \( t \):
\[ 2500t + 0.680 = \frac{\pi}{2} \].
Solving this gives us
\[ t = \frac{\frac{\pi}{2} - 0.680}{2500} = 0.00039 \text{ s} \].
So, the current reaches its maximum at \(0.00039\) seconds after \( t = 0 \).

By understanding the current function, students can better grasp how the energies in LC circuits oscillate over time.

Inductance \(L\) is a fundamental parameter in an LC circuit, representing the ability of the circuit to induce a voltage as the current changes. To find the inductance, we use the resonant angular frequency \( \omega \).

Given \( \omega = 2500 \text{ rad/s} \), and knowing that \( \omega = \frac{1}{\sqrt{LC}} \):

• First, rearrange to find \(L\): \[ L = \frac{1}{\omega^2 C} \].
• Substitute \( \omega = 2500 \text{ rad/s} \) and capacitance \( C = 64 \mu F = 64 \times 10^{-6} \text{ F} \).

So: \[ L = \frac{1}{(2500)^2 \times 64 \times 10^{-6}} \].
Calculating this, we get:
\[ L = 0.0025 \text{ H} \].

Inductance determines the circuit’s ability to oppose changes in current. It is crucial for understanding how quickly or slowly the current changes in the circuit.

The total energy in an LC circuit remains constant, as it oscillates between the capacitor and the inductor. We calculate the maximum energy stored using the formula for energy in an inductor, since current is maximum. This energy is given by:
\[ U = \frac{1}{2} L I_{max}^2 \].

Given \( L = 0.0025 \text{ H} \) and the maximum current \( I_{max} = 1.60 \text{ A} \):

• Substitute these values in: \[ U = 0.5 \times 0.0025 \text{ H} \times (1.6 \text{ A})^2 \].
• Calculating this, we get: \[ U = 0.0032 \text{ J} \].

This energy is alternately stored in the magnetic field of the inductor and the electric field of the capacitor. Understanding this energy helps in grasping how the LC circuit's energy oscillates without loss over time.

An oscillating LC circuit is one where the energy alternates between the magnetic field in the inductor and the electric field in the capacitor. This oscillation arises due to the inductance and capacitance.

Some key characteristics:

• In an ideal LC circuit (no resistance), this oscillation is perpetual and undamped.
• The oscillation frequency, \(f\), relates to \(L\) and \(C\) by: \( f = \frac{1}{2\pi \sqrt{LC}} \).
• The angular frequency \( \omega \) used in our problem is \( \omega = 2 \pi f = \frac{1}{\sqrt{LC}} \).

Understanding these oscillating behaviors helps in many practical applications such as radio transmitters, filters, and tuning circuits.

So, the oscillatory nature of LC circuits is foundational to many areas of electronics and signal processing.