Capacitance is defined as the amount of charge stored in a capacitor per volt across the capacitor. The value of the capacitance is a measure of how rapidly a capacitor stores electric potential energy as the capacitor is getting charged up.

# Finding the Electric Field produced by a Parallel-Plate Capacitor

In this lesson, we'll determine the electric field generated by a charged plate. We'll show that a charged plate generates a constant electric field. Then, we'll find the electric field produced by two, parallel, charged plates (a parallel-plate capacitor). We'll show that the electric field in between the plates has a constant magnitude \(\frac{σ}{ε_0}\). We'll also show that the direction of the electric field is a constant pointing from the positively charged plate to the negatively charged plate in a direction that is perpendicular to both plates. Lastly, we'll also prove that the electric field "outside" of the capacitor is zero everywhere.

# Calculating the amount of Electric Potential Energy Stored in a Capacitor

# Maclaurin/Taylor Polynomials and Series

In this lesson, we'll derive Maclaurin/Taylor polynomials which are used to "approximate" arbitrary functions which are smooth and continuous. More generally, they are used to give a local approximation of such functions. We'll also derive Maclaurin/Taylor series where the approximation becomes exact.

# Introduction to Integrals

# Terraforming and Colonizing Mars

*Terraforming *a world just means to make it more Earth-like. In this article, we'll discuss various techniques which have been proposed by scientists and engineers that would make Mars more like our home planet. We shall also discuss a potential scheme of future events which might occur as humans terraform and colonize Mars.

# Time Dilation

Time dilation is one of the many bizarre consequences of the speed of light being the same for everybody. For any observers in inertial frames of reference moving away from one another at a relative velocity which is close to that of the speed of light, they will see each others clocks run more slowly. In this section, we'll explore one of Einstein's original thought experiment: a train with a light clock moving past an observer standing idly at the train station. We'll see that the constancy of light speed with respect to all observers implies that time must slow down when watching events unfold in one frame of reference from another frame of reference if those two frames of reference are moving relative to each other. We'll start off by showing this for light beams bouncing off of two mirrors, but at the end we'll realize that this same argument applies to photons emitted between two atoms. The latter results in all physical clocks, chemical or biological, running more slowly.

# Time-Independant Schrodinger Equation: Free Particle and Particle in One-Dimensional Box

In this section, we'll begin by seeing how Schrodinger's time-independent equation can be used to determine the wave function of a free particle. After that, we'll use Schrodinger's time-independent equation to solve for the allowed, quantized wave functions and allowed, energy eigenvalues of a "particle in a box"; this will be useful later on as a qualitative understanding of the quantized wave functions and energy eigenvalues of atoms.

# Nuclear Fusion Engines

# Quantum Mechanics: Math Interlude

When matrix mechanics was first being developed by pioneers like Werner Heisenberg there was a "language barrier," so to speak. Most physicists of his time were accustomed to the language of differential calculus; but Heisenberg expressed many of his groundbreaking results in the abstract language of row vectors, column vectors, complex numbers, and matrices. This made it very difficult for the physicists of his time to understand his work. In order to understand quantum mechanics, we must understand the mathematical language in which it is expressed. Our goal, in this lesson, will be to familiarize ourselves with this language.