### Spherical Capacitor with Varying Permittivity

Tags: Gauss's law / Permittivity / Capacitance

We solve the problem of a spherical capacitor with (almost) arbitrary permittivity of the dielectric.

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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Electrodynamics /
Electrostatics /
Dielectrics

Tags: Gauss's law / Permittivity / Capacitance

Tags: Gauss's law / Permittivity / Capacitance

We solve the problem of a spherical capacitor with (almost) arbitrary permittivity of the dielectric.

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in
Electrodynamics /
Full Electrodynamics /
Radiation and Antennas

Tags: Directivity / Poynting Vector / Dipole / Superposition Principle

Tags: Directivity / Poynting Vector / Dipole / Superposition Principle

The simplest form of an antenna array are two short dipole antennas. In this problem you will learn how a phase difference between the currents of both antennas affects their combined radiation pattern.

Published
in
Electrodynamics /
Full Electrodynamics /
Relativistic Electrodynamics

Tags: Lagrangian / Special Relativitity / Equations of Motion

Tags: Lagrangian / Special Relativitity / Equations of Motion

An alternative Lagrangian for electrodynamics is be shown to be applicable.

Published
in
Electrodynamics /
Full Electrodynamics /
Radiation and Antennas

Tags: Directivity / Airy Disk / Dipole

Tags: Directivity / Airy Disk / Dipole

The directivity of a parabolic reflector is calculated in the Fraunhofer approximation.

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in
Electrodynamics /
Circuit Theory /
Basic AC Circuits

Tags: Harmonic Oscillator / Coupling / Resistance / Inductance / Capacitance

Tags: Harmonic Oscillator / Coupling / Resistance / Inductance / Capacitance

Coupled RLC circuits are solved as coupled harmonic oscillators. The detuning of the circuits is calculated.

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in
Electrodynamics /
Circuit Theory /
Basic AC Circuits

Tags: Resistance / Inductance / Capacitance

Tags: Resistance / Inductance / Capacitance

The classic problem of the resistance cube is solved. In addition, variations are outlined such as the C- and RLC-Cube

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in
Electrodynamics /
Circuit Theory /
Basic AC Circuits

Tags: Resonance / Kirchhoff's law / Resistance / Inductance / Capacitance

Tags: Resonance / Kirchhoff's law / Resistance / Inductance / Capacitance

The serial RLC circuit is solved for dissipation and eigenfrequencies.

Published
in
Electrodynamics /
Circuit Theory /
Transmission Lines

Tags: Resonance / Reflection Coefficient

Tags: Resonance / Reflection Coefficient

Fabry-Perot resonances are derived for transmission lines. This basic model can be applied to a wide range of physical phenomena.

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in
Electrodynamics /
Circuit Theory /
Transmission Lines

Tags: Telegrapher's equation / Reflection Coefficient / Impedance Matching

Tags: Telegrapher's equation / Reflection Coefficient / Impedance Matching

In this problem we will learn about basic principles such as the reflection of signals and impedance matching for transmission lines.

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in
Electrodynamics /
Circuit Theory /
Transmission Lines

Tags: Telegrapher's equation / Kirchhoff's law

Tags: Telegrapher's equation / Kirchhoff's law

In this problem we will derive the telegrapher equation needed to understand communication devices.