wikiworld.org - User contributions [en]

Lessepsian Migration

2021-01-16T18:39:31Z

JamesCrook: Fix link.

[[File:Canal de Suez.jpg|thumb|292x292px|The Suez Canal, through which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the migration of marine species across the Suez Canal, usually from the Red Sea to the Mediterranean Sea and, more rarely, in the opposite direction.

When the canal was completed in 1869, marine animals were exposed to a new passage between the two formerly separate bodies of water. Cross-contamination could happen between previously isolated ecosystems. This phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
===Lessepsian Migrants===
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|[[Trumpetfish]]
Image:PC040789 Moray.JPG|[[Moray eel]]
Image:PC051186 Mackerel.JPG|[[Mackerel]]
Image:PC051144 Yellowstripe Goatfish.JPG|[[Goatfish]]
</gallery>

Mackerel

2021-01-16T18:38:55Z

JamesCrook: New page.

[[Image:PC051186 Mackerel.JPG|800px]]

Trumpetfish

2021-01-16T18:38:06Z

JamesCrook: New page.

[[Image:PC041015 Trumpet Fish.JPG|800px]]

Main Page/Featured Content

2021-01-16T18:36:33Z

JamesCrook: caps

<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|[[Trumpetfish]]
Image:PC040789 Moray.JPG|[[Moray eel]]
Image:PC051186 Mackerel.JPG|[[Mackerel]]
Image:PC051144 Yellowstripe Goatfish.JPG|[[Goatfish]]
</gallery>

Main Page/Featured Content

2021-01-16T18:35:07Z

JamesCrook: Use gallery.

<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|[[Trumpet Fish]]
Image:PC040789 Moray.JPG|[[Moray Eel]]
Image:PC051186 Mackerel.JPG|[[Mackerel]]
Image:PC051144 Yellowstripe Goatfish.JPG|[[Goatfish]]
</gallery>

Lessepsian Migration

2021-01-16T18:34:17Z

JamesCrook: caps.

[[File:Canal de Suez.jpg|thumb|292x292px|The Suez Canal, through which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the migration of marine species across the Suez Canal, usually from the Red Sea to the Mediterranean Sea and, more rarely, in the opposite direction.

When the canal was completed in 1869, marine animals were exposed to a new passage between the two formerly separate bodies of water. Cross-contamination could happen between previously isolated ecosystems. This phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
===Lessepsian Migrants===
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|[[Trumpet Fish]]
Image:PC040789 Moray.JPG|[[Moray eel]]
Image:PC051186 Mackerel.JPG|[[Mackerel]]
Image:PC051144 Yellowstripe Goatfish.JPG|[[Goatfish]]
</gallery>

Lessepsian Migration

2021-01-16T18:33:53Z

JamesCrook: Links in Gallery.

[[File:Canal de Suez.jpg|thumb|292x292px|The Suez Canal, through which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the migration of marine species across the Suez Canal, usually from the Red Sea to the Mediterranean Sea and, more rarely, in the opposite direction.

When the canal was completed in 1869, marine animals were exposed to a new passage between the two formerly separate bodies of water. Cross-contamination could happen between previously isolated ecosystems. This phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
===Lessepsian Migrants===
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|[[Trumpet Fish]]
Image:PC040789 Moray.JPG|[[Moray Eel]]
Image:PC051186 Mackerel.JPG|[[Mackerel]]
Image:PC051144 Yellowstripe Goatfish.JPG|[[Goatfish]]
</gallery>

Lessepsian Migration

2021-01-16T18:25:57Z

JamesCrook: fewer links.

[[File:Canal de Suez.jpg|thumb|292x292px|The Suez Canal, through which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the migration of marine species across the Suez Canal, usually from the Red Sea to the Mediterranean Sea and, more rarely, in the opposite direction.

When the canal was completed in 1869, marine animals were exposed to a new passage between the two formerly separate bodies of water. Cross-contamination could happen between previously isolated ecosystems. This phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
===Lessepsian Migrants===
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|(Trumpet Fish)
Image:PC040789 Moray.JPG|(Moray Eel)
Image:PC051186 Mackerel.JPG|(Mackerel)
Image:PC051144 Yellowstripe Goatfish.JPG|(Yellowstripe Goatfish)
</gallery>

Bar-tailed godwit

2021-01-10T22:16:16Z

JamesCrook: New page.

