Vector magnetic induction

E is the electric field vector, B is the magnetic field vector (magnetic induction), fx, /to, , so the usual field constants (with e = /t = I in vacuo) and the Nabla operator V is defined by equation (4) ... 

In the case of a uniform magnetic induction along the z-axis, B = you will find that the vector field... 

To illustrate the use of the vector operators described in the previous section, consider the equations of Maxwell. In a vacuum they provide the basic description of an electromagnetic field in terms of the vector quantifies the electric field and 9C the magnetic field The definition of the field in a dielectric medium requires the introduction of two additional quantities, the electric displacement SH and the magnetic induction. The macroscopic electromagnetic properties of the medium are then determined by Maxwell s equations, viz. 

Zik is a measure of the zth component of the electric field produced by the Mi component of the temperature gradient (eq. (27)) and Zik is a measure of the effect of the /th component of the magnetic induction Bt on Zik. Therefore, T,ik describes the coupling between a 7)2), Zik, and an axial vector B, the components of which transform like Rx Ry Rz. The components Yk) of a 7)2) transform like binary products of coordinates, that is like the nine quantities... 

If i is the current in the windings of the solenoid and l the length of the solenoid per turn, the vector B, called the magnetic induction, is defined by... 

Electromagnetic waves combine the propagation of two vector fields, E and B. These are the electric and magnetic induction fields, respectively, and in a vacuum are governed by the Maxwell equations 1,2,3] ... 

The field vector determining the Forentz force on a current is the magnetic induction B, which is measured in teslas. In principle, B at a point P can be... 

The Bremsstrahlung can be explained by a classical analysis of the Poynting30 vector the electric field E, magnetic induction B, and Poynting vector S at a distance r from a charge q with acceleration a at a time t are given by... 

In the foregoing, B is the magnetic induction vector. For a more elementary derivation of the above results, based on a rather specialized case, see Exercise 1.6.3, while a more elegant derivation is furnished in the Appendix, Section 1.7. Again, Eq. (1.6.15a) is awkward in actual use because H and B are local fields which reflect the reaction of the medium to the applied field, and because the integration extends over all space. For practical applications it is more convenient to use Heine s (1956) formulation... 

In the above, E, D, H, B, Jt represent the electric field, electric displacement, magnetic field, magnetic induction, and free current density vectors respectively c is the velocity of light. Form the scalar product of (1.7.1a) with H and of (1.7.1b) with E and subtract to find... 

E and H being the electric and magnetic field strength vectors, D the electric displacement vector, B the magnetic induction vector, J the electric current density, and p, the electric charge density. 

A Antisymmetrizing operator A Vector potential P First hyperpolarizability P Resonance parameter in semi-empirical theory B Magnetic field (magnetic induction) X, /r, A, cr Basis functions (atomic orbitals), ab initio or semi-empirical methods rraiipp inrliiflinp basis fiinrHon 7] An infinitesimal scalar rj Absolute hardness h Planck s constant H hjl K h Core or other effective one-electron operator hap Matrix element of a one-electron operator in AO basis Matrix element of a one-electron operator in semi-empirical theory... 

These four vector relations compactly summarize the experimental laws describing all known electrical and magnetic phenomena. In these expressions, p is the electric charge density, J, the current density, E, the electric field and B, the magnetic induction. Maxwell s equations in free space (in the absence of dielectric or magnetic media) can be written... 

Light beams are represented by electromagnetic waves that are described in a medium by four vector fields the electric field E r, t), the magnetic field H r, t), the electric displacement field D r,t), and B r,t) the magnetic induction field (or magnetic flux density). Throughout this chapter we will use bold symbols to denote vector quantities. All field vectors are functions of position and time. In a dielectric medium they satisfy a set of coupled partial differential equations known as Maxwell s equations. In the CGS system of units, they give... 

If the thickness of the surface layer in which a surface exciton-polariton is localized considerably exceeds the lattice constant of a crystal, the electric and magnetic field strength vectors, i.e. vectors E and H of a wave with energy hw in both media (in vacuum and in the crystal the crystal is assumed to be nonmagnetic so that the magnetic induction vector B = H), satisfy Maxwell s equations... 

Another important quantity related to the current density distribution is the nuclear magnetic moment density distribution (or magnetization density distribution) m(r) = r x j(r), which integrates to the magnetic moment = f d r m(r) briefly mentioned above. Finally, the magnetic induction field, generated by the nuclear current density distribution, can be obtained from the vector potential or from the current density distribution as... 

---