Electrodynamics

Vectors

The cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space  is denoted by the symbol  X.
Given two linearly independent vectors a and b, the cross product, a × b (read “a cross b”), is a vector that is perpendicular to both a and b,[1] and thus normal to the plane containing them.

https://en.wikipedia.org/wiki/Cross_product

 

The exterior product or wedge product of vectors is an algebraic construction used in geometry to study  areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and  v denoted by  
{\displaystyle u\wedge v,}  is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original
space of vectors. The magnitude[4] of  \dpi{100} \large {\displaystyle u\wedge v,} can be interpreted as the area of the parallelogram with sides u  and v which in three dimensions can also be computed using the cross product of the two vectors.

https://en.wikipedia.org/wiki/Exterior_algebra#Inner_product

An eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled.

In rotational motion of a rigid body, the principal axes are the eigenvectors of the inertia matrix.

real symmetric matrices have real eigenvalues.

If A is an n × n matrix, then the sum of the n eigenvalues of A is the trace of A and the product of the n eigenvalues is the determinant of A.

 

Tensors:

A tensor is an n-dimensional array satisfying a particular transformation law.

Not every Matrix is a rank 2 Tensor.

A Rank 2 tensor can be represented as a matrix of numbers — in conjunction with an associated transformation law.

A Rank 3 tensor can be represented as a 3-dimensional array of numbers — in conjunction with an associated transformation law.

 

https://rukshanpramoditha.medium.com/real-world-examples-of-0d-1d-2d-3d-4d-and-5d-tensors-100b0837ced4

Rank 3 example: Levi-Civita_symbol

https://en.wikipedia.org/wiki/Levi-Civita_symbol

Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties.

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor).

https://rinterested.github.io/statistics/tensors2.html

https://www.mathpages.com/rr/s5-02/5-02.htm

https://math.stackexchange.com/questions/8170/intuitive-way-to-understand-covariance-and-contravariance-in-tensor-algebra

  • the nabla symbol, ∇, denotes the three-dimensional gradient operator, del,
  • the ∇⋅ symbol (pronounced “del dot”) denotes the divergence operator,
  • the ∇× symbol (pronounced “del cross”) denotes the curl operator.

 

E is the electric vector
B is the magnetic induction
ρ is the electric charge density
j is the electric current density
ε0 is the permittivity of free space
ε is the dielectric constant (or permittivity)
c is the speed of light
σ is the specific conductivity
μ is the magnetic permeability

j=σE
D=εE      D is the electric displacement
B=μH     H is the magnetic vector

Maxwell Equations

Gauss’s law
Divergence of electric field is proportional to the the volume charge density

\LARGE \vec{\nabla}\cdot\vec{E}=\rho/\epsilon_0

\LARGE \nabla\cdot \vec E = \frac{\partial \vec E_{x}}{\partial x}+\frac{\partial \vec E_{y}}{\partial y}+\frac{\partial \vec E_{z}}{\partial z}=\rho/\epsilon_0

\large \iiint_{V} (\nabla\cdot \vec E) \,\,\, dV=\oiint_{S} (\vec{E}\cdot \vec{n}) \, \vec{dS}= \iiint_{V} \rho/\epsilon_0 \,\,\, dV=Q/\epsilon_0

    \vec{n} is the outward pointing unit normal at each point on every dS on the boundary  of surface S of Volume V

 

 

 

 

Gauss’s law for magnetism
The magnetic field B has divergence equal to zero. It is equivalent to the statement that magnetic monopoles do not exist.

\LARGE \vec{\nabla}\cdot\vec{B}=0

 

 

Faraday’s law of induction
A circulating electric field is produced by a magnetic field that changes with time.

\LARGE \vec{\nabla}\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}

 

 

Ampere-Maxwell Law
A circulating magnetic field is produced by an electric current and by an electric field that changes with time.

\LARGE \vec{\nabla}\times \vec{B}=\mu_0\left(\vec{J}+\epsilon_0\frac{\partial \vec{E}}{\partial t}\right)

Deriving Electromagnetic waves from Maxwell’s equations https://www.youtube.com/watch?v=-LsXjVfucHU

 

https://web.mit.edu/sahughes/www/8.022/lec20.pdf

https://www.fiberoptics4sale.com/blogs/electromagnetic-optics/a-plain-explanation-of-maxwells-equations#:~:text=4.-,Ampere-Maxwell’s%20Law,field%20that%20changes%20with%20time.

https://en.wikipedia.org/wiki/Displacement_current

https://physics.stackexchange.com/questions/166888/the-propagation-of-electric-field

Book: https://matterandinteractions.org/

Liénard–Wiechert potential classical electromagnetic effect of a moving electric point charge

 
 
Feynman’s derivation of Liénard-Wiechert potentials revisited
 
 

What is the proof of the Biot-Savart law from Maxwell equations

https://www.khanacademy.org/science/in-in-class-12th-physics-india/moving-charges-and-magnetism


https://qr.ae/prrOcB

https://www.quora.com/Do-Maxwells-equations-imply-the-Biot-Savart-law

 
 

 

Magnetic field of a moving charge

 
In Farsi
میدان و امواج
الکترومغناطیس ١
 
 

Feynman

 

Full https://www.youtube.com/watch?v=kEx-gRfuhhk

Richard P Feynman: Quantum Mechanical View of Reality – YouTube

 
 
 
 

Explanations

This relates to electric field in that the charge moving through a circuit to light a light bulb has to be driven by some electric field, so you can reasonably ask how that field is established, and how much time it takes. Qualitatively, the necessary field is established by excess charge on the surface of the wires, with the surface charge being generally positive near the positive terminal of a battery and generally negative near the negative terminal, and dropping off smoothly from one to the other so that the electric field is more or less piecewise constant (that is, the field is the same everywhere inside a wire, and the field is the same everywhere inside a resistor, but the two field values are not the same).

When the circuit is first connected, there is a rapid redistribution of the charge on the surface of the wires which establishes the surface charge gradients that drive the steady-state current that will eventually do whatever it is you want it to do. The time required to establish the gradients and settle in to the steady-state condition is very fast, most likely on the order of nanoseconds for a normal circuit.

https://physics.stackexchange.com/questions/1747/is-there-a-travelling-speed-of-for-electric-field-if-yes-what-is-it