In this lesson, we study capacitance and introduce boundary value problems as a systematic method for analyzing electrostatic systems involving conductors and dielectrics.
We begin with the formal definition of capacitance as the ratio between charge and potential difference and show how this definition is connected to electric fields and material properties. A detailed example of the parallel-plate capacitor illustrates how geometry and permittivity determine capacitance and why capacitance is independent of the amount of stored charge.
The lesson then derives Poisson’s equation and Laplace’s equation by combining Gauss’s law with the potential gradient relation. These equations form the mathematical foundation for solving electrostatic field problems using scalar potential.
A review of the Laplacian operator in Cartesian, cylindrical, and spherical coordinates is provided, followed by a clear step-by-step solution procedure for boundary value problems. Boundary conditions at conductor and dielectric interfaces are applied to determine electric fields, charge distributions, and capacitance.
