Magnetic field energy

Содержание и основные понятия

The source's work on the current is stored in the magnetic field of the coil. When the current drops, the field returns the energy to the circuit.
GEF · 11 grade
PIF · Baseline
Interactive model
Field energy
Inductiveness
energy density
Accumulator
01
The physical picture
What happens and what boundaries the model has.
Main message
The source's work on the current is stored in the magnetic field of the coil. When the current drops, the field returns the energy to the circuit.
The energy is proportional to the inductiveness and the current square.
The energy density is distributed in magnetic field space.
The ideal LC system exchanges energy between the electric and magnetic fields.
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.
W
=
L
I
2
2
W = \frac{LI^2}{2}
W
=
2
L
I
2
​
Coating power
w
=
B
2
2
μ
μ
0
w = \frac{B^2}{2\mu\mu_0}
w
=
2
μ
μ
0
​
B
2
​
energy density
Review of the situation
Double current in the coil increases the magnetic field's energy fourfold.
04
Links and application
In the real world
Double current in the coil increases the magnetic field's energy fourfold.
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 "Master field energy" through the model, not just the formula.
Outcome 2
Clear and correctly use terms: field energy, inductiveness, energy density.
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.
Field energy
Energy contained in the physical field and capable of being transferred to the substance or other fields.
Inductiveness
The coefficient of communication between the current in the contour and the current adhesion generated by it.
energy density
Field energy or substance per unit volume.
Accumulator
A device or component designed to temporarily store energy.
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.
W
=
L
I
2
2
W = \frac{LI^2}{2}
W
=
2
L
I
2
​
model effect
The power of the coil.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
E
s
=
−
L
d
I
d
t
\mathcal{E}_s=-L\frac{dI}{dt}
E
s
​
=
−
L
d
t
d
I
​
If the current grows, the source works against the EDS of self-induction.
d
W
=
I
L
d
I
dW=I L\,dI
d
W
=
I
L
d
I
During dt, the transferred power IL(dI/dt) increases the energy of the IL DI.
W
=
∫
0
I
L
I
′
d
I
′
=
L
I
2
2
W=\int_0^I LI'\,dI'=\frac{LI^2}{2}
W
=
∫
0
I
​
L
I
′
d
I
′
=
2
L
I
2
​
The integration at constant L gives the magnetic field energy to the coil.
Conditions:
Linear coil with constant inductiveness L; current increases in a quasi-stationary way from zero to I and energy losses are neglected.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
w
=
B
2
2
μ
μ
0
w = \frac{B^2}{2\mu\mu_0}
w
=
2
μ
μ
0
​
B
2
​
model effect
Power density.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
W
=
L
I
2
2
W=\frac{LI^2}{2}
W
=
2
L
I
2
​
Magnetic field energy is found from the work on current setting.
B
=
μ
μ
0
H
B=\mu\mu_0H
B
=
μ
μ
0
​
H
In a linear environment, induction is proportional to tension.
w
=
B
H
2
=
B
2
2
μ
μ
0
w=\frac{BH}{2}=\frac{B^2}{2\mu\mu_0}
w
=
2
B
H
​
=
2
μ
μ
0
​
B
2
​
The conversion to energy of a unit of uniform field volume and the exclusion of H give the desired shape.
Conditions:
A quasi-permanent magnetic field in a linear, homogeneous and isotropic environment with constant μs.
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 change the formula to any situation just because it's a coincidence of letters. The key condition of this topic is that energy is proportional to inductiveness and the current square.
How to Check the Mind
The energy density is distributed in magnetic field space. Then check the units, vector direction, and result behaviour at zero or very large parameter.
Σ
Conclusion on the topic
Magnetic field energy
The source's work on the current is stored in the magnetic field of the coil. When the current drops, the field returns the energy to the circuit.
The energy is proportional to the inductiveness and the current square.
The energy density is distributed in magnetic field space.
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.