10-Gauss' Law.pdf - ECE 303 Sum2017 Notes Set 10 Gauss Law...

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ECE 303 - Sum2017 Notes Set 10: Gauss’ Law 1 INTRODUCTION A fundamental property we have seen is that charges create electric fields. Sometimes the charges are discrete charges or point charges, as we studied previously. However, in many physical structures it is more instructive to consider these discrete charges as being distributed over a spatial extent. This creates the concept of a distributed region in space which has a particular “charge density”. An example of such a charged region is the depletion region near the PN-junction of a semiconductor diode. If we begin with a knowledge of the charge density in a region, then Gauss’ Law gives us a way to compute the electric field E over the region. This might at first seem to be more complicated than just using Coulomb’s Law to compute the E -field given just a point charge Q . However, the region which is modeled as having a charge density may actually comprise an enormous number of individual discrete charges. To use Coulomb’s Law to find the E - field component due to each of these discrete charges may not be computationally feasable. Consequently, an alternate approach is to use Gauss’ Law, starting with the charge density due to the individual charges. There are two forms of Gauss’ Law: the Integral Form and the Differential Form. The integral form considers the E -field existing over a surface in space. However, the differential form gives us E -field at every point within the surface, as well as the E -field on the surface. For these reasons, the differential form is also often called the Point Form of Gauss’ Law. This set of Notes studies the differential form of Gauss’ Law. The PN-junction voltage from solid-state electronics is used as a motivating application to show us how Gauss’ Law can be applied to a useful problem. If you take a course in solid-state electronics you may learn about the PN-junction in much more detail. The diffusion equation and current density equations are often used to develop very quantitative results for hole and electron flow across the junction. In this notes Set, we will only examine the PN-junction from the standpoint of Gauss’ Law, which is in keeping with our study of electromagnetic phenomena.
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ECE 303 - Sum2017 Notes Set 10: Gauss’ Law 2 THE P-N JUNCTION This page describes a simple model for the hole and electron densities on the different sides of the PN-junction, prior to their being joined to actually form the junction. On the following page, we will place the sides in contact and observe the hole and electron movement. Lattice Geometry In the figure below, the bulk material on both sides of the dotted line is silicon. Atoms from column V in the periodic chart have been added to the silicon lattice to form the N-type region on the right side and atoms from column III in the periodic chart have been added to the silicon lattice to form the P-type region on the left side. These added atoms are often called the impurities.
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