EASY
What happens to the potential difference across the capacitor in an LC circuit when the current is zero?
HINT: Concept Check 10.1
The potential difference across the capacitor is at its maximum when the current is zero.
Explain why energy conservation is valid in an LC circuit with zero resistance.
Energy conservation is valid because there is no resistance to dissipate energy as heat. The total energy oscillates between the capacitor (electric energy) and the inductor (magnetic energy).
What is the total energy in an LC circuit with L=5 mH, C=3 μF, and qmax=4 mC?
2.67 J
What is the total energy in an LC circuit given by?
The total energy in an LC circuit is given by:
Utotal=q^2/2C+Li^2/2
How is the current in an LC circuit related to the charge on the capacitor during oscillations?
The current is the time derivative of the charge, and its maximum value occurs when the charge on the capacitor is zero.
If the total energy in an LC circuit is 1 mJ and ω=2000 rad/s, what is the maximum magnetic energy stored in the inductor?
The maximum magnetic energy is equal to the total energy:
U magnetic= 1 mJ= 0.001 J
In the differential equation for an LC circuit, what does q represent?
q represents the charge on the capacitor.
What is the physical significance of the term q^2/2C in the context of an LC circuit?
q^2/2C represents the electric potential energy stored in the capacitor.
At t=0.5 s, the energy in the capacitor is half of the total energy. Find the phase constant ϕ if ω=100 rad/s
50- π/4 rad
What is the angular frequency (ω) of an LC circuit in terms of inductance (L) and capacitance (C)?
ω=1/ root LC
Why does the total energy of the LC circuit remain constant over time?
The total energy remains constant because there is no resistance in the circuit, so no energy is lost as heat. Energy oscillates between the capacitor and the inductor.
What is the relationship between the total energy of an LC circuit and the angular frequency of oscillation?
The total energy does not depend on the angular frequency; it is determined by the maximum charge and capacitance, as Utotal=q2max / 2C
At what value of ϕ is the charge at its maximum value in the LC circuit?
The charge is at its maximum value when ϕ=0 or ϕ=2πn, where n is an integer.
If a capacitor with 2 μF capacitance has an initial voltage of 50 V calculate the maximum charge stored in the capacitor.
q= C*V = 2×10-6 ⋅50=1×10-4 =0.1 mC
Explain how the total energy in an LC circuit remains constant, even as the forms of energy (electric and magnetic) change.
The total energy alternates between electric energy in the capacitor and magnetic energy in the inductor. At any moment, the sum of these energies is constant because the circuit is lossless and has no resistance.