[[File:BartailedGodwit24.jpg|800px]]

Bar-tailed godwit.

New Zealand Wood Pigeon

2021-01-10T22:12:06Z

JamesCrook: New page.

[[File:New Zealand Wood Pigeons are important for maintaining and spreading native bushes and forests. They are the largest seed eating bird remaining in New Zealand, so are the only bird that can still eat (34494347080).jpg|800px]]

New Zealand Wood Pigeon.

Kaka

2021-01-10T22:08:18Z

JamesCrook: New page.

[[File:New Zealand kaka, Mount Bruce National Wildlife Centre, New Zealand 09.JPG|600px]]

Kaka bird of New Zealand.

Blue Duck

2021-01-10T22:03:31Z

JamesCrook: New page.

[[File:Whio_(Blue_Duck)_at_Staglands,_Akatarawa,_New_Zealand.jpg|800px]]

The blue duck, 'whio' in Maori, of New Zealand.

Melatonin

2020-10-25T18:39:22Z

JamesCrook: formatting.

<div class='hide'>[[Image:Melatonin.png|Melatonin]]</div>
{{#widget:WikiDiagram|page=Molecule|init=yes}}

'''Melatonin''' is a [[hormone]] found in animals, plants, and microbes. In animals, levels of melatonin have a daily cycle. It drives the [[circadian rhythm]]s of several biological functions.

Melatonin is produced in the [[pineal gland]]. The pineal gland is on the blood side of the blood-brain barrier. It secretes melatonin "directly into the systemic circulation", thus melatonin is not affected by the blood-brain barrier

==See also==
{{Dig|WP|Melatonin}}

Melatonin

2020-10-25T16:28:26Z

JamesCrook: blood-brain barrier

<div class='hide'>[[Image:Melatonin.png|Melatonin]]</div>
{{#widget:WikiDiagram|page=Molecule|init=yes}}

'''Melatonin''' is a [[hormone]] found in animals, plants, and microbes. In animals, levels of melatonin have a daily cycle. It drives the [[circadian rhythm]]s of several biological functions.

Melatonin is produced in the [[pineal gland]]. The pineal gland is on the blood side of the blood-brain barrier.
It secretes melatonin "directly into the systemic circulation", thus melatonin is not affected by the blood-brain barrier

==See also==
{{Dig|WP|Melatonin}}

WikiWorld:About

2020-10-01T21:39:07Z

JamesCrook: Yet more words.

This wiki fuses art, technology and education, to make a website about our planet and its future. We share plans that show the beauty and complexity of how things work. We have fun collaborating to better show ''"what is"'', with a long term goal to make the important things on this planet work better.

This is both a tribute site to the awesome work done at [[wikipedia:Main_Page|Wikipedia]], and it is a feeder site, where we can be more artful, more technological and more educational. We're not an encyclopaedia. Our pages don't try to cover all the bases. Our pages tell stories. We're a place to have fun with ideas, art and technology.

A lot of the motivation, and some seed content, comes from Wikipedia. Like trout swimming up river, some of what we put together here will find its way back to Wikipedia.

==Knowledge Structure==

On this site, good explanations take centre stage. Information is not enough. Emotion and understanding are needed too. How you feel about knowledge is often as important as the knowledge itself. Learning, sharing, understanding are all mixed in with fun here. It's [[Hard Fun (Papert)|hard fun]], in the Seymour Papert sense.

Individual voices, rather than just neutral encyclopaedic content can sometimes be heard here. In this we are different from Wikipedia. Sometimes we are wrong, or we have strayed too far from the usual consensus in order to tell the story we want to tell. So please engage brain when reading what is here. We offer vistas and technology, lenses and images for the future. We offer tools for understanding and creating a better future. We also showcase and share some new tools and new ways of making content which we invite others to use too.

==How it Works==

* Everything here must be licensed on terms that are compatible with Wikipedia, so that content created here can be freely re-used and changed and included in Wikipedia itself.
* Wikiworld makes extensive use of the [[WikiWorld:WikiDiagrams|Wikidiagrams]] mediwiki extension, to make data driven, interactive diagrams.
* Read about [[WikiWorld:Good Article|Good Article]] and [[WikiWorld:Featured Article|Featured Article]] status to understand what we aim for.

