#current density

[Click here for a PDF version of this post] Motivation I’ve been assembling a table of all the Green’s functions that can be used in electrodynamics. There’s one set of those Green’s functions left to fill in, the Green’s functions for the spacetime gradient: \begin{equation}\label{eqn:spacetimeGradientGreens:20} \lr{\spacegrad + \inv{c}\PD{t}{}} G(\Bx, \Bx’, t, t’) = \delta(\Bx – \Bx’)\delta(t –…

Introduction Current density is a measure of the amount of electric current that flows through a particular cross-sectional area of a material per unit time. It is a fundamental concept in electromagnetism and is particularly important in the fields of electronics and materials science. Definition The current density $\vec{J}$ is defined as the electric current $\vec{I}$ per unit…

Gauge freedom and four-potentials in the STA form of Maxwell’s equation.

[If mathjax doesn’t display properly for you, click here for a PDF of this post] Motivation. In a recent video on the tensor structure of Maxwell’s equation, I made a little side trip down the road of potential solutions and gauge transformations. I thought that was worth writing up in text form. The initial point of that side trip was just to point out that the Faraday tensor can be expressed in…

Generalizing Ampere’s law using geometric algebra.

[Click here for a PDF of this post with nicer formatting, and oriented integrals. All oriented integrals in this post have a clockwise direction.].

The question I’d like to explore in this post is how Ampere’s law, the relationship between the line integral of the magnetic field to current (i.e. the enclosed current) \begin{equation}\label{eqn:flux:20} \oint_{\partial A} d\Bx \cdot \BH =…

Magnetostatic force and torque

[Click here for a PDF of this post with nicer formatting]

In Jackson [1], the following equations for the vector potential, magnetostatic force and torque are derived

\begin{equation}\label{eqn:magnetostaticsJacksonNotesForceAndTorque:20} \Bm = \inv{2} \int \Bx’ \cross \BJ(\Bx’) d^3 x’ \end{equation} \begin{equation}\label{eqn:magnetostaticsJacksonNotesForceAndTorque:40} \BF = \spacegrad( \Bm…

Green’s function inversion of the magnetostatic equation

[Click here for a PDF of this post with nicer formatting]

A previous example of inverting a gradient equation was the electrostatics equation. We can do the same for the magnetostatics equation, which has the following Geometric Algebra form in linear media

\begin{equation}\label{eqn:biotSavartGreens:20} \spacegrad I \BB = – \mu \BJ. \end{equation}

The Green’s inversion of this is \begin{equation…

[Click here for a PDF of this post with nicer formatting]

Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti.

Where we left off

\begin{equation}\label{eqn:qmLectur…