Question

Explain the production of the electrical oscillation in LC circuits. Under what conditions is the oscillation produced in the circuit?

Answer 1

In order to solve this question we need to understand the inductor and capacitor. Inductor is defined as device in which back emf induces, actually when a loop is subjected to potential difference then a current flows in it but if the current suddenly starts to decrease so there would be flux change as the magnetic field across loop changes so a back emf induces in device and this back emf is directly proportional to rate of decrease of current and proportionality constant is known as inductance. An inductor uncharged behaves as an open circuit while an inductor fully charged behaves as a closed circuit.

Complete step by step answer:
The steady state condition tells us that, full charge inductor behaves as a closed circuit while a full charge capacitor behaves as an open circuit. Consider a capacitor and inductor connected to form a circuit. If the capacitor of capacitance $C$ connected with an inductor of inductance $L$. Suppose the capacitor is initially fully charged with $+ Q$ charge on its positive plate and $- Q$ charge on its negative plate.

So due to the inductor connected it would draw current in it and hence the magnetic field across the inductor begins to rise now the inductor opposes this behavior and generates a back emf across it. Now due to back emf induced in inductor current starts to decrease, so energy stored in inductor begins to decrease, since we know energy cannot be created neither can be destroyed so this energy begin to store on capacitor in form of charges store but this time charge stored on opposite plates

Means $+ Q$ charge stored on plate which is negative earlier and $- Q$ charge stored on plate which is positive earlier, so charge configuration across capacitor changes.In this way charge and magnetic field oscillate across the inductor and capacitor connected to form a circuit.
Energy stored by the capacitor is, $U = \dfrac{{{Q^2}}}{{2C}}$.
And energy that stores across the inductor is, $U = \dfrac{1}{2}L{i^2}$.

Conditions on which oscillation depends:
-How much charge is stored on plates of capacitor initially, because this would decide the magnitude of oscillation.
-Is there any dissipative force in the circuit or not? Like if we assume the resistance of wire to be considered then while oscillating energy would be lost in heating so after some time oscillation will stop.

Note: It should be remembered that both the capacitor and inductor are ideal in this case, so no dissipative or damping force present. It must be noted that capacitor charged capacitor behaves as an open circuit, thus it acts as a potential difference across an inductor shaped in solenoid form similarly a fully charged inductor behaves as a closed circuit or potential difference across a capacitor.