### Providing a Homogeneous Magnetic Field - The Helmholtz Coil

Tags: Biot-Savart's law / Taylor Expansion

Helmholtz coils are a devices that can provide a very homogeneous magnetic field. Let's find an approximation for their magnetic field!

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!

Published
in
Electrodynamics /
Magnetostatics /
Currents and Magnetic Fields

Tags: Biot-Savart's law / Taylor Expansion

Tags: Biot-Savart's law / Taylor Expansion

Helmholtz coils are a devices that can provide a very homogeneous magnetic field. Let's find an approximation for their magnetic field!

Through physical reasoning we find that the spin of the electron cannot be explained by a classical rotation.

The Proca-formulation of electrodynamics allows to account for a hypothetical massive photon. This formulation is formally equivalent to the London theory of superconductivity

Published
in
Electrodynamics /
Magnetostatics /
Magnetic Fields in Matter

Tags: Magnetostatic Potential / Legendre Polynomials / Boundary Conditions

Tags: Magnetostatic Potential / Legendre Polynomials / Boundary Conditions

Using the magnetostatic potential can be extremely useful to calculate magnetostatic problems. In this example we use it to derive the magnetic field of a sphere with surface current.

Published
in
Electrodynamics /
Electrostatics /
Charges and Movement

Tags: Equations of Motion / Metric / Christoffel Symbols

Tags: Equations of Motion / Metric / Christoffel Symbols

The movement of a charged particle along a surface subject to an external electric field is considered. We find the same equations of motion as for geodesic movement in general relativity.

A derivation of the boundary integral equation in electrostatics. The relation to the Boundary Element Method is explained.

The electrostatic field of a point charge close to a grounded metallic sphere is derived.

Published
in
Electrodynamics /
Electrostatics /
Dielectrics

Tags: Permittivity / Legendre Polynomials / Boundary Conditions / Fröhlich Condition

Tags: Permittivity / Legendre Polynomials / Boundary Conditions / Fröhlich Condition

The electrostatic potential of a dielectric sphere subject to an external field is calculated.

The capacitance of a coaxial capacitor with two embedded dielectrics is calculated.

Published
in
Electrodynamics /
Electrostatics /
Charges and Electric Fields

Tags: Near Field / Far Field / Green's Function

Tags: Near Field / Far Field / Green's Function

The potential of a finite line charge is calculated. Approximations are performed and shown to correspond to known results.