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					Optical Equations 1 Maxwell’s equations – A historical review for better understanding (continued)

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Acceleration (AC Current)
• If the charge stream is accelerated – the magnetic flux (swirl) becomes dynamic; what makes it more important is that an electric field swirl will be generated by the dynamic magnetic flux, which leads to Faraday’s law. • Why would a dynamic magnetic flux generate the electric field swirl? • Implication - the system is attempting to stay still: once a change is found in magnetic flux, an electric field is excited in such a way that it tries to cancel out the magnetic flux change by generating a new magnetic flux against the original one; i.e.,
     s B  ds  l E  dl  0 t
  B  E   t

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Extended Summary beyond Static Fields - A Wrap up of Experimental Observations
• Stay still charge distribution generates divergence driven, swirl free electric field (which can be sensed by any charged object, hence we have the name “electric”). • Charge in static motion generates not only the above mentioned electric field, but also swirl driven, divergence free magnetic field (which differs from the electric field as it can only be sensed by the charged moving object, hence we have the name “magnetic”). - So far, the fields are static (with spatial dependence only, no temporal dependence) and non-coupled (between the electric and magnetic ones).
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Extended Summary beyond Static Fields - A Wrap up of Experimental Observations
• Accelerated charge generates dynamic (time-varying) magnetic field, which induces the swirl to the electric field. • Consequently, the electric field will be driven by both divergence and curl sources; the former comes from the stay still or constantly moving charges, whereas the latter is induced by the time-varying of the magnetic field which comes from the charge acceleration. Also, the electric and magnetic fields becomes coupled, but still in one way (from magnetic to electric only).

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Extended Summary beyond Static Fields - A Wrap up of Experimental Observations
• We can express these conclusions mathematically to obtain the governing equations for any electromagnetic effect in vacuum
  B  E   t     B  0 J    E   /0  B  0
Faraday’s law Ampere’s law Gauss’ law (E) Gauss’ law (M)

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Maxwell’s Equations - Power of Logical Thinking and Math
• Maxwell found inconsistency in the 2nd equation if the charge accelerates
      B  0  J  0     0 t

• He then mended the 2nd equation by
       B   0 (  J  ) t   0  0

   E   B   0 J   0 0 t
with Gauss’ law applied to the last term on the RHS

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Maxwell’s Equations - Power of Logical Thinking and Math
• Implication of the added term – time-varying electric field, similar to the current, also generates magnetic field.

• Hence we name the time change rate of the electric field the displacement current (more accurately, the timederivative of the displacement vector), and the conventional current the conduction current.

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Maxwell’s Equations - Power of Logical Thinking and Math
• As a consequence
– 1. charge acceleration generates dynamic magnetic field in its neighborhood (Ampere’s law); – 2. dynamic magnetic field induces dynamic electric field (Faraday’s law); – 3. dynamic electric field in its neighborhood generates dynamic magnetic field (Maxwell’s displacement current equivalence + Ampere’s law), such sequence repeats endlessly in a area which is not necessarily limited to the location of the source where the charge accelerates

• This process describes the electromagnetic wave generation and propagation.

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Maxwell’s Equations - The Ultimate Form in Vacuum
  B  E   t    E   B   0 J   0 0 t    E   / 0  B 0

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Maxwell’s Equations - The Ultimate Form in Media
  B  E   t    D  H  J  t  D    B  0

  D  E   B  H   J  E

There are 16 scalar variables, but 17 equations. One equation is redundant. Normally, we don’t need the last one (the magneto Gauss’ law ), as it is hidden in the 1st equation. The carrier continuity equation is hidden in the 2nd equation.

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Home Work 1 (continued)
• Design a simple electromagnetic wave generation and radiation system (antenna system). • If you move along with the propagation of the electromagnetic wave, what do you expect to observe? Use equations to describe the electromagnetic effect.

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