Dimension and Electric Force

A friend recently reminded me of something I thought years ago. I have a little note here.

The Google Docs implementation of a delta function is here.

As you can read in the first note, I got the idea from John Kogut.

This is Electrodynamics, which is a Gauge Theory  forcing fields to change in the presence of charges to conserve them. 

Remember this is a 1-D world, electric field lines cannot leave and come out or back into the source (positive) or sink (negative). Everything has to happen on the line. I use  Gauss's law for this field here. The charge is in the origin, it is represented by a Kronecker Delta function, in my discrete version of this problem, not by a distribution function like the Dirac Delta function.

In Gauss's version of the Law of Conservation of Electric Charge, we have a surface, where field lines representing the amount of charge cross. In 1-D these surfaces, or boundaries, are points to the left and right of the delta function. One electric field line points to the left, and other to the right of the charge. You can see in the Google Docs interpretation, how the value of the field changes sign, once one passes the charge. After passing the charge the value of the field stays constant, until another charge is inside the surface.

Gauss's Law is then, an accountant view of the charges we have inside the bag. Since charge is conserved, the field value remains constant also.

In other dimensions, the field has to decrease as one gets away from the charge, as  in two dimensions, and  in three. There are more directions where the field has to go, to keep charge reckoning alright.