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Gauss' Law

By now you are familiar with GAUSS' LAW in its integral form,

$$\epsilon_\circ \; \oSurfIntS \, \Vec{E} \cdot d\Vec{A} = Q_{\rm encl}$$ (22.1)

where $Q_{\rm encl}$ is the electric charge enclosed within the closed surface ${\cal S}$. Except for the "fudge factor" $\epsilon_\circ$, which is just there to make the units come out right, GAUSS' LAW is just a simple statement that electric field "lines" are continuous except when they start or stop on electric charges. In the absence of "sources" (positive charges) or "sinks" (negative charges), electric field lines obey the simple rule, "What goes in must come out." This is what GAUSS' LAW says.

There is also a GAUSS' LAW for the magnetic field $\Vec{B}$; we can write it the same way,

$$\hbox{\rm (some constant)} \; \oSurfIntS \, \Vec{B} \cdot d\Vec{A} = Q_{\rm Magn}$$ (22.2)

where in this case $Q_{\rm Magn}$ refers to the enclosed magnetic charges, of which (so far) none have ever been found! So GAUSS' LAW FOR MAGNETISM is usually written with a zero on the right-hand side of the equation, even though no one is very happy with this lack of symmetry between the electric and magnetic versions.

  

Figure: An infinitesimal volume of space.

Suppose now we apply GAUSS' LAW to a small rectangular region of space where the z axis is chosen to be in the direction of the electric field, as shown in Fig. 22.1.^22.1 The flux of electric field into this volume at the bottom is $E_z(z) \, dx \, dy$. The flux out at the top is $E_z(z+dz) \, dx \, dy$; so the net flux out is just $[E_z(z+dz) - E_z(z)] \, dx \, dy$. The definition of the derivative of E with respect to z gives us $[E_z(z+dz) - E_z(z)] = (\dbyd{E_z}{z}) \, dz$ where the partial derivative is used in acknowledgement of the possibility that E_z may also vary with x and/or y. GAUSS' LAW then reads $\epsilon_\circ (\dbyd{E_z}{z}) \, dx \, dy \, dz = Q_{\rm encl}$. What is $Q_{\rm encl}$? Well, in such a small region there is some approximately constant charge density $\rho$ (charge per unit volume) and the volume of this region is $dV = dx \, dy \, dz$, so GAUSS' LAW reads $\epsilon_\circ (\dbyd{E_z}{z}) \, dV = \rho \, dV$ or just $\epsilon_\circ \; \dbyd{E_z}{z} = \rho$. If we now allow for the possibility of electric flux entering and exiting through the other faces (i.e. $\Vec{E}$ may also have x and/or y components), perfectly analogous arguments hold for those components, with the resultant "outflow-ness" given by

$${\partial{E}_x \over \partial x} \, + \, {\partial{E}_y \ov . . . . . . tial z} \; = \; \Div{E} \; \equiv \; \hbox{\rm div} \, \Vec{E}$$

where the GRADIENT operator $\Vec{\nabla}$ is shown in its cartesian representation (in rectangular coordinates x,y,z). It has completely equivalent representations in other coordinate systems such as spherical ( $r,\theta,\phi$) or cylindrical coordinates, but for illustration purposes the cartesian coordinates are simplest.

We are now ready to write GAUSS' LAW in its compact differential form,

$$\hbox{\fbox{ ${\displaystyle \epsilon_\circ \; \Div{E} = \rho }$\space } }$$ (22.3)

and for the magnetic field, assuming no magnetic charges ( MONOPOLES),

$$\hbox{\fbox{ ${\displaystyle \Div{B} = 0 }$\space } }$$ (22.4)

These are the first two of MAXWELL'S EQUATIONS.