Vector held flux

Alternatively we can say that the divergence of a vector v is the number of vector lines originating in an infinitely small volume or, which is the same thing, the flux of the vector held through the surface of this volume. 

Figure 5.43. Heteronuclear (as well as homonuclear cf. Pig. S.42) molecules can be partitioned into atoms. S represents a slice through the zero-flux surface that deflnes the atoms A and B in a molecule AB. The lines with arrows are the trajectories of the gradient vector Held. S passes through the bond critical point C and is not crossed by any trajectory lines.

Both the electric Held (r,t) and the magnetic flux density B(rj) may be derived from the vector potential A(r.r). Using the so-called Coulomb gauge one obtains... 

Gauss law The total electric flux normal to a closed surface in an electric held is proportional to the algebraic sum of the electric charges within the surface. A similar law applies to surfaces drawn in a magnetic field and the law can be generalized for any vector field through a closed surface. It was first stated by Karl Gauss. 

Gauss s theorem equates the flux of a vector field through a closed surface with the divergence of that same held throughout its volume. This result will be useful in the following chapters. 

These generalizations now lead to a remarkable result called Gauss s law. Because the electric held E(r) from any constellation of hxed charges is always the vector sum of the holds from the component charges, E = Ei -fE -tE -1- -1-E , the flux through any closed surface around any constellation of charges is... 

Equation (20.17) is Gauss s law. it says that a very complex quantity can be computed by a very simple recipe. If you want to hnd the flux of the electrostatic held through any bounding balloon, no matter how complex its shape, you don t need to compute the electrostatic held vector at each point in space, hnd its dot product with all the surface elements, and integrate. Instead, you can compute the flux simply by counting up the total net charge contained within the bounding surface, and divide by fo, a constant. 

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