Advanced chapters of theoretical electroengineering.
SPbTU, IE, Prof. A.G. Kalimov 2022 |
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Lecture 6
Time dependent fields
Gunther Lehner
Electromagnetic Field Theory for Engineers and Physicists
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Faraday’s Law
Закон электромагнитной индукции
The Faraday’s Law is based on Maxwell equations.
The main idea of this law is:
changing in time magnetic field induces the electric field.
The origin of the induced voltage:
time varying magnetic fields;
moving of the coil in stationary magnetic field
Lenz's law (правило Ленца) climes that the electric field induced in a circuit due to a change in a magnetic field is directed to oppose the change of the flux.
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Induction by a temporal change of B
Consider a time dependent magnetic field described by the magnetic flux density or also called the magnetic induction and a contour l, fixed in space.
The Maxwell equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-conservative electric field:
B curlE t
The magnetic flux:
B ds
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s |
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B |
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Edl |
curlE |
dS |
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dS |
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S t |
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- electromotive force (EMF) or voltage |
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Induction through the motion of the conductor.
Inside the magnetic field the force is: |
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F Q v B |
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The same effect is produced by the electric field: |
F Q E |
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This effect may be explained by assuming |
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E v B |
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Moving a thin conducting contour in static magnetic field we shall get:
E dl |
v B dl B v dl |
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a b c c a b c b a |
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l |
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Displacement vector covers the area during the time period of Δt
dS v dl dt
dl
v dt
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Induction through the motion of the conductor.
The EMF induced in the contour:
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dS |
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U E |
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The magnetic field distribution does not depend on time!
If the conductor is not closed
E v B
If the last 3 vectors are normal to each others than
U v B l
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Induction through the motion of the conductor.
If the contour moves in the uniform magnetic field no EMF is induced because of mutual canceling of partial electromotive forces:
If only one side moves, than
Uv B l
Bla(t) Bl a0 vt
U d v B l dt
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Induction through the motion of the conductor.
During this important process of energy transformation, mechanical energy is transformed into electrical energy.
Consider a frame rotating in the uniform magnetic field
If ω is the angular velocity, then the flux encompassed during the time period t is
B a l cos t
U d B a l sin t dt
This is the principle of alternating current generators.
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Induction by simultaneous temporal change of B and motion of the conductor.
The two effects discussed above can also occur simultaneously and in such case need to be added.
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E dl |
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B dS |
B v dS |
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Using the vector magnetic potential: |
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E dl |
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A dS |
v B dl |
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A dS |
v B dS (the Gauss theorem for curl of vector) |
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dS 0 |
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or: |
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v |
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Induction by simultaneous temporal change of B and motion of the conductor.
The last expression is true for any surface so: |
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Introducing a suitable scalar |
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function φ, we may write |
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The field intensity E here |
is the one which an |
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observer would “see” when moving with the conductor.
An observer at rest then sees the field
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