Goatfish

2020-10-01T21:32:15Z

JamesCrook: add sand-and

[[Image:PC051144 Yellowstripe Goatfish.JPG|800px|(Yellowstripe Goatfish)]]

'''Goatfish''' have elongated bodies, forked tails and are often brightly coloured. They get their name from their 'goatee', a pair of chin barbels which contain chemoreceptors and which are used to probe sand and the reef for food.

==See also==

{{Dig|WP|Goatfish}}

Lessepsian Migration

2020-10-01T17:25:25Z

JamesCrook: Label the migrants.

[[File:Canal de Suez.jpg|thumb|292x292px|The [[Suez Canal]], through which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the [[migration (ecology)|migration]] of marine species across the [[Suez Canal]], usually from the [[Red Sea]] to the [[Mediterranean Sea]] and, more rarely, in the opposite direction.

When the canal was completed in 1869, marine animals were exposed to a new passage between the two formerly separate bodies of water. Cross-contamination could happen between previously isolated [[ecosystem]]s. This phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
===Lessepsian Migrants===
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|(Trumpet Fish)
Image:PC040789 Moray.JPG|(Moray Eel)
Image:PC051186 Mackerel.JPG|(Mackerel)
Image:PC051144 Yellowstripe Goatfish.JPG|(Yellowstripe Goatfish)
</gallery>

Lessepsian Migration

2020-10-01T17:22:41Z

JamesCrook: Remove redundant enumeration.

[[File:Canal de Suez.jpg|thumb|292x292px|The [[Suez Canal]], across which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the [[migration (ecology)|migration]] of marine species across the [[Suez Canal]], usually from the [[Red Sea]] to the [[Mediterranean Sea]] and, more rarely, in the opposite direction.

When the canal was completed in 1869 marine animals were exposed to a new passage between the two formerly separate bodies of water. Cross-contamination could happen between previously isolated [[ecosystem]]s. This phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|(Trumpet Fish)
Image:PC040789 Moray.JPG|(Moray Eel)
Image:PC051186 Mackerel.JPG|(Mackerel)
Image:PC051144 Yellowstripe Goatfish.JPG|(Yellowstripe Goatfish)
</gallery>

Lessepsian Migration

2020-10-01T17:20:48Z

JamesCrook: Shorter sentence.

[[File:Canal de Suez.jpg|thumb|292x292px|The [[Suez Canal]], across which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the [[migration (ecology)|migration]] of marine species across the [[Suez Canal]], usually from the [[Red Sea]] to the [[Mediterranean Sea]] and, more rarely, in the opposite direction. When the canal was completed in 1869, [[fish]], [[crustacean]]s, [[mollusk]]s, and other marine animals were exposed to a new passage between the two separate bodies of water. Cross-contamination could happen between formerly isolated [[ecosystem]]s. The phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|(Trumpet Fish)
Image:PC040789 Moray.JPG|(Moray Eel)
Image:PC051186 Mackerel.JPG|(Mackerel)
Image:PC051144 Yellowstripe Goatfish.JPG|(Yellowstripe Goatfish)
</gallery>

Lessepsian Migration

2020-10-01T17:19:50Z

JamesCrook: Shorter.

[[File:Canal de Suez.jpg|thumb|292x292px|The [[Suez Canal]], across which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the [[migration (ecology)|migration]] of marine species across the [[Suez Canal]], usually from the [[Red Sea]] to the [[Mediterranean Sea]] and, more rarely, in the opposite direction. When the canal was completed in 1869, [[fish]], [[crustacean]]s, [[mollusk]]s, and other marine animals were exposed to a new passage between the two separate bodies of water, and cross-contamination was made possible between formerly isolated [[ecosystem]]s. The phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|(Trumpet Fish)
Image:PC040789 Moray.JPG|(Moray Eel)
Image:PC051186 Mackerel.JPG|(Mackerel)
Image:PC051144 Yellowstripe Goatfish.JPG|(Yellowstripe Goatfish)
</gallery>

Lessepsian Migration

2020-10-01T17:17:42Z

JamesCrook: Fewer distracting asides.

