1.5 Gauss' law

Now we have to review some field theory. We will be usingfields from time to time in this course, and when we need some aspect of field theory, we will introduce what we need at thatpoint. This seems to make more sense than spending several weeks talking about a lot of abstract theory without seeing howor why it can be useful.

The first thing we need to remember is Gauss' Law . Gauss' Law, like most of the fundamental laws ofelectromagnetism comes not from first principle, but rather from empirical observation and attempts to matchexperiments with some kind of self-consistent mathematical framework. Gauss' Law states that:

just says that if you add up the surface integral of the displacement vector $D$ over a closed surface $S$ , what you get is the sum of the total charge enclosed by that surface. Useful as it is, theintegral form of Gauss' Law, (which is what is) will not help us much in understanding the details of the depletion region. We will have to convertthis equation to its differential form. We do this by first shrinking down the volume $V$ until we can treat the charge density $\rho (v)$ as a constant $\rho$ , and replace the volume integral with a simple product. Since weare making $V$ small, let's call it $\Delta (V)$ to remind us that we are talking about just a small quantity.

If $E$ only varies in one dimension, which is what we are working with right now, the expression forthe divergence is particularly simple. It is easy to work out what it is from a simple picture. Looking at we see that if $E$ is only pointed along one direction (let's say $x$ ) and is only a function of $x$ , then the surface integral of $E$ over the volume $\Delta (V)=\Delta (x)\Delta (y)\Delta (z)$ is particularly easy to calculate.

Thus, in our case, the rate of change of $E$ with $x$ , $\frac{d E}{d x}$ , or the slope of $E(x)$ is just equal to the charge density, $\rho (x)$ , divided by $\epsilon$ .