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Gauss' Law

Module by: Bill Wilson. E-mail the author

Summary: Introduction of Gauss' Law, one of the field theories.

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

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

s,Dd S = Q encl =v,ρvd V s S D Q encl v V ρ v
(1)
where D D is the electric displacement vector, which is related to the electric field vector, E E, by the relationship D=εE D ε E . εε is called the dielectric constant. In silicon it has a value of 1.1×10-12Fcm 1.1-12 F cm . (Note that D D must have units of Coulombscm2 Coulombs cm 2 to have everything work out OK.) Qencl Qencl is the total amount of charge enclosed in the volume V V, which is obtained by doing a volume integral of the charge density ρv ρ v .

Figure 1: Pictorial representation of Gauss' Law.
Figure 1 (2_31.png)

Equation 1 just says that if you add up the surface integral of the displacement vector DD over a closed surface SS , what you get is the sum of the total charge enclosed by that surface. Useful as it is, the integral form of Gauss' Law, (which is what Equation 1 is) will not help us much in understanding the details of the depletion region. We will have to convert this equation to its differential form. We do this by first shrinking down the volume VV until we can treat the charge density ρv ρ v as a constant ρρ, and replace the volume integral with a simple product. Since we are making VV small, let's call it ΔV Δ V to remind us that we are talking about just a small quantity.

Δv,ρvd V ρΔv Δ v V ρ v ρ Δ v
(2)
And thus, Gauss' Law becomes:
s,Dd S =εs,Ed S =ρΔV s S D ε s S E ρ Δ V
(3)
or
1ΔV(s,Ed S )=ρε 1 Δ V s S E ρ ε
(4)
Now, by definition the limit of the LHS of Equation 4 as ΔV0 Δ V 0 is known as the divergence of the vector EE, divE E . Thus we have
limit   ΔV 0 1ΔV(s,Ed S )=divE=ρε Δ V 0 1 Δ V s S E E ρ ε
(5)
Note what this says about the divergence. The divergence of the vector EE is the limit of the surface integral of EE over a volume VV, normalized by the volume itself, as the volume shrinks to zero. I like to think of as a kind of "point surface integral" of the vector EE.

Figure 2: Small volume for divergence
Figure 2 (2_32.png)

If EE only varies in one dimension, which is what we are working with right now, the expression for the divergence is particularly simple. It is easy to work out what it is from a simple picture. Looking at Figure 2 we see that if EE is only pointed along one direction (let's say xx) and is only a function of xx, then the surface integral of EE over the volume ΔV=ΔxΔyΔz Δ V Δ x Δ y Δ z is particularly easy to calculate.

s,Ed S =Ex+ΔxΔyΔzExΔyΔz s S E E x Δ x Δ y Δ z E x Δ y Δ z
(6)
Where we remember that the surface integral is defined as being positive for an outward pointing vector and negative for one which points into the volume enclosed by the surface. Now we use the definition of the divergence
divE=limit   ΔV 0 1ΔV(s,Ed S )=limit   ΔV 0 (Ex+ΔxEx)ΔyΔzΔxΔyΔz=limit   ΔV 0 Ex+ΔxExΔx=Ex x E Δ V 0 1 Δ V s S E Δ V 0 E x Δ x E x Δ y Δ z Δ x Δ y Δ z Δ V 0 E x Δ x E x Δ x x E x
(7)
So, we have for the differential form of Gauss' law:
Ex x =ρxε x E x ρ x ε
(8)

Thus, in our case, the rate of change of EE with xx, dd x E x E , or the slope of Ex E x is just equal to the charge density, ρx ρ x , divided by εε.

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