Electrodynamics

Vector Field Visualization

https://www.geogebra.org/m/QPE4PaDZ

 
  • 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 charge density inside volume

\LARGE \vec{\nabla}\cdot\vec{E}=\rho/\epsilon_0    This is an abuse of notation

\LARGE \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      This has  abuse of notation

\iiint_{V} (\frac{\partial \vec E_{x}}{\partial x}+\frac{\partial \vec E_{y}}{\partial y}+\frac{\partial \vec E_{z}}{\partial z}) \,\,\, 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 This is an abuse of notation

\dpi{120} \large \frac{\partial \vec B_{x}}{\partial x}+\frac{\partial \vec B_{y}}{\partial y}+\frac{\partial \vec B_{z}}{\partial z}=0

 

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

Curl of E , the circulation of electric field , around an axis perpendicular to each plane at point (x,y,z) is equal to the rate of change in magnetic field

\LARGE \vec{\nabla}\times \vec{E}=-\frac{\partial \vec{B}}{\partial t} This is an abuse of notation

\dpi{120} \large {\left({\frac {\partial E_{z}}{\partial y}}-{\frac {\partial E_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial E_{x}}{\partial z}}-{\frac {\partial E_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial E_{y}}{\partial x}}-{\frac {\partial E_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}=-\frac{\partial \vec{B}}{\partial t}=-\frac{\partial \vec{B_x}}{\partial t}-\frac{\partial \vec{B_y}}{\partial t}-\frac{\partial \vec{B_z}}{\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) This is an abuse of notation

Curl of B , the circulation of magnetic  field , around an axis perpendicular to each plane at point (x,y,z) is proportional  to the rate of change in Electric field + electrical current

if  J=0 then

\large {\left({\frac {\partial B_{z}}{\partial y}}-{\frac {\partial B_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial B_{x}}{\partial z}}-{\frac {\partial B_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial B_{y}}{\partial x}}-{\frac {\partial B_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}=\mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}=\mu_0\epsilon_0(\frac{\partial \vec{E_x}}{\partial t}+\frac{\partial \vec{E_y}}{\partial t}+\frac{\partial \vec{E_z}}{\partial t})

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

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