[[File:Canal de Suez.jpg|thumb|292x292px|The [[Suez Canal]], across which marine species migrate in the so-called '''Lessepsian migration''']]

The '''Lessepsian migration''' is the [[migration (ecology)|migration]] of marine species across the [[Suez Canal]], usually from the [[Red Sea]] to the [[Mediterranean Sea]] and, more rarely, in the opposite direction. When the canal was completed in 1869, [[fish]], [[crustacean]]s, [[mollusk]]s, and other marine animals and plants were exposed to an artificial passage between the two naturally separate bodies of water, and cross-contamination was made possible between formerly isolated [[ecosystem]]s. The phenomenon is still occurring today. It is named after Ferdinand de Lesseps, the France diplomat in charge of the canal's construction.

----<br clear=all>
<gallery mode="packed-hover">
Image:PC041015 Trumpet Fish.JPG|(Trumpet Fish)
Image:PC040789 Moray.JPG|(Moray Eel)
Image:PC051186 Mackerel.JPG|(Mackerel)
Image:PC051144 Yellowstripe Goatfish.JPG|(Yellowstripe Goatfish)
</gallery>

WikiWorld:About

2020-10-01T17:15:29Z

JamesCrook: what is

This wiki fuses art, technology and education, to make a website about our planet and its future. We share plans that show the beauty and complexity of how things work. We have fun collaborating to better show ''"what is"'', with a long term goal to make the important things on this planet work better.

This is both a tribute site to the awesome work done at [[wikipedia:Main_Page|Wikipedia]], and it is a feeder site, where we can be more artful, more technological and more educational. We're not an encyclopaedia. Our pages don't try to cover all the bases. Our pages tell stories. We're a place to have fun with ideas, art and technology.

A lot of the motivation, and some seed content, comes from Wikipedia. Like trout swimming up river, some of what we put together here will find its way back to Wikipedia.

==Knowledge Structure==

On this site, good explanations take centre stage. Information is not enough. Emotion and understanding are needed too. How you feel about knowledge is often as important as the knowledge itself. Learning, sharing, understanding are all mixed in with fun here. It's [[Hard Fun (Papert)|hard fun]], in the Seymour Papert sense.

Individual voices, rather than just neutral encyclopaedic content can sometimes be heard here. We offer vistas and technology, lenses and images for the future. We offer tools for understanding and creating a better future. We showcase some new tools and new ways of making content.

==How it Works==

* Everything here must be licensed on terms that are compatible with Wikipedia, so that content created here can be freely re-used and changed and included in Wikipedia itself.
* Wikiworld makes extensive use of the [[WikiWorld:WikiDiagrams|Wikidiagrams]] mediwiki extension, to make data driven, interactive diagrams.
* Read about [[WikiWorld:Good Article|Good Article]] and [[WikiWorld:Featured Article|Featured Article]] status to understand what we aim for.

WikiWorld:About

2020-10-01T17:14:43Z

JamesCrook: Less is more.

This wiki fuses art, technology and education, to make a website about our planet and its future. We share plans that show the beauty and complexity of how things work. We have fun collaborating to better show what is, with a long term goal to make the important things on this planet work better.

This is both a tribute site to the awesome work done at [[wikipedia:Main_Page|Wikipedia]], and it is a feeder site, where we can be more artful, more technological and more educational. We're not an encyclopaedia. Our pages don't try to cover all the bases. Our pages tell stories. We're a place to have fun with ideas, art and technology.

A lot of the motivation, and some seed content, comes from Wikipedia. Like trout swimming up river, some of what we put together here will find its way back to Wikipedia.

==Knowledge Structure==

On this site, good explanations take centre stage. Information is not enough. Emotion and understanding are needed too. How you feel about knowledge is often as important as the knowledge itself. Learning, sharing, understanding are all mixed in with fun here. It's [[Hard Fun (Papert)|hard fun]], in the Seymour Papert sense.

Individual voices, rather than just neutral encyclopaedic content can sometimes be heard here. We offer vistas and technology, lenses and images for the future. We offer tools for understanding and creating a better future. We showcase some new tools and new ways of making content.

==How it Works==

* Everything here must be licensed on terms that are compatible with Wikipedia, so that content created here can be freely re-used and changed and included in Wikipedia itself.
* Wikiworld makes extensive use of the [[WikiWorld:WikiDiagrams|Wikidiagrams]] mediwiki extension, to make data driven, interactive diagrams.
* Read about [[WikiWorld:Good Article|Good Article]] and [[WikiWorld:Featured Article|Featured Article]] status to understand what we aim for.

