Conference rusure::math

617.0. "Magnetic Fields do not exist" by CURIUM::PETERSON () Sun Nov 30 1986 22:23

    Here's a quick proof that magnetic fields do not exist (i.e., are zero 
    everywhere).

To begin, note that one of Maxwell's equations states that

(1)             Div B = 0

where B is the vector equation describing the magnetic field.  Because B has
zero divergence there exists some vector function, A, such that

(2)             B = Curl A

Now, by the divergence theorem we know that

(3)             II (B)(n) dS = III Div B dV = 0
                  S               V
                  
                II  = Double integral over the closed surface, S
                  S
                III = Triple integral over the volume, V, enclosed by S.
                   V
                (B)(n) = Dot product of the magnetic field, B, and the
                         unit normal vector of the surface, S.
                         
By substituting (2) into (3) we obtain

                II (n)(Curl A) dS = 0
                  S
                  
                II = double integral as before
                  S
                (n)(Curl A) = Dot product between unit vector, n, and the
                              Curl of A.
                              
We now apply Stoke's theorem to find that

                I (A)(t) dS = II (n)(Curl A) dS = 0
                 C              S
                 
                I = Line integral around the closed curve, C
                 C
                (A)(t) = Dot product between vector function A and t, the
                         tangent vector at any point along C.
                         
In otherwords, the circulation of A is path independent!  Therefore, it
follows that

                A = Grad a
                
Where 'a' is some scalar function.  Since the Curl of the gradient of a function
is zero we arrive at the remarkable fact that

                B = Curl A = Curl (Grad a) = 0
                
or, that all magnetic fields are zero. Q.E.D!
    
    Gizbah?

    The surface integrals used when invoking the divergence theorem and
    then Stokes theorem were not the same.  The first was closed, while
    the latter was not (else the line integral could be shrunk to a point
    and would indeed vanish.)

    - Jim

    
    re: .0  Moreover, Classical Electomagnetic Theory is just an
    approximation.  Try getting to QED (quod erat demonstrandum) with
    QED (quantum electrodynamics).  :-)
    
    					Monty