### Taming Ions: the Paul Trap

Tags: Paul trap / Mathieu differential equation

We will study the movement of charged particles in a so-called Paul trap by solving the equations of motion using approximative methods.

To understand classical electrodynamics (or classical electromagnetism) is so old and fundamental to mankind that it is worthwhile to think about the importance of it a little while.

In the Book of Genesis already the third verse says "**And God said, Let there be light: and there was light.**". And in the Qu'ran we can find "**Allah is the Light of the heavens and the earth.**"

Of course the **mythological meaning of light** in these ancient books is extremely deep. Light is, among other things, meant as a force of creation.

One thing is clear: light itself (and its absence, darkness) has fascinated mankind ever since.

Nevertheless it took us almost our entire existence on this planet to figure out how to describe light as a phenomenon. **Static electricity** has been known for thousands of years.

Already the Greek knew that rubbing amber would lead to an attraction of some other materials, such as feathers; or iron pieces.

Amber, in Greek is called ἤλεκτρον - "**electron**".

It may be that our first usage of a **magnetic** compass before 1000 BC by the Olmecs, a now extinct civilization in Mexico's Gulf coast.

Full-scale **electromagnetic** phenomena such as lightning were also well known to our ancestors.

However, a thorough understanding of electromagnetism took us thousands of years. It was very hard for us to figure out a connection between phenomena of static electricity, magnetism and electromagnatic ones.

In 1873 James Clerk Maxwell published "A Treatise on Electricity and Magnetism". In this two-volume treatise on electromagnetism the Scottish mathematician provided a unified description of electricity, magnetism and light.

Maxwell introduced the use of vector fields for different physical fields such as the electric and magnetic fields. Foremost, Maxwell outlined their mathematical relation in now so-called **Maxwell's Equations**.

All phenomenon of classical electromagnetism are governed by Maxwell's Equations.

Maxwell's Equations are hard to understand. They are a set of coupled partial differential equations. To describe physical phenomena, however, these equations often need to be boiled down to a specific **approximation**.

For example: if the electric and magnetic fields do not vary fastly, their mutual dependency can neglected. In this case we are able to understand phenomenon of electrostatics and magnetostatics.

Therefore, an understanding of electrodynamics usually starts with these very approximations. Our website is structured in the same way:

First, we learn how to correctly describe **static effects**. We will learn how charges interact, what currents cause and how to mathematically describe the electric and magnetic fields.

Afterwards we relax our approximations a little, which opens a whole no world to electromagnetic circuit theory. Yes, circuits are described by electromagnetism as well! We will be able to enter this field that is the backbone of electronics, the field that has had a deep impact on mankind since the 20th century.

Finally, our understanding can widen to the entire theory of electrodynamics / electromagnetism. In this part we can finally describe light and also understand relativistic phenomena.

The photonics101 approach to learning electromagnetism is two-fold. At first a **theoretical description** of the theory is given. Then you are invited to solve **worksheets / tasks** based on the outlined theory. **Hints** are given to guide you along the way. A **complete solution** to the task is also outlined.

We hope that you find a reasonable source for your understanding of electromagnetism on our website! We have been putting a lot of effort in these pages and are looking forward to your advancements!

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in
Electrodynamics /
Electrostatics /
Charges and Movement

Tags: Paul trap / Mathieu differential equation

Tags: Paul trap / Mathieu differential equation

We will study the movement of charged particles in a so-called Paul trap by solving the equations of motion using approximative methods.

Published
in
Electrodynamics /
Full Electrodynamics /
Relativistic Electrodynamics

Tags: Gauge Invariance / Charge Conservation / Noether's theorem

Tags: Gauge Invariance / Charge Conservation / Noether's theorem

Usually, gauge transformations are used to simplify certain calculations. In this problem, we want to use them to derive charge conservation from the invariance of the action S.

Published
in
Electrodynamics /
Magnetostatics /
Currents and Magnetic Fields

Tags: Superposition Principle

Tags: Superposition Principle

What happens if we take a wire, drill a hole in it and set some current in motion? Will we encounter a magnetic field?

Published
in
Electrodynamics /
Magnetostatics /
Currents and Magnetic Fields

Tags: Rotational Symmetry / Cylindrical Coordinates / Ampere's law / Stoke's theorem / Biot-Savart's law

Tags: Rotational Symmetry / Cylindrical Coordinates / Ampere's law / Stoke's theorem / Biot-Savart's law

The magnetic field of a thin and infinitely long wire is calculated using Biot-Savart's law.

Published
in
Electrodynamics /
Full Electrodynamics /
Relativistic Electrodynamics

Tags: Lorentz Invariance

Tags: Lorentz Invariance

Lorentz invariance is at the core of any (special) relativistic theory. Sometimes, however, we encounter expressions that seem to transform in a wrong way. This is especially true for the four-potential calculated from the retarded four-current. Find out why the four-potential is indeed Lorentz invariant!

Published
in
Electrodynamics /
Electrostatics /
Boundary Value Problems

Tags: Image Charges / Dipole / Green's Function

Tags: Image Charges / Dipole / Green's Function

The potential of a dipole in front of a flat metallic surface is calculated. A Taylor expansion is used to understand the resulting electric field.

Published
in
Electrodynamics /
Electrostatics /
Multipole Moments

Tags: Rotational Symmetry / Quadrupole Moment / Legendre Polynomials

Tags: Rotational Symmetry / Quadrupole Moment / Legendre Polynomials

Starting from specific examples, we find the general form of the multipole moments for charge distributions with rotational symmetry.

Published
in
Electrodynamics /
Magnetostatics /
Currents and Magnetic Fields

Tags: Rotational Symmetry / Cylindrical Coordinates / Ampere's law / Stoke's theorem / Biot-Savart's law

Tags: Rotational Symmetry / Cylindrical Coordinates / Ampere's law / Stoke's theorem / Biot-Savart's law

We calculate the magnetic field of a wire with constant current. Using symmetries we verify the right-hand rule.

Published
in
Electrodynamics /
Magnetostatics /
Currents and Magnetic Fields

Tags: Rotational Symmetry / Cylindrical Coordinates

Tags: Rotational Symmetry / Cylindrical Coordinates

The magnetic field for a current distribution in form of a cylinder shell is determined.

Published
in
Electrodynamics /
Electrostatics /
Charges and Electric Fields

Tags: Gauss's law / Laplace equation

Tags: Gauss's law / Laplace equation

We find the solution to the electrostatic problem of a homogeneously charged sphere. The electrostatic potential and fields are calculated.