WikiWorld:About

2020-10-01T17:13:05Z

JamesCrook: Updates.

This wiki fuses art, technology and education, to make a website about our planet and its future. We share plans that show the beauty and complexity of how things work. We have fun collaborating to better show what is, with a long term goal to make the important things on this planet work better.

This is both a tribute site to the awesome work done at [[wikipedia:Main_Page|Wikipedia]], and it is a feeder site, where we can be more artful, more technological and more educational. We're not an encyclopaedia. Our pages tell stories. We're a place to have fun with ideas, art and technology.

A lot of the motivation, and some seed content, comes from Wikipedia. Like trout swimming up river, some of what we put together here will find its way back to Wikipedia.

==Knowledge Structure==

On this site, good explanations take centre stage. Information is not enough. Emotion and understanding are needed too. How you feel about knowledge is often as important as the knowledge itself. Learning, sharing, understanding are all mixed in with fun here. It's [[Hard Fun (Papert)|hard fun]], in the Seymour Papert sense.

Individual voices, rather than just neutral encyclopaedic content can sometimes be heard here. We offer vistas and technology, lenses and images for the future. We offer tools for understanding and creating a better future. We showcase some new tools and new ways of making content.

==How it Works==

* Everything here must be licensed on terms that are compatible with Wikipedia, so that content created here can be freely re-used and changed and included in Wikipedia itself.
* Wikiworld makes extensive use of the [[WikiWorld:WikiDiagrams|Wikidiagrams]] mediwiki extension, to make data driven, interactive diagrams.
* Read about [[WikiWorld:Good Article|Good Article]] and [[WikiWorld:Featured Article|Featured Article]] status to understand what we aim for.

Maxwell's equations

2020-10-01T17:05:42Z

JamesCrook: Permissivity.

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The history of how the equations came about is actually helpful in understanding them. Originally the equations were separate observations. Pass a magnet through a coil and you get an electric current. Pass an electric current through a coil, get a magnetic field.

* Equations (1) and (2) in essence state that electric and magnetic fields spread outwards. The lines of force of a magnetic field don't suddenly stop (except at the North or South pole of a magnet). Similarly for an electric field.
* Equation (3), Faraday's law, relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circulation of electric field, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.
* Equation (4), Ampere's law, relates the rate at which electric field is changing, <math>\frac{\partial\mathbf{E}}{\partial t}</math> to circulation of magnetic field, <math>\nabla\boldsymbol{\times}\mathbf{B}</math>.

Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

It is astonishing that formulae originally created for quantifying relationships between electricity and magnetism led on to an understanding of light and radio transmission, and that both are electromagnetic waves. In the formulae above, the speed of light <math>c</math> is explicit. In the original arrangement of the equations and using the units of measurement in use at that time, <math>c</math> was a little more hidden, derived from two quantities called 'permissivity' and 'permitivity'.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then <math>\mathbf{F}</math> has combined electromagnetic and magnetic fields into one complex valued vector field that represents both. Do the same thing for <math>\rho</math> and <math>\mathbf{J}</math>, i.e. <math>\rho=\rho_m + i\rho_e</math> and <math>\mathbf{J}=\mathbf{J}_m+i\mathbf{J}_e</math> and the four equations now reduce down to two:

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

An important detail is how the minus sign has been absorbed into the multiplications by <math>i</math>. In the original equations the second derivative of <math>\mathbf{B}</math> (or indeed of <math>\mathbf{E}</math>) appears from combining both equations (3) and then (4), and is related to the original with a minus sign, picked up from equation (3). Now instead one picks up two multiplications by <math>i</math>. It's the same thing, but done more symmetrically.

In one sense this reformulation in terms of complex numbers does not simplify anything. The four equations in real values are still there 'behind' the two equations in complex values. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of the same field.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Melatonin

2020-10-01T16:29:34Z

JamesCrook: Redundant information removed.

Main Page

2020-10-01T16:27:35Z

JamesCrook: Add link.

==Topics==

===Natural World===

----
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{{:Main Page/Featured Content}}
----
----
===World of Physics===

----
----
===World of Ideas===

----
----
===Other Worlds===

----
----

===Pages===

This wiki is new, and there aren't many pages yet. Here are some:

* [[Lessepsian Migration]]
* [[Biochemical Pathways]]
* [[Melatonin]]
* [[Myelin]]
* [[Maxwell's equations]]
* [[Moray eel]]
* [[Goatfish]]

Myelin

2020-10-01T16:27:11Z

JamesCrook: New page.

