Maxwell’s Equations


Equation 1

Gausses law – the divergence of the electric vector field foverned by material permittivity parameters (i.e., the electric flux density) is equal to the volume-charge density

\nabla \cdot \overrightarrow{D} = e_V

  • Variables and parameters
    • \overrightarrow {B} = electric flux density
    • \varepsilon = material permittivity parameter
    • \overrightarrow{E} = electric field
    • e_V = volume-charge density
    • \overrightarrow {D} =\varepsilon \overrightarrow {E}
  • Note: by convention, positive point charges act as sources for electric fields and negative point charge act as sinks for electric fields

Equation 2

Gausses law of magnetism – the divergence of the magnetic field is zero

\nabla \cdot \overrightarrow{B} = 0

  • Variables and parameters
    • \overrightarrow {B} = magnetic flux density
    • \overrightarrow {H} = magnetic field
    • \mu = proportionality constant between the magnetic flux density and the magnetic field
    • \overrightarrow {B} =\mu\overrightarrow {H}
  • Note: magnetic monopoles do not exist, magnetic fields always point of of the south pole and towards the north pole of a single magnet. For example consider of a bar magnet with its magnetic fields projecting out of the north pole of the magnet and termining in towards the south pole of the magnet (Figure 1).
magnetic field
Figure 1 (source link)

Equation 3

Faraday’s law – the curl of the electric field is equal to the rate of change of the magnetic flux density (and thus, the magnetic field)

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

Equation 4

Ampere’s law – the curl of the magnetic field is equal to the change in electrix flux density over time plus the electric/conductive current density

\nabla \times \overrightarrow{H} = -\frac{\partial \overrightarrow{D}}{\partial t} + \overrightarrow{J}

  • Variables and parameters
    • \overrightarrow{J} = electric/conductive current
    • \sigma = material conductivity
    • \overrightarrow{J} = \sigma \overrightarrow{E}


  • Gausses law implies charges give rise to diverging electric currents
  • Diverging electric fields give rise to current
  • Current gives rise to rotating magnetic fields
  • Changes in magnetic fields (and thus, changes in current) give rise to rotating electric fields
  • This is how electromagnetic waves propagate!!!!

Note: I just wanted to say that I did not come up with these conclusions myself. Once I find the source I used for these conclusions I will post it 🙂

Helpful Diagrams from External Links

Click on the figure label for the source tikz code used to generate this file

eletromagnetic waves

My own thoughts:

Note: all these thoughts/ideas are based on material presented on the wikipedia page for Coulumb’s constant

  • It turns out that the magnetic permeability in a vacuum is \mu_0 = 4\pi \cdot 10^-7 \mathrm{N/A^2}
    • Does this have anything to do with the fact that 4\pi is equivalent to two revolutions?
  • Electic permittivity in a vacuum is related to \mu_0 as follows:

\varepsilon_0 = \frac{1}{\mu_0 c^2}  where c = the speed of light

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