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 weíre 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 weíll 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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