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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.