I-C Oscillations. A capacitor with capacitance 6.00 ×10^-5 F is charged by connecting it to a 1

I-C Oscillations. A capacitor with capacitance 6.00 ×10^-5...

I-C Oscillations. A capacitor with capacitance $6.00 \times 10^{-5} \mathrm{~F}$ is charged by connecting it to a $12.0 \mathrm{~V}$ battery. The capacitor is disconnected from the battery and connected across an inductor with $L=1.50 \mathrm{H}$. (a) What are the angular frequency $\omega$ of the electrical os cillations and the period of these ascillations (the time for one oscillation)? (b) What is the initial charge on the capacitor? (c) Ilow much energy is initially stored in the capacitor? (d) What is the charge on the capacitor $0.0230 \mathrm{~s}$ after the connection to the inductor is made? Interpret the sign of your answer. (c) At the time given in part (d), what is the current in the inductor? Interpret the sign of your answer. (f) At the time given in part (d), how much electrical energy is stored in the capacitor and how much is stored in the inductor?

Key Concepts

Energy Distribution in LC Oscillations

In an ideal LC circuit, the total energy remains constant as it oscillates between the capacitor's stored electrostatic energy and the inductor's magnetic energy. At any moment, the capacitor stores energy U_C = 1/2 C V², and the inductor stores energy U_L = 1/2 L I². This continuous energy exchange underlines the conservation of energy in the circuit.

LC Circuit

An LC circuit is a fundamental electrical circuit that contains an inductor (L) and a capacitor (C) connected together. It can exhibit oscillatory behavior where energy continually transfers between the magnetic field of the inductor and the electric field of the capacitor. This circuit is ideal for studying pure oscillations without resistive losses.

Angular Frequency and Period in LC Oscillations

The angular frequency (?) of an LC oscillator is determined by the inductance and capacitance of the circuit, given by the formula ? = 1/?(LC). The period (T) of oscillation, which is the time for one complete cycle, is related to the angular frequency by T = 2?/?, making it directly dependent on the values of L and C.

Capacitor Initial Charge

The initial charge on a capacitor is calculated using the relation Q = C × V, where Q is the charge, C is the capacitance, and V is the voltage across the capacitor. This relationship is crucial for understanding how much energy is stored before the capacitor is connected to other circuit elements.

Electrostatic Energy in a Capacitor

The energy stored in a capacitor, represented by U = 1/2 C V², indicates the amount of energy held in the electric field between the capacitor’s plates. This stored energy is central to the dynamics of an LC circuit, as it converts to magnetic energy in the inductor during oscillations.

Charge and Current Oscillations in an LC Circuit

In an LC circuit, the charge on the capacitor and the current in the inductor vary sinusoidally with time. The charge can be expressed as Q(t) = Q_max cos(?t) and the corresponding current as I(t) = -? Q_max sin(?t). These expressions highlight the phase difference between the voltage (related to charge) and the current, which is a key feature of oscillatory behavior in LC circuits.

Transcript

00:01 In the first part of this problem, we are going to calculate the angular frequency at the period of the circuit, that is omega and capital t.

00:13 For lc circuit, we can wide the angular frequency as omega is equal to 1 divided by square root of l multiply by c where this l is l is the inductance and a c is the capacitance of the circuit.

00:24 Let's insert the values into this equation.

00:26 So it will be 1 divided by square root of l, which is equals to 1 .50.

00:33 Into the c which is equal to 6 .00 multiply by 10x power minus 5 ferret and we have the square root up to this point so from here we can write the omega as omega is equal to 1005 .4 radiant per second so this is the angular frequency now we can write the period of oscillation as a t is equals to 2 pi radiance divided by omega so by inserting values into this equation we can write here two into 3 .14 radiance divided by 105 .4 radiance per second so from here we can write the value for this capital t as capital t is equals to 59 .6 milliseconds in the second part of this problem we have to calculate the initial charge on the capacitor, so at time t equals to zero, the charge on the capacitor will be capital q.

01:55 This can be calculated through the capacitor equation as capital q is equals to c multiplied by v, where this v is the potential difference between the blades of the capacitor.

02:04 Let's put the values into this equation, so it will be q is equals to 6 .00 multiplicity by a turner's power minus 5 ferret into 12 .0 volt so from here we can write the value for this capital q as 72 .72 .0 multiply by 10xbar minus 5 column so this is the maximum charge on the plate of the capacitor in part c of this problem we have to calculate the energy stored in the capacitor initially that is you see so at time t equals to 0, the energy stored in the capacitor will be uc and this is equals to 1 divided by 2c v.

03:03 Square.

03:04 So by inserting values into the equation, we can write here, uc is equal to 1 divided by 2 into 6 .00, multiply by 10x power minus 5 ferratt into 12 .0 volt whole square.

03:19 So from here we can write uc is equal to 4 .32 multiply by terrestrial power minus 3 javls or it can be written as 4 .32 millijal.

03:38 So this is the answer to the part c of this problem.

03:43 Let's move toward part d of this problem.

03:47 So in d, we have to calculate the charge on the capacitor at the given instant of time.

03:52 So we need to calculate small q...

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