Talk:MyOpenMath/Electromagnetic momentum

Editor's scrapbook
https://en.wikiversity.org/wiki/File:E14-V20-B1.gif  File:Wave_packet.gif




 * scienceworld-wolfram: $S/c^2$ =momentum density (cgs units)
 * Wikipedia: Momentum density is $S/c^2$

Ranked from most elementary to most advanced: Phasor algebra MyOpenMath/Complex phasors Poynting's theorem Poynting_vector Maxwell_stress_tensor

- - Links


 * https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves
 * https://openstax.org/books/university-physics-volume-2/pages/16-2-plane-electromagnetic-waves
 * https://openstax.org/books/university-physics-volume-2/pages/16-3-energy-carried-by-electromagnetic-waves
 * https://openstax.org/books/university-physics-volume-2/pages/16-4-momentum-and-radiation-pressure
 * http://www.physics.smu.edu/~scalise/P7311fa13/HiddenMomentum.pdf
 * https://arxiv.org/pdf/0903.0210.pdf F. E. M. Silveira and J. A. S. Lima. "Attenuation and damping of electromagnetic fields: Influence of inertia and displacement current" (2 Mar 2009)
 * https://arxiv.org/pdf/1408.4144.pdf On electromagnetic momentum of an electric dipole in a magnetic field. Discusses Fringing Field.
 * MIT open courseware https://ocw.aprende.org/courses/physics/8-02sc-physics-ii-electricity-and-magnetism-fall-2010/maxwells-equations/poynting-vector-and-energy-flow-in-a-capacitor/MIT8_02SC_challenge_prob27.pdf
 * Feynmann disc paradox https://www.feynmanlectures.caltech.edu/II_17.html
 * Phasor algebra
 * Maybe put something in Momentum?
 * MIT complex notation only source of good explanation
 * String wave energy
 * Wrong? https://users.physics.ox.ac.uk/~palmerc/Wavesfiles/Energy_Handout.pdf
 * EJP Google Scholar ResearchGate iopscience.iop.org try this

List of equations in table
For future reference and editing:


 * $$B=B_x$$
 * $$J=J_y=\sigma E_y$$
 * $$f=f_x=J_yB_z=\sigma E B$$
 * $$\sigma = 2\varepsilon_0\alpha$$
 * $$S_x=\frac 1 \mu_0 E_yB_z$$ intensity = power/area
 * $$c^2=\frac{1}{\mu_0\varepsilon_0}$$

Removing discussion about dimensional analysis
Complicated calculations should always begin with an effort to reduce the number equations and symbols involved. We begin with the six expressions shown the table. Electric conductivity, $$\sigma$$, appears in the expression,

where $$\alpha$$ is the exponential damping rate, which is measured in inverse-seconds. One reason to deemphasize $$\sigma$$ is that it is not expressed in one of the. We remove $$\sigma$$ from the force equation as follows:

where,

is the Poynting vector. While it may seem more natural to express the force density in terms of current and magnetic field, as $$\vec J \times \vec B$$, the bother of trying to remember $$\alpha$$ in (2) as some constant time conductivity, is rewarded by the simplicity of verifying that the units check. Using the textbooks notation, we use square brackets $$[\quad]$$ to denote dimensions, with $$(\mathsf L,\mathsf M, \mathsf T)$$ denoting (length, mass, time.) To obtain the dimensions of the LHS of (2), use the fact that force density, $$f$$, is force, $$F$$, per unit volume:

Turning our attention to the RHS of (2), we first note that $$[2\alpha/c^2]=\mathsf L^2/\mathsf T$$. The Poynting vector vector, $$\vec S,$$ equals the electromagnetic power that flows through a unit area. Energy, $$U$$, has the same units as work (force X distance) and, power, $$P=dU/dt $$, is the time derivative of energy. Therefore:

We shall soon see that with (4), the variable for conductivity $$\sigma$$ can be effectively removed from consideration.

Put this somewhere in an introduction
It is not customary to model a transverse electromagnetic wave in a conductor using a real-valued wavenumber and a complex angular frequency. Usually the wave is usually created the system with a real frequency and calculating the decay with respect to distance into the medium, as shown in the figure. The driving (real-valued) frequency might be created by microwaves or an ac voltage source. We adopt this unorthodox model because the mathematical solution was stated in OpenStax Physics as the solution to the damped harmonic oscillator.

Fun modification of lorem ipsum
Homework assignment Replace Lorem ipsum with an abstract and introduction to this essay. "Non levis momentum habent?" Examinatin electricae atque magneticae in wiki vniuersitatis.

Ac momentum habent ordinem per impulse: