The magnetic field generated from a single thin wire can be calculated by the Biot-Savart Law:

                                        (1)

where r is the distance from the wire, I is the current in the wire, and  is the permeability constant which has a value of  in free space.  The magnetic field is therefore measured in units of  as required by the magnetic force law.  These units are often called tesla (T). [3]  However, since the current in the wire is deemed constant during the pulse, the current, I can be taken out of the integral.  Therefore equation 1 becomes,

                                                (2)

Figure 2:  The geometry of the line segment problem

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     The generalized equation for a line segment at any point in space was determined to be (see Figure 2):

                                 (3)

where ,  and are all quantities defined as in Figure 2 where the endpoints of the line segment are A(a₁,a₂,a₃) and B(b₁,b₂,b₃) and the point of interest is P(x,y,z) and  is the distance between points A and B,  bar is the distance between points P and B, and  is the distance between points P and A.  is defined to be the shortest path from the point of interest, P, to the line segment.  Equation 7 gives the direction of the magnetic field by taking the cross product of the line segment unit vector in the direction of the current and the unit vector in the direction of the shortest path to the point of interest from the line segment.  Appendix I contains the complete derivation. ,  and  are defined by the following equations.

                                                                                                                          (4)
                                                                                         (5)
             (6)
                                                                                                                                                        (7)

    The summation of each component of the magnetic field at the point P contributed by each small line segment along the coil is the total magnetic field at that point.  The total magnitude of the magnetic field at point P is simply the square root of the sum of squares of each component magnitude.

Figure 3:  A virtual rod and an MRI image of a stimulated brain

Figure 3 depicts a Magnetic Resonance Image of a human brain which was stimulated by a TMS coil.  The virtual rod shown in the figure is the placement of the TMS coil's center on the subject.  This image is of importance since oncethe distribution is determined, the rod will be replaced with the exact location of the peak focused magnetic field.  This will enable the researcher to find the correlation between the induced change in brain activity with the 3-dimensional magnetic field distribution produced by the TMS coil.

    The magnetic flux density B is similar to the electric flux density D.  As in free space, the magnetic flux density B is related to the magnetic field intensity H according to:

(8)

where  is the permeability of free space.

    The magnetic flux through a surface S is given by

                           (9)

where  is the magnetic flux in webers (Wb), B is the magnetic flux density in Wb/m²  or teslas and S is the surface area. [4]  L is the inductance given in henry for a conductor and I is the current given in ampere. [5]  The magnetic flux line is the path to which B is tangential at every point in a magnetic field.  Each flux line is closed and therefore has no beginning or ending.  The flux lines don't cross each other, regardless of the circuit geometry.  The direction of the magnetic field line follows the right hand rule.  That is, if you place your right hand thumb in the direction of the current through the wire, then your fingers will curl in the direction of the magnetic field lines.  Figure 4 illustrates the direction of the magnetic field lines for a wire carrying current out of the page.

Figure 4:  Magnetic flux lines due to a straight wire with current coming out of the page

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Micheal Faraday in London and Joseph Henry in New York discovered that a time-varying magnetic field would produce an electric current.  A static magnetic field produces no current flow, but a time-varying field produces and induced voltage (called electromotive force or simply emf) in a closed circuit, which induces a current flow.  Faraday discovered that the induced emf, Vemf (in volts), in any closed circuit is equal to the time rate of change of the magnetic flux linkage by the circuit.  This is called Faraday's Law and it can be expressed as:

                                        (10)

where N is the number of turns in the circuit and  is the flux through each turn.  The negative sign shows that the induced voltage acts in such a way as to oppose the flux producing it.  This is known as Lenz's Law and it emphasizes the fact that the direction of current flow in the circuit is such that the induced magnetic field produced by the induced current will oppose the original magnetic field. [4]

    The induced voltage for a conductor system C is given by [5]:

                                                                               (11)

or

                                                                            (12)

    Any physical effect whose magnetic field dependence is precisely known can be used for the measurement of magnetic fields.  Many methods are used to measure steady fields but for pulsed field, only a few of them remain of practical interest.  Many of the methods used to measure steady fields cannot be used in pulsed operation because their frequency response is insufficient.  The Spherical Magnetic Probe (SMP) used here for the measurement of the magnetic field intensities produced by TMS was designed according to the Inductive Magnetic Probe design [5].  The method most used for measuring pulsed fields is based on the simple pickup coil.  When a solenoidal coil with an effective area S is placed in a time-varying magnetic field H, the induced voltage Vi(t) at its terminals is:

                                                                (13)

    The coil has N turns, each of cross-sectional area A, and is placed with its axis parallel to the field lines.  Therefore, we can write

                                                                       (14)

where Sa is an additional active area formed by the leads and connections; in carefully prepared probes, Sa can be limited to less than 1 mm². [5]

    From equation 13, it follows that the magnetic field is proportional to .  This integral can evaluated from the measured Vi(t) signal through numerical or graphical integration, or measured directly by coupling the probe to a ballistic galvanometer or an electronic integrating circuit. [5]