OSCILLATION IN AN LC – CIRCUIT
LC- oscillation —-
An LC circuit consists an inductor and a capacitor connected in parallel . Its another name is tank circuit.
At t= 0 – – capacitor is first charged . The whole energy is stored in the capacitor which is U= ½ CV2 , at that instant charge on the capacitor is Q , and there is no current through the circuit and energy in the inductor at that instant is zero .
At t=T/4 (T= time period of oscillation ) – – the capacitor begins to discharge and a growing current starts flowing through the inductor , which setup magnetic field around it which produces induced current in the inductor in opposite direction and hence there is delay in discharge . As the capacitor discharges its energy starts decreasing which converted as magnetic energy in the inductor , and at an instant total charge in capacitor is zero , and hence energy stored in the capacitor is zero and energy stored in the inductor at that time is ½ LI2‑.
At t=T/2 – as soon as capacitor is completely discharged , the magnetic field around the inductor starts decreasing , it produced an induced emf in opposite direction , due to which capacitor start charging it self in opposite direction and at an instant inductor discharged completely and capacitor charged completely again which is ½ CV2 .
This process repeated again and again then energy obtained once keep on oscillating between capacitor and inductor and to and fro pulses of current are produced .
WORKING OF LC-OSCILLATION AND DERIVATION OF ITS FREQUENCY – Watch Video Below
Derivation for frequency of LC- oscillation
– Suppose an electrical circuit having a capacitor of capacitance C and an inductor of inductance L connected as shown in fog. Below-
. Let the capacitor is charged Q initially and when key is closed emf induced in the inductor is L (di / dt) and potential difference across the capacitor is Q/C . Then from Kirchhoff’s law L (dI/dt) +Q/C =0 ………..(i)
But I = dQ /dt
And hence eq. (i) may be written as
L(d2Q/dt2)+ Q/C =0
Or , d2Q/dt2 + Q/LC = 0 ………………………..(ii)
But as we know eq . of SHM. May be written as d2x/dt2+ ω2x = 0 ………..(iii)
Comparing eq. (ii) and (iii) we get
ω2 = 1/LC
or, ω= 1/√LC .
but , ω=2Πѵ ( where ѵ=frequency)
so , frequency , ѵ= 1/2Π√LC