Physics Problem: Oscillating LC Circuit Analysis
Question:
In an oscillating LC circuit, when 80.8% of the total energy is stored in the inductor’s magnetic field:
- What multiple of the maximum charge is on the capacitor?
- What multiple of the maximum current is in the inductor?
Answer:
We start by using the fundamental relationships in an LC circuit. The total energy at any moment is split between the capacitor and the inductor:
E = 1⁄2 Q²/C = 1⁄2 L I²
At a particular instant, energy in each component is:
EC = 1⁄2 q²/C
EL = 1⁄2 L i²
EL = 1⁄2 L i²
Given: 80.8% of the total energy is in the inductor, so:
EL = 0.808E
EC = 1 – 0.808 = 0.192E
EC = 1 – 0.808 = 0.192E
Step (a): Multiple of the Maximum Charge on the Capacitor
We use the ratio of energy in the capacitor to the total energy:
1⁄2 q²/C = 0.192 × 1⁄2 Q²/C
Canceling common terms:
q² = 0.192 Q² ⟹ q/Q = √0.192 ≈ 0.438
(a) The charge on the capacitor is approximately 0.438 times the maximum charge (Q).
Step (b): Multiple of the Maximum Current in the Inductor
Similarly, for the inductor’s energy:
1⁄2 L i² = 0.808 × 1⁄2 L I²
Canceling common terms:
i² = 0.808 I² ⟹ i/I = √0.808 ≈ 0.899
(b) The current in the inductor is approximately 0.899 times the maximum current (I).
Final Answer Summary:
- (a) Charge on the capacitor = 0.438 Q
- (b) Current in the inductor = 0.899 I