Michael Mozina quoted an excerpt from Wikipedia that contains an actual equation. How cute.
Here's the other "flaw" in your "proof":
http://en.wikipedia.org/wiki/Magnetic_field#B-field_lines_never_end
B-field lines never end
Main article: Gauss's law for magnetism
Field lines are a useful way to represent any vector field and often reveal sophisticated properties of fields quite simply.
One important property of the B-field revealed this way is that magnetic B field lines neither start nor end (mathematically, B is a solenoidal vector field); a field line either extends to infinity or wraps around to form a closed curve.[nb 8]
To date no exception to this rule has been found. (See magnetic monopole below.)
Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet B-field lines continue through the magnet from the south pole back to the north.[nb 9] If a B-field line enters a magnet somewhere it has to leave somewhere else;
it is not allowed to have an end point. Magnetic poles, therefore, always come in N and S pairs. Cutting a magnet in half results in two separate magnets each with both a north and a south pole.
More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the 'number'[nb 10] of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent to:
where the integral is a surface integral over the closed surface S (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since dA points outward, the dot product in the integral is positive for B-field pointing out and negative for B-field pointing in.
There is also a corresponding differential form of this equation covered in Maxwell's equations below.
Michael Mozina says he's found a "flaw" in my "proof". According to the rules of mathematical and scientific discourse, he is now obliged to identify the flaw in my proof, which proceeds by proving the differential form that, according to the text
Michael Mozina quoted, is equivalent to the equation
Michael Mozina quoted above.
For easy reference, here's the proof outline I gave earlier, followed by the calculation that confirms that
the experiment I've been suggesting satisfies the differential form of Gauss's Law for Magnetism:
When you calculate its divergence, you will find that Gauss's Law for Magnetism is satisfied. If you vary the current slowly throughout the experiment, the time derivative will be negligible. By superposition, you may then conclude that Gauss's Law is satisfied throughout
the experiment I've been suggesting. Since that freshman-level exercise also demonstrates (slow!) magnetic reconnection, you will have proved to yourself that magnetic reconnection is consistent with Maxwell's equations.
If you have trouble calculating the divergence, I suggest you use cylindrical coordinates (with the rod at r=0) and consider
[latex]
\[
V_\delta(r, \theta, z) = \left\{ \langle r^\prime, \theta^\prime, z^\prime \rangle
\; | \;
|r-r^\prime| \leq \delta \; \& \;
|\theta-\theta^\prime| \leq \delta \; \& \;
|z-z^\prime| \leq \delta \right\}
\]
[/latex]
for small δ > 0.
Reality Check has been trying to give you an even more basic hint:
Starting from the above, plus
Reality Check's hint and the five equations I quoted in an earlier message: Given any r > 0, define
[latex]
\[
S^{(r+)}_\delta(r, \theta, z) = \left\{ \langle r + \delta, \theta^\prime, z^\prime \rangle
\; : \;
|\theta-\theta^\prime| \leq \delta \; \& \;
|z-z^\prime| \leq \delta \right\}
\]
[/latex]
[latex]
\[
S^{(r-)}_\delta(r, \theta, z) = \left\{ \langle r - \delta, \theta^\prime, z^\prime \rangle
\; : \;
|\theta-\theta^\prime| \leq \delta \; \& \;
|z-z^\prime| \leq \delta \right\}
\]
[/latex]
[latex]
\[
S^{(\theta+)}_\delta(r, \theta, z) = \left\{ \langle r^\prime, \theta + \delta, z^\prime \rangle
\; : \;
|r-r^\prime| \leq \delta \; \& \;
|z-z^\prime| \leq \delta \right\}
\]
[/latex]
[latex]
\[
S^{(\theta-)}_\delta(r, \theta, z) = \left\{ \langle r^\prime, \theta - \delta, z^\prime \rangle
\; : \;
|r-r^\prime| \leq \delta \; \& \;
|z-z^\prime| \leq \delta \right\}
\]
[/latex]
[latex]
\[
S^{(z+)}_\delta(r, \theta, z) = \left\{ \langle r^\prime, \theta^\prime, z + \delta \rangle
\; : \;
|r-r^\prime| \leq \delta \; \& \;
|\theta-\theta^\prime| \leq \delta \right\}
\]
[/latex]
[latex]
\[
S^{(z-)}_\delta(r, \theta, z) = \left\{ \langle r^\prime, \theta^\prime, z - \delta \rangle
\; : \;
|r-r^\prime| \leq \delta \; \& \;
|\theta-\theta^\prime| \leq \delta \right\}
\]
[/latex]
[latex]
\begin{align*}
S_\delta(r, \theta, z) =
& S^{(r+)}_\delta(r, \theta, z) \cup
S^{(r-)}_\delta(r, \theta, z) \\
&\cup
S^{(\theta+)}_\delta(r, \theta, z) \cup
S^{(\theta-)}_\delta(r, \theta, z) \\
&\cup
S^{(z+)}_\delta(r, \theta, z) \cup
S^{(z-)}_\delta(r, \theta, z)
\end{align*}
[/latex]
For the magnetic field of a single current-carrying rod (which
Michael Mozina has been unable to calculate or even to describe, so I'm not going to give that away here) and any δ such that 0 < δ < r:
[latex]
\begin{align*}
\oint_{S_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} =
& \int_{S^{(r+)}_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} &+
\int_{S^{(r-)}_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} \\
&+ \int_{S^{(\theta+)}_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} &+
\int_{S^{(\theta-)}_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} \\
&+ \int_{S^{(z+)}_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} &+
\int_{S^{(z-)}_\delta(r,\theta,z)} \hbox{{\bf B}} \cdot d \hbox{{\bf A}} \\
= & \; 0
\end{align*}
[/latex]
where I left out the next-to-last step as an exercise for
Michael Mozina.
Taking the limit as δ goes to zero (which is trivial) yields the differential form of Gauss's Law for Magnetism. By superposition, Gauss's Law holds for the entire experiment.
Since magnetic reconnection does indeed occur within the experiment, and
Michael Mozina's denials of that fact are based on nothing more than his total lack of understanding of magnetic reconnection, this shows that magnetic reconnection is consistent with Gauss's Law for Magnetism, despite repeated protests from people who literally do not know what they're talking about.
!
!