Faradays Law and Inductance

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Original poster: "Jared Dwarshuis" <jdwarshuis@xxxxxxxxx>


Faradays Law and Inductance:



For an air cored solenoid:



Closed integral E dot ds =  d (flux B)/dt



Closed integral E dot ds =  (Area)   d(B)/dt



Closed integral E dot ds =  (pi  r(sqrd) )   d(B)/dt



Using B for a solenoid from Amperes law:



B = u   Ienclosed / height of solenoid



We get:



Closed integral E dot ds  =  u (pi  r(sqrd) ) / h          d( Ienclosed)/ dt



Multiplying numerator and denominator by 4 pi we get:



Closed integral E dot ds  =  u (2pi  r)sqrd / 4 pi h     d (Ienclosed) /dt



Since:  Ienclosed = (N)  Io          ( Io is the current in a given turn)



Closed integral E dot ds  = u (2pi r)sqrd / 4 pi h      ( (N)   d (Io)/dt )



Since:



u = 1 / e Csqrd      from:  1/ sqrt( u e ) = C



We can equivalently write:



Closed integral E dot ds  =



1/( 4 pi  e h)     (circumference/ the speed of light) sqrd       ( 
(N)   d (Io)/dt )



Since: Voltage = (N)  d (flux B)/dt



Then:



V =   [u (2pir)sqrd (N)sqrd /4pi h]   d(Io)/dt



Of course:  (2pi r)sqrd (N)sqrd = (2 pi r N)sqrd



And since (2 pi r N) is the physical wire length of the solenoid:



V =   u (wire length)sqrd / 4pi h         d(Io)/dt



Or:



V = (wire length/ speed of light)sqrd       1/ (4pi e h)       d(Io)/dt



Both forms are identical to:



V = L d(Io)/dt



Observations:



We have acquired a factor of (N) from the law: V = (N)  d(flux B)/dt

  (This part demonstrates how each loop contributes voltage).



 We acquired a second (N) from the current component of the equation:

 Ienclosed  = Io (N)



Since:



 [(2pi  r)sqrd ] [Nsqrd] = (2pi r N)sqrd = (wire length)sqrd.  We 
can  clearly see that wire length is a fundamental geometry for 
describing air cored solenoid inductance in the framework of 
Maxwell's equations.



( Interestingly; in the second form for voltage, we  find an inverse 
capacitance in the form: 1/ (4pi e h). This strongly suggests that a 
shell theorem for solenoids may exist. )



Sincerely: Jared Dwarshuis, Larry Morris      Oct. 2006