Question
Download Solution PDFThe maximum charge on the capacitor in the LC oscillation circuit is Qo. Find the current in the circuit when the energy stored in the capacitor is equal to the energy stored in the inductor.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCONCEPT:
LC Oscillations:
- We know that a capacitor and an inductor can store electrical and magnetic energy, respectively.
- When a capacitor (initially charged) is connected to an inductor, the charge on the capacitor and the current in the circuit exhibit the phenomenon of electrical oscillations similar to oscillations in mechanical systems.
- Let a capacitor and an inductor are connected as shown in the figure.
- Let a capacitor be charged Qo at t = 0 sec.
- The moment the circuit is completed, the charge on the capacitor starts decreasing, giving rise to a current in the circuit.
- The angular frequency of the oscillation is given as,
\(⇒ ω_o=\frac{1}{\sqrt{LC}}\)
Where L = self-inductance and C = capacitance
- The charge on the capacitor varies sinusoidally with time as,
⇒ Q = Qocos(ωot)
- The current in the circuit at any time t is given as,
⇒ I = Iosin(ωot)
Where Io = maximum current in the circuit
- The relation between the maximum charge and the maximum current is given as,
⇒ Io = ωoQo
EXPLANATION:
Given UC = UI
Where UC = energy stored in the capacitor and UI = the energy stored in the inductor
- As the maximum charge on the capacitor in the LC oscillation circuit is Qo.
- So the total energy stored in the LC oscillation circuit is given as,
\(⇒ U=\frac{Q_o^2}{2C}\) -----(1)
Let the current in the circuit be I, when the energy stored in the capacitor is equal to the energy stored in the inductor.
- We know that the energy stored in the inductor is given as,
\(⇒ U_I=\frac{1}{2}LI^2\) -----(2)
Where L = self-inductance and I = current
We know that for an LC oscillation circuit,
⇒ UC + UI = U
⇒ UI + UI = U
⇒ 2UI = U
\(\Rightarrow U_I=\frac{U}{2}\) -----(3)
By equation 1, equation 2, and equation 3,
\(⇒ \frac{1}{2}LI^2=\frac{1}{2}\times\frac{Q_o^2}{2C}\)
\(⇒ I=\frac{Q_o}{\sqrt{2LC}}\)
- Hence, option 1 is correct.
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