The lack of oxygen in Mars' atmosphere and running liquid water on its surface is very inconveniant for any humans living their since oxygen and liquid water are necessary for humans to survive. Fortunatelly, there is an abundance of *frozen water* on Mars' surface. In this lesson, we'll discuss various techniques which can be used to extract all of this water. Once the water is obtained, by performing electrolysis on the water we can distill all of the oxygen from that water we need.

# Capacitance

# 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.

# Alcubierre Warp Drive

A spaceship using an Alcubierre warp drive would involve assembling a ring of negative energy around the spaceship which would distort spacetime in this particular way: the spaceship sits in a "bubble" of flat, Minkowski spacetime which is "pushed" by expanding space behind it and "pulled" by contracting space infront of it. The spaceship does not move through space, but rather space itself moves and carries along the spaceship for the ride. Since general relativity places no limit on how fast space can move, the space can "carry" the warp bubble and spaceship away at faster than light (FTL) speeds.

# Introduction to Limits and Derivatives

Limits describe what one quantity approaches as some other quantity approaches a given value. This concept is the basis of calculus because it is used to define both derivatives and integrals. In this lesson, we'll try to wrap our minds around what the notion of a limit is and use it to define the derivative function.

# Introduction to Special Relativity

When Einstein was first working on developing his special theory of relativity, he was studying both Newtonian mechanics and Maxwell's equations: the two pillars of modern physics of his time. He noticed that these two theories contradicted one another in a very deep way. One of these pillars must fall. Newtonian mechanics implies the Galilean transformation equations—something which we are all already familiar with. These equations are consonant with our everyday intuitions: they tell us that velocities add. This makes sense: If I watch someone standing on top of semi throw a baseball at 5 m/s (in the same direction the truck is moving) and the truck wizzes by me at 30 m/s, then I'll see the baseball travel away from me at 35 m/s. But Maxwell's equations predict something very strange. These equations predict that a light beam will move away from you at the same speed whether your standing still or, as in another one of Einstein's thought experiments, "chasing" the light beam at 99% the speed of light. The reconciliation between Newtonian mechanics and Maxwell's equations is called the special theory of relativity. When we modify Newtonian mechanics to allow for the constancy of light speed for all observers, we get some very strange, bizarre, but also very profound consequences.

# 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.

# The Eigenvalues of any Observable \(\hat{L}\) must be Real

The eigenvalues are the values that you measure in an experiment: for example, the position or momentum of a particle. Because the eigenvalues are what you measure, it wouldn't make physical sense if the eigenvalue of an observable had an imaginary part. In this lesson, we'll prove that the eigenvalue of any observable is a real number.

# Pauli Matrices

The three operators—\(\hat{σ}_x\), \(\hat{σ}_y\), and \(\hat{σ}_z\)—are associated with the measurements of the \(x\), \(y\), and \(z\) components of spin of a quantum particle, respectively. In this lesson, we'll represent each of these three operators as matrices and solve for the entries in each matrix. These three matrices are called the Pauli matrices.

# The Einstein Equivalence Principle

When Einstein first realized that someone falling in an elevator near Earth's surface would experience all the same effects as another person riding in a rocket ship accelerating at 9.8 meters per second, he described it as "the happiest thought of his life." He realized that all the laws of physics and any physical experiment done in either reference frame would be identical and completely indistinguishable. This is because the effects of gravity in a constant gravitational field are identical to the effects of constant acceleration. This lead Einstein to postulate that gravity and acceleration are equivalent. Analogous to how all the physical consequences of special relativity could be derived from the postulation of the constancy of light speed and the sameness of physical laws in all inertial reference frames, all of the physical consequences of general relativity are derived from the postulate that acceleration and gravity are equivalent and that the laws of physics are the same in all reference frames. The former has been called the Einstein Equivalence Principle. There are various different forms that this statement can take, but in this lesson we shall describe the strong version of the Einstein Equivalence Principle.