'''Myelin''' is a lipid-rich material that surrounds nerve cell axons. It insulates them, allowing faster transmission of signals, using less energy. It is particularly important for long nerves in the spinal column.

[[Image:MyelinByGoodsell.png|500px]]

File:MyelinByGoodsell.png

2020-10-01T16:20:39Z

JamesCrook: Acknowledgement: Illustration by David S. Goodsell, RCSB Protein Data Bank http://pdb101.rcsb.org/sci-art/goodsell-gallery/myelin

== Summary ==
Acknowledgement: Illustration by David S. Goodsell, RCSB Protein Data Bank

http://pdb101.rcsb.org/sci-art/goodsell-gallery/myelin
== Licensing ==
{{cc-by-4.0}}

Maxwell's equations

2020-09-30T13:01:57Z

JamesCrook: Clearer?

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The history of how the equations came about is actually helpful in understanding them. Originally the equations were separate observations. Pass a magnet through a coil and you get an electric current. Pass an electric current through a coil, get a magnetic field.

* Equations (1) and (2) in essence state that electric and magnetic fields spread outwards. The lines of force of a magnetic field don't suddenly stop (except at the North or South pole of a magnet). Similarly for an electric field.
* Equation (3), Faraday's law, relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circulation of electric field, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.
* Equation (4), Ampere's law, relates the rate at which electric field is changing, <math>\frac{\partial\mathbf{E}}{\partial t}</math> to circulation of magnetic field, <math>\nabla\boldsymbol{\times}\mathbf{B}</math>.
* Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

It is astonishing that electricity, magnetism, light and radio waves are interrelated.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then <math>\mathbf{F}</math> has combined electromagnetic and magnetic fields into one complex valued vector field that represents both. Do the same thing for <math>\rho</math> and <math>\mathbf{J}</math>, i.e. <math>\rho=\rho_m + i\rho_e</math> and <math>\mathbf{J}=\mathbf{J}_m+i\mathbf{J}_e</math> and the four equations now reduce down to two:

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

An important detail is how the minus sign has been absorbed into the multiplications by <math>i</math>. In the original equations the second derivative of <math>\mathbf{B}</math> (or indeed of <math>\mathbf{E}</math>) appears from combining both equations (3) and then (4), and is related to the original with a minus sign, picked up from equation (3). Now instead one picks up two multiplications by <math>i</math>. It's the same thing, but done more symmetrically.

In one sense this reformulation in terms of complex numbers does not simplify anything. The four equations in real values are still there 'behind' the two equations in complex values. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of the same field.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

MediaWiki:Tagline

2020-09-30T12:12:57Z

JamesCrook: Less pretentious

From Wikiworld's pages

Maxwell's equations

2020-09-30T11:54:11Z

JamesCrook: /* Using Complex Numbers */

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The history of how the equations came about is actually helpful in understanding them. Originally the equations were separate observations. Pass a magnet through a coil and you get an electric current. Pass an electric current through a coil, get a magnetic field.

* The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.
* Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

It is astonishing that electricity, magnetism, light and radio waves are interrelated.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then <math>\mathbf{F}</math> has combined electromagnetic and magnetic fields into one complex valued vector field that represents both. Do the same thing for <math>\rho</math> and <math>\mathbf{J}</math>, i.e. <math>\rho=\rho_m + i\rho_e</math> and <math>\mathbf{J}=\mathbf{J}_m+i\mathbf{J}_e</math> and the four equations now reduce down to two:

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

An important detail is how the minus sign has been absorbed into the multiplications by <math>i</math>. In the original equations the second derivative of <math>\mathbf{B}</math> (or indeed of <math>\mathbf{E}</math>) appears from combining both equations (3) and then (4), and is related to the original with a minus sign, picked up from equation (3). Now instead one picks up two multiplications by <math>i</math>. It's the same thing, but done more symmetrically.

In one sense this reformulation in terms of complex numbers does not simplify anything. The four equations in real values are still there 'behind' the two equations in complex values. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of the same field.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

2020-09-30T11:47:40Z

JamesCrook: Spell it out.

