Ah yes, the lecture of yesterday. Who remembers it?
We covered divergence, which involved a derivation for the divergence expressed in Cartesian co-ordinates. Then we combined the divergence concept with the integral form of Gauss's Law, and arrived at the differential form of Gauss's Law (also known as the point form). After that, we used the fact that the electric field vector is given by the negative gradient of the voltage potential, substituted that into the point form of Gauss's Law, and presto mundo we had Poisson's equation. Setting the volume charge density to zero then led to Laplace's equation.
We tried one practice problem, which was to use Laplace's equation to find an equation for the voltage in the region between two parallel plate conductors of known potential and known separation distance. From the voltage equation we found the electric field between the plates.
I pointed out that the electrical field is constant, and increases with increasing plate voltage difference and decreasing plate separation. This has important ramifications for capacitor design (e.g. allowable dielectric, voltage rating).
We covered divergence, which involved a derivation for the divergence expressed in Cartesian co-ordinates. Then we combined the divergence concept with the integral form of Gauss's Law, and arrived at the differential form of Gauss's Law (also known as the point form). After that, we used the fact that the electric field vector is given by the negative gradient of the voltage potential, substituted that into the point form of Gauss's Law, and presto mundo we had Poisson's equation. Setting the volume charge density to zero then led to Laplace's equation.
We tried one practice problem, which was to use Laplace's equation to find an equation for the voltage in the region between two parallel plate conductors of known potential and known separation distance. From the voltage equation we found the electric field between the plates.
I pointed out that the electrical field is constant, and increases with increasing plate voltage difference and decreasing plate separation. This has important ramifications for capacitor design (e.g. allowable dielectric, voltage rating).
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