Below are my notes taken from MIT 8.02x Electricity and Magnetism (with experiments) taught by Dr. Walter Lewin back in the early 2000's.
Playlist: https://www.youtube.com/playlist?list=PLyQSN7X0ro2314mKyUiOILaOC2hk6Pc3j
Lecture 1: Coulomb's Law, Electric Charges and Forces
Lecture 2: Electric Field Lines, Superposition, Inductive Charging, Induced Dipoles Lecture 3: Electrostatics, Electric Flux, Gauss's Law Lecture 4: Electrostatic Potential, Equipotential Surfaces Lecture 5: E = - Grad(V), Conductors, Electrostatic Shielding Lecture 6: High voltage breakdown, lightning, sparks, St. Elmo's fire Lecture 7: Capacitance, Electric Field Energy Lecture 8: Polarization, Dielectrics, Van de Graaff Generator, Capacitors [*] Lecture 9: Electric Currents, Resistivity, Conductivity, Ohm's Law Lecture 10: Batteries, Kirchoff's Rule, Circuits Lecture 11: Magnetic Fields, Lorentz Force, Electric Motors Lecture 12: Exam 1 Review Lecture 13: Moving charges in B-fields, Cyclotrons Lecture 14: Biot-Savart, Div(B)=0, High-Voltage Power Lines Lecture 15: Ampere's Law, Solenoids Lecture 16: Electromagnetic Induction, Faraday's Law, Lenz's Law Lecture 17: Motional EMF, Dynamos, Eddy Currents, Magnetic Braking Lecture 18: Displacement Current, Synchronous Motors Lecture 19: Magnetic LEvitation, Superconductivity, Aurora Borealis Lecture 20: Inductance, RL Circuits, Magnetic Field Energy Lecture 21: Magnetism... Diamagnetism, Paramagnetism, Ferromagnetism Lecture 22: Magnetism... Magnetic field effect on materials, historicis Lecture 23: Exam 2 Review Lecture 24: RC Circuits and Transformers Lecture 25: Driven LRC circuits, metal detectors Lecture 26: Travelling waves, standing waves, musical instruments Lecture 27: Destructive Resonance, Electromagnetic Waves, Speed of Light Lecture 28: Poynting Vector, Oscillating Charges, Polarization, Radiation Pressure Lecture 29: Snell's Law, Index of Refraction, Huygen's Principle Lecture 30: Polarizers, Malus' Law, Light Scattering, Blue Skies Lecture 31: Rainbows, Halos, Sun Dogs Lecture 32: Exam 3 Review Lecture 33: Double-Slit Interference, Interferometers Lecture 34: Diffraction Gratings, Angular Resolution Lecture 35: Doppler Effect, Big Bang, Cosmology Lecture 36: Last Lecture
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A couple of years back I went through the MIT Open Courseware videos from Dr. Arthur Mattuck's Spring 2003 Differential Equations course. I watched the videos in the morning while eating breakfast and would re-derive the results from the video when I got to school. Unfortunately, I did not keep very detailed notes. I have found myself going back to the videos to refresh my knowledge on certain subjects, but I can't always remember which video contains which topic. For future reference I will start keeping track of keywords for each video and putting them below.
Playlist: https://www.youtube.com/playlist?list=PLEC88901EBADDD980
Lecture 1: First order ODEs of the form y' = f(x,y)
Lecture 2: Numerical solutions to differential equations by Euler's Method Lecture 3: First-order linear equations and the separation of variables Lecture 4: Change of variables and substitutions Lecture 5: Autonomous differential equations, integral curves, critical points Lecture 6: Complex numbers, polar form Lecture 7: Linear first order ODEs
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Here are my notes from Gilbert Strang's MIT Open Courseware Class in Linear Algebra. Here is a link to the YouTube playlist of all lectures. Lecture 1: Column and Row Space Visualization of Matrix [Really awesome lecture]
Lecture 2: Elimination, Back-Substitution, Elimination Matrices, Matrix Multiplication Lecture 3: Matrix Multiplication, Matrix Inversion, Gauss-Jordan Lecture 4: Inverse of AB, Product of Elimination Matrices, A=LU Lecture 5: PA=LU, Vector Spaces and Subspaces Lecture 6: Vector Spaces and Subspaces Lecture 7: Algorithm for Finding the Nullspace [Ax=0] Lecture 8: Complete Solution of Ax=b Lecture 9: Linear Independence, Spanning a Space, Basis, Dimension Lecture 10: The Four Fundamental Subspaces Lecture 11: Bases of Vector Spaces, Rank 1 Matrices Lecture 12: Graphs and Networks Lecture 13: Review Practice Problems Lecture 14: Orthogonal Vectors and Subspaces Lecture 15: Projection, Least Squares, Projection Matrix Lecture 16: Projection Matrices, Least Squared continued Lecture 17: Orthogonal Basis, Gram-Schmidt, Orthogonal Matrix Lecture 18: Determinants 1/2 Lecture 19: Determinants 2/2, Cofactor Expansion Lecture 20: Matrix Inversion and Determinants [Great Lecture, Lost Notes...] Lecture 21: Introduction to Eigenvalues and Eigenvectors