Physics

# Maxwell’s Equations

[latexpage]

### 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).

### 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
• $\overrightarrow{J} = \sigma \overrightarrow{E}$

## Conclusions

• 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 🙂

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

#### 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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