Capacitors, Magnetic Circuits, and Transformers is a free introductory textbook on the physics of capacitors, coils, and transformers. See the editorial for more information.... 
Home Capacitance and Related Effects Gauss's Theorem  
See also: Electric Flux  
Search the VIAS Library  Index  
Gauss's Theorem
Gauss's theorem states that the net electric flux emanating from an imaginary closed surface is equal to the net electric charge enclosed within that surface.
Thus in Fig. 26 the electric flux that emanates from surface (1) is equal to +q and that from surface (2) is equal to q. The flux that emanates from surface (3) is zero because that surface does not enclose any charge. The flux that emanates from surface (4) is likewise zero because that surface encloses equal but opposite charges, the net charge being zero.
Consider a differential area da of a closed surface, as shown in Fig. 27, with an electric flux density of D coulombs per square meter. The flux emanating from the surface represented by this differential area is expressed by
It must be kept in mind that the vector da, which represents the area of a surface, is normal to that surface. The product in Eq. 218 is a scalar product. The scalar product of the infinitesimal area and the normal component of flux density integrated over the entire enclosing surface yields the total flux emanating from this surface, i.e.
This flux may also be expressed in terms of the surface integral of electric field intensity. If the enclosing surface lies completely within a medium having a fixed relative dielectric constant k_{r}, the relationship in Eq. 219 can also be expressed by
because
On the other hand, if the enclosing surface lies in a medium in which the relative dielectric constant k_{r} is not fixed, then the flux is expressed in terms of electric field intensity by
An example of a variation in k_{r} is the case of a medium in which k_{r} is a function of the electric field intensity E. Another is one in which the enclosing surface might be partly in air, partly in oil, and partly in porcelain. Equations 219, 220, and 221 are valid regardless of the medium within the enclosing surface.


Home Capacitance and Related Effects Gauss's Theorem 