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The history of how the equations came about is actually helpful in understanding them. Originally the equations were separate observations. Pass a magnet through a coil and you get an electric current. Pass an electric current through a coil, get a magnetic field.

* The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.
* Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

It is astonishing that electricity, magnetism, light and radio waves are interrelated.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then F has combined electromagnetic and magnetic fields into one complex valued vector field that represents both. Do the same thing for <math>\rho</math> and <math>\mathbf{J}</math>, i.e. <math>\rho=\rho_m + i\rho_e</math> and <math>\mathbf{J}=\mathbf{J}_m+i\mathbf{J}_e=</math> and the four equations reduce down to two.

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

An important detail is how the minus sign has been absorbed into the multiplications by <math>i</math>. In the original equations to go from <math>\mathbf{B}</math> to <math>\mathbf{E}</math> and back via (3) and then (4), one picks up one minus sign. Now instead one picks up two multiplications by <math>i</math>. It's the same thing, but done more symmetrically.

In one sense this reformulation in terms of complex numbers does not simplify anything. The four equations in real values are still there 'behind' the two equations in complex values. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of the same field.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

2020-09-30T11:31:00Z

JamesCrook: Framing statement.

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The history of how the equations came about is actually helpful in understanding them. Originally the equations were separate observations. Pass a magnet through a coil and you get an electric current. Pass an electric current through a coil, get a magnetic field.

* The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.
* Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

It is astonishing that electricity, magnetism, light and radio waves are interrelated.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then F has combined electromagnetic and magnetic fields into one vector, and the four equations reduce down to two.

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

In one sense this does not simplify anything. The four equations are still there. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of a field, which is best thought of as being represented with complex numbers.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

2020-09-30T11:09:10Z

JamesCrook:

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The history of how they came about is actually helpful in understanding them. Originally the equations were separate observations. Pass a magnet through a coil and you get an electric current. Pass an electric current through a coil, get a magnetic field.

* The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.
* Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then F has combined electromagnetic and magnetic fields into one vector, and the four equations reduce down to two.

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

In one sense this does not simplify anything. The four equations are still there. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of a field, which is best thought of as being represented with complex numbers.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

2020-09-30T11:05:52Z

JamesCrook: Teasers.

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.

Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
Rather than treating magnetic field and electric field as separate things, you can combine them, using complex numbers.

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then F has combined electromagnetic and magnetic fields into one vector, and the four equations reduce down to two.

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

In one sense this does not simplify anything. The four equations are still there. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of a field, which is best thought of as being represented with complex numbers.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

2020-09-30T11:02:10Z

JamesCrook: Neater.

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.

Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

==Integral Version==
{{collapse top|1=Same Equations - Using Integrals}}

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]]. You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

==Using Complex Numbers==
{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then F has combined electromagnetic and magnetic fields into one vector, and the four equations reduce down to two.

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

In one sense this does not simplify anything. The four equations are still there. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of a field, which is best thought of as being represented with complex numbers.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

2020-09-30T10:40:09Z

JamesCrook: Complex number version.

[[Image:MaxwellEqns.png]]

'''Maxwell's equations''' are partial differential equations that state the relationships between electric and magnetic fields. They underlie the wave nature of light. Equation (1) and (2) are Gauss' laws, (3) Faraday's law of induction, (4) Ampere's circuital law.

The four equations were originally formulated based on observations. Equation 3, for example, comes from Faraday's law. It relates the rate at which magnetic field is changing, <math>\frac{\partial\mathbf{B}}{\partial t}</math> to circular flow of electric current, <math>\nabla\boldsymbol{\times}\mathbf{E}</math>.

Maxwell's genius was in completing the four equations and from the combination conjecturing that light and radio waves were electromagnetic phenomenon. In particular, the speed at which light travels can be deduced from the equations.

==Prerequisities==

The maths of these equations will make no sense to you whatsoever without the following foundations:

* Understanding of [[Simple harmonic motion]]. The key idea is that differential equations relating position, velocity and acceleration can lead to a solution that oscillates.
* Familiarity with vectors in 3 dimensions, [[Vector product]] and [[Cross product]]. The vector product allows you to calculate the angle between two vectors. The cross product, given two vectors, finds another vector that is perpendicular to both.
* The [[Div Grad and Curl]] differential operations in 3D that are shown here, and in particular why <math>\nabla\cdot(\nabla\times\mathbf{B}) = 0</math>. Div, Grad and Curl are differential operations in 3D space. Maxwell's 3rd and 4th equation relate the results of a spatial derivative of one field (magnetic or electric) with a temporal derivative of the other kind of field.

