Summary

General Principles

Maxwell’s Equations

These equations govern electromagnetic fields:

$\oint \vec{E} \cdot d \vec{A} = \frac{Q_{in}}{ϵ_{0}}$	Gauss’s law
$\oint \vec{B} \cdot d \vec{A} = 0$	Gauss’s law for magnetism
$\oint \vec{E} \cdot d \vec{s} = - \frac{d Φ_{m}}{d t}$	Faraday’s law
$\oint \vec{B} \cdot d \vec{s} = μ_{0} I_{through} + ϵ_{0} μ_{0} \frac{d Φ_{e}}{d t}$	Ampère-Maxwell law

Maxwell’s equations tell us that:

An electric field can be created by

Charged particles
A changing magnetic field

A magnetic field can be created by

A current
A changing electric field

Lorentz Force

This force law governs the interaction of charged particles with electromagnetic fields:

\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})

An electric field exerts a force on any charged particle.
A magnetic field exerts a force on a moving charged particle.

Field Transformations

Fields measured in reference frame A to be $\vec{E}_{A}$ and $\vec{B}_{A}$ are found in frame B to be

\begin{array}{l} {\vec{E}}_{B} & = & {\vec{E}}_{A} + \vec{v}_{BA} \times {\vec{B}}_{A} \\ {\vec{B}}_{B} & = & {\vec{B}}_{A} - \frac{1}{c^{2}} \vec{v}_{BA} \times {\vec{E}}_{A} \end{array}

The figure illustrates field transformation of a frame to form a new frame.

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Important Concepts

Induced fields

An induced electric field is created by a changing magnetic field.

The figure illustrates an induced electric field depicted as concentric circles due to a change in the magnetic field.

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An induced magnetic field is created by a changing electric field.

The figure illustrates an induced magnetic field due to a change in the electric field depicted as concentric circles.

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These fields can exist independently of charges and currents.

An electromagnetic wave is a self-sustaining electromagnetic field.

An em wave is a transverse wave with $\vec{E}$ , $\vec{B}$ , and $\vec{v}_{em}$ mutually perpendicular.
An em wave propagates with speed $v_{em} = c = 1 / \sqrt{ϵ_{0} μ_{0}}$ .
The electric and magnetic field strengths are related by $E = c B$ .
The Poynting vector $\vec{S} = (\vec{E} \times \vec{B}) / μ_{0}$ is the energy transfer in the direction of travel.
The wave intensity is $I = P / A = (1 / 2 c μ_{0}) {E_{0}}^{2} = (c ϵ_{0} / 2) {E_{0}}^{2}$ .

The figure illustrates an electromagnetic wave in an x y z plane.

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Applications

Polarization

The electric field and the Poynting vector define the plane of polarization. The intensity of polarized light transmitted through a polarizing filter is given by Malus’s law:

I = I_{0} {cos}^{2} θ

where $θ$ is the angle between the electric field and the polarizer axis.

The figure illustrates intensity of a polarized light arrow transmitted through a polarizing filter.

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