need help
msteryou
Lecture_4_2.pptxGauss’s Law
The total of the electric field flux out of a closed surface is equal to the charge enclosed.
Here is the integral form of Gauss’s Law, which is one of Maxwell’s four equations governing classical electromagnetics. This equation says that the total flux of D out of a closed surface is equal to the charge enclosed
This so-called Gaussian surface doesn’t have to correspond to a physical surface, it can be some defined closed make-believe surface.
I am assuming that have seen Gauss’s Law before in your physics courses on electricity and magnetism, so we are going to quickly cover this topic here.
1
Applying Gauss’s Law
Construct an imaginary Gaussian cylinder of radius r and height h:
SOLUTION:
[[[read and complete example]]]
2
Prove Coulomb’s Law using Gauss’s Law
[[[prove it]]]
3
Differential form of Gauss’s Law
Shrink down the Gaussian surface to an infinitesimally small volume (any shape), and you get the differential form of Gauss’s Law:
Can be proved using the divergence theorem.
This differential form is a local law, which tells us what is happening at a given point.
Each of the four Maxwell’s equations can be written in integral form and in differential form. Here is the differential form of Gauss’s Law.
Conceptually, we can view this equation as imagining that we shrink down an arbitrary Gaussian surface until the volume is infinitesimally small. The flux (per volume) flowing out of the resulting differential surface is equal to the volumetric charge density located within the Gaussian surface.
Differential forms are useful because they describe the field at a given point.
Both the differential and integral formulations are useful. The integral formulation can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential formulation is a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using numerical techniques like finite element analysis.
[[[prove using divergence theorem]]]
4
