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Electrostatics - LC Oscillations

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LC Oscillations

A capacitor of capacitance C is connected to an inductor of inductance L through a key K2. A cell is connected to C through key k1.

When plug of K1 is put in, the cell charges the capacitor to a potential V = q/C where q is the charge on capacitor plates and V is voltage across the plates. Some energy form the cell is strode in the dielectric medium between the plates of capacitor in the form of electrostatic energy [ u E = q2/2C].

On removing plug of K1 and putting in plug of K2. The charged capacitor is connected to L and starts discharging through L. an induced develop in the circuit which opposes the growth of current in L and hence delays it. When capacitor is completely discharged the energy stored in the capacitor papers in the form of magnetic field energy around L [uB = 1/2 LI2].

As soon as discharge of the capacitor is complete, current stops and magnetic flux linked with L starts collapsing. Therefore an induced develops which starts recharging the condenser in the opposite direction. The recharging is also opposed and hence delayed when condenser is recharged completely; the magnetic field energy around L reappears in the form of electric filed energy between the plates. The entire process is repeated. Thus energy taken once from the cell and given to capacitor keeps on oscillating between Cand L. if the circuit has no resistance no loss of energy would occur. The oscillations produced will be of constant amplitude, fig (2) these are called undamped oscillations. 

Figure.3 shows eight stages in a single cycle of oscillation of a resistanceless LC circuit. The bar graphs in each figure represent the stored electric energy (uE = q20/2C) and stored magnetic energy (uB = 1/2 LI20). The magnetic field lines of inductor and electric field lines of capacitor are also shown.

Fig.  shows capacitor with maximum charge and no current in the circuit. Fig. shows discharging of capacitor with increasing current. Fig.  shows capacitor fully discharged and maximum current. Fig.  shows charging of capacitor in opposite direction and current decreasing fig.  shows capacitor recharged fully with polarity opposite to that of (a) and no current fig.  shows discharging  capacitor, current increasing in opposite direction fig.  shows fully discharged capacitor and maximum current in opposite direction shows charging of capacitor with current decreasing as fig. shows. This cycle is being repeated.

In actual practices however thereof occur some losses of energy. Therefore amplitude of oscillations goes on decreasing. These are called damped oscillations as shown in fig.

To obtain undamped oscillations, the energy loss is duly compensated in proper phase.

The frequency of electrical oscillations can be obtained mathematically as follows.

In fig. the moment the circuit is completed charge on condenser starts decreasing giving rise to current in the circuit. As dI/dt is positive, the induced in L will be such that Vb < Va.

∴ q/C – L (dI/dt) = 0

As q decreases ; I increases

∴ I = - dq/dt

Eqn. (49) becomes q/c + L d2 (q/dt2) = 0

Or d2 (q/dt2) + (1/LC) q = 0

This equation has the form d2 x / dt2 + w2. X = 0 

For a simple harmonic oscillator,

The charge therefore oscillated with a natural frequency. 

W = (1/LC) = 2πv 
∴ V = 1 / 2 π√LC

The variation of charge with time is represented as q = q0 cos wt

The current, I = - dq / dt = q0 w sin wt = I0 sin wt

Where I0 = w q0 = maximum value of current

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