Show that in free oscillation of an lc circuit. The sum of energies stored inthe capacitor and inductor is constant i time
Answers
Energy flow occurs between between an inductor and capacitor when they are connected together, energy flows from capacitor to inductor and then back to capacitor. Since there is no resistance, energy can not dissipate via inductor and capacitor and the whole energy of the system continues to swing back and forth between inductor and capacitor.This process continues forever in the absence of resistance in a cyclic oscillatory manner.
We know that from the properties of inductor and capacitor,
Energy stored in an inductor = LI²/2
Energy stored in the capacitor = Q²/2C
Let the initial current in the inductor = 0
Let the initial charge of the capacitor = Q₀
So the initial energy of the system,
U₀ = energy stored in the capacitor + energy stored in the inductor
= Q₀²/2C + 0
= Q₀²/2C
when the switch is closed current starts flowing in the circuit and capacitor starts discharging, this current flows in the circuit until the capacitor is discharged completely. Let the final maximum current be I₀.
So final energy stored in the system now
Uₙ = energy stored in the capacitor + energy stored in the inductor
= 0 + LI₀²/2
= LI₀²/2
Since the inductor opposes the change in current, so this current continues to flow in the circuit even after the capacitor is discharged totally. So this current now starts charging the capacitor until it is fully charged in the reverse polarity. This cycle of energy flow continues forever.
At any point the energy stored in the system will be the sum of energy stored in the capacitor and inductor.
U = Q²/2C + LI²/2
In this process energy dissipation does not occur because there is no resistance in the circuit. Hence the total energy of the system remains constant and jumps from capacitor to inductor and vice versa in an oscillatory manner forever.
Hence ,
U = Q²/2C + LI²/2 = LI₀²/2 = Q₀²/2C