Lecture 22: Matrix Diagonalization [S-1 * A * S = L], Powers of A, Fibonacci Example [*] Lecture 23: Application of Linear Algebra to Differential Equations Lecture 24: Markov Matrices and Fourier Series Lecture 24b: Review and Practice Problems Lecture 25: Symmetric Matrices Facts Lecture 26: Complex Vectors and Matrices, Fourier Matrix, FFT [Matrix Factorization] Lecture 27: Positive Definite Matrices Lecture 28: Positive Definite Matrices, Similar Matrices Lecture 29: Singular Value Decomposition [SVD] Lecture 30: Linear Transformations Lecture 31: Change of Basis, Image Compression Lecture 32: Review and Practice Problems Lecture 33: Inverses, Course Review Lecture 34: Review I have been covering the topic of plane waves quite a bit recently. After my post on the "phase circle" I got to thinking about wave polarization. In this post I will describe the polarization of plane waves. For now, I am going to upload a few simple videos, then I will come back to provide detail in a bit. Initial DetailsThe examples shown below display the x-component (red), y-component (yellow), and total field (purple) of a time varying signal of the form: F = A*cos(wt - Bz + phi) Where we are fixed at a location in space (say, z=0). If we decompose F into x and y, we have F = A*cos(wt + phi_x)*x_hat + A*cos(wt + phi_y)*y_hat where the amplitude, A, is a complex number. Now we can enforce a difference in phase between the x and y component of our field. It is this difference in phase between our components that gives us a sense of polarization. If phi_x equals phi_y then the wave is said to be linearly polarized. If phi_x equals phi_y +- 90 degrees then the wave is circularly polarized. Otherwise, in the most general case, the sum of the components trace out an ellipse as a function of time, called elliptical polarization. Linear PolarizationCircular PolarizationElliptical Polarization
This morning as I was browsing the internet i came across this neat graphic. Every point is moving back and forth along a single axis in a sinusoid manner. This person incrementally adds more points to the plot and keeps track of the path of motion in a very colorful way!
I saw the graphic below a few years back. It is much simpler but illustrates the same sinsusoidal motion along a fixed direction where each point is offset in phase from its neighbor rendering the interesting rotation effect we see.
My Implementation
I couldn't resist putting together my own implementation. Below is a gif I created and the code I used to generate it. The sinusoidal variation along an axis is relatively straightforward, the only part that requires much thought is the initial phase offset. If you don't get that right, things get weird, but they also get very cool!
What's Actually Happening?
To understand what's actually happening, try focusing on one single point. Notice that it only moves back and forth along a single line, it does not rotate around the page. The rotation effect is a result of all the points moving at once. I can tell a point to move back and forth along a line using the standard equation for a line, y = mx + b, or I can decompose my line into x- and y-directed components. I chose to define unit vectors x and y that together always point me in the correct direction but whose amplitude I can vary sinusoidally. Look back at that point from before. Notice as the point gets closer to the edge of the frame it appears to slow down and as it moves to the center it speeds up. Using trigonometry, the amplitude varies like:
A = cos(theta) and theta takes on values between 0 and 2*pi so that A varies from -1 to 1. The amplitude at each point varies in this same way, but I include another argument in my sinusoid to offset each point in phase. A = cos(theta + phi) where phi is called the phase offset and allows me to start each point at a different initial position along the axis of motion. MATLAB Code
% Simple code to plot a phase circle
deltapi = pi/8; theta = 0 : deltapi : pi-deltapi; x = cos(theta); % x-directed vector y = sin(theta); % y-directed vector fig1 = figure(1); for angle = 0: pi/64 : 2*pi A = cos(angle+theta); % Amplitude scaling factor with phase offset figure(1); plot(A.*x, A.*y, 'ok', 'MarkerSize', 10, 'MarkerFaceColor','k'); hold on; % Plot straight line paths for i=1:length(theta) plot(2*[-cos(theta(i)) +cos(theta(i))], 2*[-sin(theta(i)) +sin(theta(i))],'color','k'); end hold off; xlim([-1 1]); ylim([-1 1]); drawnow; % pause(.15); end |
AcademicsHere I will share material relating to my work in the classroom, both as a student and instructor. Archives
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