Free fluctuations in LC

Содержание и основные понятия

The charged capacitor is discharged through the coil: the electrical energy becomes magnetic, then the process goes back. In the ideal contour, the fluctuations are inexhaustible.
GEF · 11 grade
PIF · Baseline
Interactive model
LC-consequence
own frequency
phase
The blackout.
01
The physical picture
What happens and what boundaries the model has.
Main message
The charged capacitor is discharged through the coil: the electrical energy becomes magnetic, then the process goes back. In the ideal contour, the fluctuations are inexhaustible.
The charge and current are in phase for a quarter of a period.
The resistance is fading.
The frequency depends on L and C.
How to reason
First, name the system and the interaction, then select the values, and only then write down the law. The formula works within the model's assumptions.
02
Laws and values
Symbols should be read as a relationship of values, not as a recipe for substitution.
T
=
2
π
L
C
T = 2\pi\sqrt{LC}
T
=
2
π
L
C
​
Thomson formula
q
=
q
m
cos
⁡
ω
t
q=q_m\cos\omega t
q
=
q
m
​
cos
ω
t
Capacitor charge
Review of the situation
The increase in capacity quadruples the period of its own fluctuations by half.
04
Links and application
In the real world
The increase in capacity quadruples the period of its own fluctuations by half.
Model verification
Compare the prediction with observation, evaluate the units, the order of magnitude and the influence of the factors not taken into account. Disparity does not always mean an error of law: often idealization is no longer sufficient.
C
What needs to be understood
Outcome 1
Explain the physical meaning of the topic "Free fluctuations in LC conturbation" through the model and not just through the formula.
Outcome 2
Distinguish and correctly use terms: LC-construction, own frequency, phase.
Outcome 3
Indicate the conditions for the applicability of the law, verify units and limit cases.
T
Terms of the topic
The definition should indicate the physical meaning, not a substitute for a domestic synonym.
LC-consequence
The electrical circuit of capacitor and coil, in which the energy is periodically passed between the electric and magnetic fields.
own frequency
The frequency of free fluctuations of the system after a slight deviation from equilibrium.
phase
A value that determines the status of the periodic process at the selected time.
The blackout.
Reduced amplitude of fluctuations due to energy losses.
F
Origin and conclusion of the formulas
The fundamental law cannot be "proven" by algebras: for each record, it is stated whether it is a definition, an experimental law, or a model effect.
T
=
2
π
L
C
T = 2\pi\sqrt{LC}
T
=
2
π
L
C
​
model effect
The tomson formula.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
q
C
+
L
d
I
d
t
=
0
,
I
=
d
q
d
t
\frac{q}{C}+L\frac{dI}{dt}=0,\qquad I=\frac{dq}{dt}
C
q
​
+
L
d
t
d
I
​
=
0
,
I
=
d
t
d
q
​
In the ideal LC-consequence, the capacitor voltage is balanced by the EDS of self-induction.
d
2
q
d
t
2
+
q
L
C
=
0
\frac{d^2q}{dt^2}+\frac{q}{LC}=0
d
t
2
d
2
q
​
+
L
C
q
​
=
0
The current setting results in an equation of the harmonic oscillator.
ω
0
=
1
L
C
,
T
=
2
π
ω
0
=
2
π
L
C
\omega_0=\frac{1}{\sqrt{LC}},\qquad T=\frac{2\pi}{\omega_0}=2\pi\sqrt{LC}
ω
0
​
=
L
C
​
1
​
,
T
=
ω
0
​
2
π
​
=
2
π
L
C
​
The actual frequency of the equation gives you a period of hesitation.
Conditions:
The ideal rolling contour with constant L and C, zero resistance, and no radiation loss.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
q
=
q
m
cos
⁡
ω
t
q=q_m\cos\omega t
q
=
q
m
​
cos
ω
t
model effect
The capacitor charger.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
d
2
q
d
t
2
+
ω
2
q
=
0
\frac{d^2q}{dt^2}+\omega^2q=0
d
t
2
d
2
q
​
+
ω
2
q
=
0
The balance of the pressures of the ideal LC conturbation gives an equation of the harmonic variations of the charge.
q
(
t
)
=
q
m
cos
⁡
(
ω
t
+
φ
)
q(t)=q_m\cos(\omega t+\varphi)
q
(
t
)
=
q
m
​
cos
(
ω
t
+
φ
)
The overall harmonic solution contains an amplitude and an initial phase.
q
(
0
)
=
q
m
,
I
(
0
)
=
0
⇒
φ
=
0
q(0)=q_m,\ I(0)=0\quad\Rightarrow\quad\varphi=0
q
(
0
)
=
q
m
​
,
I
(
0
)
=
0
⇒
φ
=
0
If at the beginning the capacitor is loaded to the maximum, the cosine shall be selected without the initial phase.
Conditions:
The ideal LC output; time is selected at the time of the maximum charge, × = 1/ G(LC).
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
!
Model boundaries and typical errors
Error: Forget model condition
You can't move the formula to any situation just because it's a coincidence of letters. The key condition of this topic is that the charge and current are in phase for a quarter of a period.
How to Check the Mind
Resistance causes blackout. Then check the units, vector direction, and behavior of the result at zero or very large parameter.
Σ
Conclusion on the topic
Free fluctuations in LC
The charged capacitor is discharged through the coil: the electrical energy becomes magnetic, then the process goes back. In the ideal contour, the fluctuations are inexhaustible.
The charge and current are in phase for a quarter of a period.
The resistance is fading.
Any numerical result shall be accompanied by a unit, a sign and an explanation of its physical meaning.
The content is consistent with the FRP on Physics of the OS/OSS 2025 and the subject matter of the current lines of the PNP: I. M. Peryškin and co-authors (7–9), G. J. Mäkishev and coauthors (10–11), and with the lines "Engineers of the Future" and V. A. Kasyanov. The original texts are not reproduced, the text of the textbooks is not reproduced.