Biot and Savart law

According to the Biot and Savart law of electromagnetic theory, the magnetic field B at the fixed electron due to the revolving positively charged nucleus is given in SI units to first order in y/c by... 

The specific rotation angle [a] is related to the experimental rotation angle of fight a through the so-called Biot and Savart law ... 

The absence of magnetic monopole implies the conditions V = 0 and B = -Vx = 0, which are consistent with the relations given above, since B =VAA =0. However, as with the A vector potential, the equality Bx = = Vx / 0 enables us to define a scalar potential x that can be calculated on the basis of Biot-Savart law for a filiform (filament-shaped) circuit... 

The magnetic field of the split solenoid was calculated by numerically solving the Biot-Savart law for the geometry as described in Fig. 7 and a field of about... 

The starting point for the calculation of magnetic field distributions for static fields, field gradients, and if fields is the Biot-Savart law. It determines the magnetic field B at a distance r from the wire element dl created by a current of strength /... 

The spatial distribution of Bi. completely describes the receptivity pattern of the surface coil. This distribution can he obtained for any coil geometry from the Biot-Savart law (2.3.8) l.etl). Field distributions simulated for a one-turn receiver coil are shown in Fig. 9.2.3 [Bos 11. Contours of constant field are shown for the j v plane al c = 0 (Fig. 9.2.3(a)) and for the yz plane at x =0 (Fig. 9.2.3(b)). High values of B, . and thus high signal sensitivity is found near the coil wires. With increasing distance y from the coil the transverse magnetic field falls off rapidly. The shape of the Biyy distribution in the. vy plane is distinctly different from that in they plane. This results from the fact that two transverse components of B exist in the jtv plane but only one in the yz plane. In... 

Each of them expresses the same fact, that is, that magnetic charges do not exist. Equations 1.109 have been derived by algebraic manipulation of Biot Savart law for direct currents, but they actually remain valid for alternating electromagnetic fields, and are in effect the fourth of Maxwell s equations. 

Starting with Biot-Savart law, we can say that the magnetic field has axial symmetry and is represented by a single component From the principle of superposition, one can say that the total field is the sum of a number of fields contributed by current elements / dz. Then we have ... 

Suppose that a current electrode is placed in a uniform conducting medium so that the distribution of currents possesses the spherical symmetry (Fig. 1.32a). It is then a simple matter to realize that the magnetic field is zero everywhere in the medium. This follows directly from Biot-Savart law and the symmetry of the model. In other words, one can always find two current elements which are located symmetrically with respect to the observation point and of which the magnetic field differ by sign only. Let us notice that Ampere s law does not apply here because the current lines are not closed. 

Starting with Biot-Savart law and making use of the principle of conservation of the charge, we were able to derive two equations describing constant magnetic fields,... 

Here p is the resistivity of the ring, I is its circumference, and S its cross-sectional area. According to Biot-Savart law, it is clear that the magnetic flux ips caused by the current flow in the ring is directly proportional to I, and can therefore be written as ... 

The inphase component of the field, as follows from Biot-Savart law, is caused by the current in the transmitter and the inphase component of induced currents in a medium, while the quadrature component of the magnetic field is generated by the quadrature component of induced currents only. Therefore, one can write ... 

It is appropriate to emphasize again that according to the Biot-Savart law the quadrature component of the magnetic field arises from currents induced in a medium for which the phase is shifted by 90° with respect to the current in the dipole source, while the inphase component is the algebraic sum of the primary and secondary fields. The inphase component of the secondary field is contributed by induction currents in the medium shifted by 180° or 0° with respect to the source current. 

Now let us suppose that a vertical magnetic dipole is located on the borehole axis and the medium possesses axial symmetry (Fig. 3.4). In accord with the Biot-Savart law the current of the magnetic dipole creates the primary magnetic field and its change with time generates the primary vortex electric field. Due to the axial symmetry this electric field does not intersect boundaries between media with different conductivities. Because of this no electric charges develop and as a result of the existence of the vortex electric field currents arise at every point, of the conductive medium with a density given by ... 

The quadrature and the inphase components of the magnetic field on the borehole axis are defined by the distribution of the quadrature and the inphase components of current density, respectively. It follows directly from Biot Savart law. 

Omitting intermediate transformations related with the calculation of the magnetic field and making use of the Biot-Savart law we obtain for a ratio of EMF caused by currents in shells, the radius of which exceeds R2 to EMF in a measuring coil, located in a uniform medium at distance R from the dipole, the following expression ... 

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