Transcranial Magnetic Stimulator 
The 3-Dimensional Mapping of the Magnetic Field
 
Appendix I

The Derivation of the Magnetic Field Equation for a Finite Straight Line Segment

    The following is the complete derivation that was completed to determine the generalized equation for the magnetic field induced due to a finite line segment in terms of only three coordinates, A(a1, a2, a3), B(b1, b2, b3) and P(x,y,z).  The geometry of the line segment in terms of the line segment end points A and B and a point in space P that was used to determine the generalized equation is shown below.

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    For a line segment:

                                                                                                                                 (19)

    Considering the contribution  from at point P:

                                                                                                                                            (20)

    But and .  Therefore,

                                                                                                                                         (21)
                                                                                                                (22)
 
Let  and 

Therefore,

                                                                                          (23)
                                                                                                         (24)

The final general equation for the magnetic field for a finite line segment was determined to be:


                                                                                                 

(25)
where the direction is determined by equation 26.
                                                                                                                                                            (26)

Considering points A=(a1, a2, a3), B=(b1, b2, b3) and P=(x, y, z):

                                                (27)
                                                        (28)
                                                            (29)
            (30)
                                                                                             (31)
 
                                                                                                                                      (32)

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