In an oscillating L C circuit, L=25.0 mH and C= 7.80μF . At time t=0 the current is 9.20 mA , th

In an oscillating L C circuit, L=25.0 mH and C= 7.80μF . At...

In an oscillating $L C$ circuit, $L=25.0 \mathrm{mH}$ and $C=$ 7.80$\mu \mathrm{F} .$ At time $t=0$ the current is 9.20 $\mathrm{mA}$ , the charge on the capacitor is 3.80$\mu \mathrm{C}$ and the capacitor is charging. What are (a) the total energy in the circuit, (b) the maximum charge on the capacitor and (c) the maximum current? (d) If the charge on the capacitor is given by $q=Q \cos (\omega t+\phi),$ what is the phase angle $\phi$ ? (e) Suppose the data are the same, except that the capacitor is discharging at $t=0 .$ What then is $\phi$ ?

Key Concepts

Phase Angle in Oscillations

The phase angle is a parameter in the sinusoidal expression for oscillatory motion that accounts for the initial condition of the system. It shifts the sine or cosine curve along the time axis and is crucial for matching the model to the observed initial charge and current values. Different initial scenarios, such as charging versus discharging, lead to different phase angles.

Application of Initial Conditions

The initial conditions of an oscillatory system, such as the initial charge and current, determine the specific solution for the system's time evolution. By applying these conditions to the general sinusoidal solution, one can derive the phase angle and fully describe the behavior of the circuit over time. This approach is fundamental in solving differential equations that arise in oscillatory systems.

Angular Frequency

The angular frequency (?) of an LC circuit is given by ? = 1/?(LC) and is a fundamental parameter that describes how fast the circuit oscillates. This frequency appears in the sinusoidal equations that represent the time-dependent behavior of charge and current, and it connects the physical properties of the circuit elements to the dynamics of the system.

Energy Conservation in LC Circuits

In an ideal LC circuit, the total energy is conserved as it oscillates between the magnetic field of the inductor and the electric field of the capacitor. Energy is stored in the capacitor as electrical potential energy and in the inductor as magnetic energy. By equating these energies at different points of the oscillation, one can determine the maximum charge and maximum current in the circuit.

LC Circuit Oscillations

An LC circuit comprises an inductor (L) and a capacitor (C) that exchange energy, resulting in oscillatory behavior. The oscillations are determined by the circuit’s natural frequency, which is calculated as the reciprocal of the square root of the product of L and C. This concept forms the basis for many applications in resonant circuits and signal processing.

Transcript

00:04 So in this problem, we have an lc circuit, and we're also told that the capacitor is charging.

00:19 We are given that the inductance is equal to 25 millie henries, that the capacitance is equal to 7 .80 microfarats, and that when the time is equal to zero, the current is equal to 9 .20 ampers, and the charge is equal to 3 .80 microcolums.

00:46 Part a, what is the total energy of the system? now, the total energy is going to be the sum of the electric energy plus the magnetic energy, where the electric energy is equal to the charge square over two times the capacitance, and the magnetic potential energy is the current square times the inductance over two.

01:13 So if we put the numbers in, so this is equal to the charge square.

01:19 So 3 .80 times 10 to the minus 6 columns square over 2 times the capacitance.

01:26 So over 2 times 7 .80 times 10 to the minus 6 ferats plus the current square, 9 .20 ampers square, times the inductance, 25 times 10 to the minus 3 henry's.

01:43 Over 2, and this give us equal to 1 .98 times 10 to the minus 6 joules.

01:52 So this is the total energy of this lc circuit.

02:01 In part b, we need to calculate what is the maximum charge.

02:07 For this, we use the formula that the electric energy is equal to q square over 2 times the capacitance.

02:16 So the electric energy is a maximum when the electric energy is a maximum, when all of this energy is electric, okay? and in that case, the charge is going to be a maximum.

02:33 So that means that the charge is equal to the square root of two times the capacitans times the maximum electric energy, which is equal to the square root of two times the capacitance 7 .80 times 10 to the minus 6 ferats times the electric potential energy when it's maximum, that is 1 .98 times 10 to the minus 6 joules.

03:05 And that will tell us that the maximum charge of the capacitor is going to be 5 .56 times 10 to the minus 6 columns.

03:21 In part c we need to find what is the maximum current...

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