Physics 201- Introduction to Electricity & Magnetism
December 4- Maxwell's Equations & Electromagnetic Waves

 

 

 

 
 

 

  Maxwell's Equations

Here are our EM equations that we've worked with so far:






How do they look?  Scientists are always looking for the beauty or symmetry in nature.  For the most part, we have not been disappointed.  Do these equations look symmetric?
 
 
 

Recall what we saw before with the charging capacitor:

Let's look at a current carrying wire.  For the most part it is just like the infinite straight wires we've seen before.  The only difference is that were going to put a capacitor in the middle.  Essentially this creates a gap in the wire.



Let's now try to use Ampere's Law to find the magnetic field near the wire.  The main hurdle, as always, is find the enclosed current.  We draw our Amperian loop around the wire, and them ask ourselves how much current goes through the wire.  You can imagine that we take an elastic membrane and stretch it over the loop of integration.
 

Simple, right?

Okay, let's do something different.  Let's pick a different surface.  After all while the one shown above may be the simplest, it is not the only one we can use with Ampere's Law.   If you notice the current used in Ampere's Law is defined to be "the total current passing through any open surface whose parameter is the path of integration."  Consider the following surface-



The path of integration is the same- a circular loop of radius R that encircles the wire, but now well look at a surface that is "ballooned" out a bit.  It is no longer in the plane of the loop.  Remember, it is an open surface (only the gray portion) that is stretched around the wire and plate.  What is the current that passes through this new surface?  Using Ampere's Law, what does this give for the magnetic field along the path of integration?
 
 

This example seems to hint a problem with Ampere's Law (as does the lack of symmetry among the quations). 
 
 

 

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Jeff Phillips
phillips@lmu.edu
Loyola Marymount University
Fall 2002