Physics · 17
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Use Gauss’s law to find the electric field of a point charge.
Timer not started. Twenty minutes of real work before a solution.
Spherical symmetry: \(\mathbf E\) is radial and constant in magnitude on a sphere of radius \(r\). Flux is \(E\cdot 4\pi r^2=q/\varepsilon_0\), so \(E=q/(4\pi\varepsilon_0 r^2)\).
Attack 2
Find the field of an infinite line charge with density \(\lambda\).
Timer not started. Twenty minutes of real work before a solution.
Cylindrical symmetry. A Gaussian cylinder of radius \(r\) and length \(L\) has flux \(E\cdot 2\pi r L\) through the curved wall and none through the caps. Enclosed charge \(\lambda L\). Thus \(E=\lambda/(2\pi\varepsilon_0 r)\).
Attack 3
Why is the electric field inside the material of a conductor, in electrostatics, equal to zero?
Timer not started. Twenty minutes of real work before a solution.
If it were not, free charges would accelerate. Electrostatics means they are not accelerating, so the field in the conducting material is zero. The excess charge sits on the surface.
Electromagnetism in this course is Maxwell’s equations in integral and differential form, and the special cases you can compute: electrostatics, magnetostatics, induction, and electromagnetic waves in vacuum. The mathematical content is the divergence theorem and Stokes’ theorem applied to the electric and magnetic fields. Gauss’s law computes the field when the symmetry kills the flux through most of a surface. Ampère’s law does the same for steady currents. Faraday’s law is induction. The Lorentz force is how fields act on charges.
OpenStax Volume 2 is the free working text. Griffiths, Introduction to Electrodynamics, ISBN 978-1-108-42041-9, is the paid book you should buy when you want the subject at the level of a physics major; the ISBN was catalogued and is the standard fourth edition’s number in that catalog entry. MIT 8.02 is the course. Irodov’s electrodynamics problems are the ceiling alongside Griffiths’ harder exercises. Feynman Volume II is the preview, not the drill.
A Gaussian surface that does not match the symmetry is a calculation you invented. State the symmetry before you integrate.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Electrostatics. \(\nabla\cdot\mathbf E=\rho/\varepsilon_0\), \(\nabla\times\mathbf E=0\) in electrostatics, so \(\mathbf E=-\nabla V\). Gauss’s law in integral form is the flux statement. The potential of a localized charge distribution that vanishes at infinity is the usual Coulomb integral.
The curl-free claim is Faraday’s law with no changing magnetic field. Existence of the potential is the vector-calculus theorem you proved on the calc-3 page, on a simply connected domain. Gauss’s law is Maxwell’s equation, taken as a law, checked on a point charge by the attack.
Magnetostatics. \(\nabla\cdot\mathbf B=0\). For steady currents, \(\nabla\times\mathbf B=\mu_0\mathbf J\). The Biot–Savart law is the integral solution. The Lorentz force is \(q(\mathbf E+\mathbf v\times\mathbf B)\).
These are the laws in the steady case. Ampère’s law in integral form follows from Stokes’ theorem. Use it for a wire, a sheet, and a solenoid, and say why the symmetry lets the field come out of the integral.
Induction. Faraday: the circulation of \(\mathbf E\) around a closed loop equals minus the derivative of the magnetic flux through a surface bounded by the loop. Lenz’s law is the minus sign. Motional emf for a rod is the Lorentz force on charges in the rod.
The flux rule is the law. For a moving rod on rails, computing \(\oint (\mathbf v\times\mathbf B)\cdot d\mathbf l\) reproduces the flux rule. Do that once so the two pictures are the same fact.
Maxwell and waves. The displacement current \(\mu_0\varepsilon_0\partial\mathbf E/\partial t\) repairs Ampère’s law. In vacuum the fields satisfy the wave equation with speed \(1/\sqrt{\mu_0\varepsilon_0}\). A plane wave has \(\mathbf E\), \(\mathbf B\), and the direction of propagation mutually perpendicular, with \(E=cB\) in SI.
Take the curl of Faraday, use the vacuum Maxwell equations, and the vector identity for \(\nabla\times(\nabla\times\mathbf E)\), together with \(\nabla\cdot\mathbf E=0\). The speed is read off the wave equation. The relation \(E=cB\) is the content of Faraday or Ampère applied to a monochromatic plane wave.
Circuits, as lumped Maxwell. Kirchhoff’s voltage law is Faraday with no changing flux through the circuit. Kirchhoff’s current law is charge conservation. \(RC\), \(RL\), and \(RLC\) transients are linear ODEs.
Write the ODE for a series RLC from the loop rule and solve it with the methods of the ODE page. The underdamped, critically damped, and overdamped cases are the discriminant of that characteristic polynomial.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Feynman Lectures, Volume II, from the official contents
Read once, do not study from this. Open Volume II from that page and read the contents. It is the conceptual preview. It is not where you learn to do Gauss’s law under a clock.
The thing you work through.
OpenStax University Physics Volume 2 and MIT 8.02
Free text plus the course https://ocw.mit.edu/courses/8-02-physics-ii-electricity-and-magnetism-spring-2019/ . Griffiths, ISBN 978-1-108-42041-9, is the paid upgrade when the OpenStax problems feel short on theory.
The problem book.
8.02 problem sets and Griffiths’ problems, if you own the book
The public hard layer is 8.02. Griffiths is paid. Do the exam problems before you buy anything else.
The hardest questions that exist at this level.
Irodov, electrodynamics
The ceiling. Krotov’s relevant problems sit in the same archive item as the mechanics olympiad set: https://archive.org/details/ptitudetestproblemsinphysics
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: one 8.02 exam, plus written derivations of the point-charge field, the line-charge field, and the vacuum wave speed. Ten Irodov electrodynamics problems. Fail if a Gaussian surface is chosen without a symmetry sentence.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
State Gauss’s law, derive the line-charge field, and say what Faraday’s law asserts. No notes.