Show that in the free oscillations of an lc circuit the sum of energies stored in the capacitor and inductor is constant in time
Answers
When an inductor and capacitor are connected together , energy flows from capacitor to inductor and then back to capacitor. This process continues forever in the absence of resistance in a cyclic oscillatory manner. 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.
Let a capacitor which is initially charged as Q₀ and an inductor with initial current 0 are connected together.
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
This current continues to flow in the circuit even after the capacitor is discharged totally because the inductor opposes the change in current. 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
Since there is no resistance in the circuit energy dissipation does not takes place and the total energy of the system always remains constant.
Hence ,
U = Q²/2C + LI²/2 = LI₀²/2 = Q₀²/2C
Answer:
hope this answer helps you.
