Magnetic Pressure (fwd)

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---------- Forwarded message ----------
Date: Sun, 24 Jun 2007 21:12:21 -0400
From: Jared Dwarshuis <jdwarshuis@xxxxxxxxx>
To: Pupman <tesla@xxxxxxxxxx>
Subject: Magnetic Pressure

Magnetic pressure in an 'ideal' air cored solenoid:



Derivation by: Lawrence Morris,  Jared Dwarshuis

June 2007



We will use the following:



F = M Vsqrd / r

E = M Csqrd

Sa = 2pi r h

Eb = 1/2 L Isqrd

L = u Nsqrd Area / h

C = 1/sqrt(u e)

W = 2pi r N (wire length)

B = (u Io N) / h  =  (u W Io) / Sa          (Where I total = Io N)



First we will multiply the numerator and denominator of L by
4pi.andregroup. We will also use the fact that C = 1 / sqrt( u e)



Then equivalently we can express L as:



L = ((2pi r N)/ C) sqrd       1/ (4pi e h)



Recognizing that (2pi r N) = wire length (labeled W)



L = ( W/C)sqrd   1/ (4pi e h)



We will now assume that waves of energy are traveling at C down the entire
length of  wire in an inductor and proceed.



We can describe the magnetic energy of the inductor as:  Eb = 1/2  L Isqrd

Conservation principles demand that the energy contained by the magnetic
fields must be equal to the energy of the currents Ei within the inductor.
Thus the total energy equals: Eb + Ei     Then:  Etotal = L Isqrd



We are describing waves of energy traveling at C down the entire length of
wire in an inductor. Their must be a mass associated with this energy: E/
Csqrd = M

The mass does not travel in a straight line but must follow a circular path.
A force is required to change the direction of this mass.

We can use Newton's shell theorem to demonstrate that the entire mass
distribution  can be considered at a single point.



Then:



F = M Vsqrd / r



Substituting we get: F = E / Csqrd   Vsqrd / r



Since  we have assumed that V = C, it cancels leaving:



F = E / r   or     F = L Isqrd / r



Since:



Pressure = Force /  Surface area:



Magnetic pressure = (L Isqrd) / ( 2pi  (r)sqrd   h)



Substituting in L and solving we get:



Magnetic pressure = 1/2e   [ W I / C Sa ]sqrd



After reducing this yields:  u /2   (N/h)sqrd  Isqrd



Classic equation: Magnetic pressure = Bsqrd / 2u.



After reducing this also yields:  u /2   (N/h)sqrd  Isqrd



The forms are equivalent.