Unipolar generator.
(Униполярный генератор)
Consider the wheel made of conductive material rotating with a constant angular frequency ω in a uniform magnetic field.
Two brushes provide connectivity between the stationary outside part of the circuit that includes a voltmeter and the rotating part inside.
The result of this closed circuit is an induced EMF
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Unipolar generator.
Velocity is a function of the |
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Electric field intensity: |
E B r |
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Br02 |
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U E dr |
Er dr |
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The flux passing through the contour is constant! |
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Unipolar generator.
Another consideration may take place.
The circuit shown here is also a possibility.
The magnetic flux in this case is:
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r2 t |
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The induced EMF is: |
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However, this explanation is not very persuasive.
What is always true: |
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Using the flux in this certain |
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Hering’s paradox. (Парадокс Геринга)
Consider a magnet without leakage. It has an air gap that is penetrated by a magnetic field.
Outside the magnet, there is a conducting loop with elastic contacts that ensure a closed circuit at all times. The circuit also includes a voltmeter
Movement of the coil causes a voltage impulse |
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The loop is |
Finally, the loop is |
further moved to |
pulled from the |
a new position |
magnet to the outside |
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Hering’s paradox.
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The spring contacts open, but stay in contact with the magnet, so the circuit is still closed at all times
Experiment demonstrated – there is no induced EMF.
This may appear like a paradox which represents a contradiction to Faraday’s Law. Essential is, however, that the flux change is not related to the motion of the conductor.
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Inside the magnet: field exists but there is no movement |
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Outside the magnet: conductor moves, but there is no field. |
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Diffusion of electromagnetic fields.
Consider electromagnetic field in conductor.
The displacement currents are neglected. No free charges.
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divB 0 |
B H |
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Basic equations: |
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divJ 0 |
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curlH J |
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divE 0 |
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Differential equations for the electric field intensity:
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Mathematical |
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transformation: |
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Diffusion of electromagnetic fields.
Diffusion equations for the electromagnetic field characteristics:
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Such equations (applied to scalar variables) describe processes of the particle diffusion, thermal processes.
One-dimensional equations: (here we assume that only x- component of the E and y- component of H exist)
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Periodic electromagnetic field in the conductors.
Quasi-stationary approach:
Ex Em sin( t E ); H y Hm sin( t H )
Applying the complex method:
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Em Eme |
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Differential equations in complex form:
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Periodic electromagnetic field in the conductors.
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Equation: |
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Solution for the complex field intensity: |
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Parameter α: |
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e j |
2 e j 4 |
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Using designation: |
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Solution for the field intensity: |
H Hm0e kz sin( t H0 |
kz) |
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Penetration of the electromagnetic field into a conductor
1 |
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k |
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Amplitude of the electromagnetic wave dumps according to exponential dependence:
H Hm0e kz sin( t H0 kz 4 )
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