==No Magnetic Monopoles==

To-Be-Written.

{{collapse top|1=Same Equations - Using Integrals}}
==Integral Version==

The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]]. You can view the differential equations as answering the question 'what happens at each point in space?' and the integral equations as answering 'what happens in each volume of space?'.

The four differential equations are equivalent to these four integral equations. If you differentiate these integral equations, you get the differential ones.

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle\partial \Omega } &
\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_e \,\mathrm{d}V\tag{1}
\end{align}
</math>

<math>
\begin{align}
{\oint}\!\!\!\!{\oint}_{\scriptstyle \partial \Omega } &
\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 4 \pi \iiint_\Omega \rho_m \,\mathrm{d}V\tag{2}
\end{align}
</math>

<math>
\begin{align}
- \oint_{\partial \Sigma} & \mathbf{E} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_m} \cdot \mathrm{d}\mathbf{S} \right) \tag{3}\\
\end{align}
</math>

<math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \frac{1}{c} \left(\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} + 4 \pi \iint_{\Sigma} \mathbf{J_e} \cdot \mathrm{d}\mathbf{S} \right) \tag{4}\\
\end{align}
</math>

A few words first on the meanings of the integral symbols...
* The number of integral signs indicates whether you're integrating over one, two or three dimensions. To calculate the treble integral in equation (1) you might integrate first on <math>x</math>, then on <math>y</math>, then on <math>z</math>. The integral doesn't have to be calculated that way. You can get the volume integral perfectly well in other ways. For example, if the volume you are integrating over is spherical, one of your integrations might be over a radial distance.
* The subscript to the integrals, such as <math>\Omega</math> is giving a name to the volume you are integrating over, so you can describe its shape elsewhere.
** The <math>\scriptstyle \partial \Omega</math> subscript on the integral is representing the surface of the volume <math>\Omega</math>.
** In these equations, <math>\Sigma</math> is a convenient Greek letter name to give to a surface. These equations are ''not'' using the common convention that <math>\Sigma</math> means 'a sum'.
** The <math>\scriptstyle \partial \Sigma</math> subscript on the integral is representing the edge of a, typically disk shaped, surface <math>\Sigma</math>.
* The circular symbol superimposed over the middle of an integral is a hint that the line or area being integrated over is a loop in the case of a line, or a closed surface like the surface of a sphere in the case of an area.

In equation (1) you consider any volume of space, <math>\Omega</math>. The right hand side is integrating all charge <math>\rho_e</math> in that volume. <math>\scriptstyle \partial \Omega</math> represents the surface of that volume. The left hand side is a surface integral, over the surface of <math>\Omega</math>, integrating electric field. The dot product with <math>E</math> inside the integral ensures it is measuring how much electric field crosses the surface.

Equation (2) is the same thing for magnetic field and 'magnetic charge'. As <math>\rho_m</math> is zero everywhere, the right hand side evaluates to zero.

In equation (4) you consider a 'loop' in space that is the edge of some surface <math>\Sigma</math>. <math>\scriptstyle \partial \Sigma</math> is the edge of the surface <math>\Sigma</math>. The left hand side with its dot product is measuring the magnetic field around the loop. The right hand side is measuring the change in electric field through the surface, and adding in the electric current through the surface (electric current density integrated over the surface <math>\Sigma</math>).

Equation (3) does the same thing for electric field around the loop and magnetic field crossing the surface, and as <math>J_m = 0</math> the 'magnetic current' term disappears.

{{collapse bottom}}

{{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}
==Using Complex Numbers==

Put:

<math>\begin{align}
\mathbf{F} = \mathbf{B} + i\mathbf{E}
\end{align}</math>

Then F has combined electromagnetic and magnetic fields into one vector, and the four equations reduce down to two.

<math>\begin{align}
\nabla \cdot \mathbf{F} = 4\pi\rho
\end{align}</math>

<math>\begin{align}
\nabla \times \mathbf{F} = \frac{i}{c}\left( \frac{\partial \mathbf{F}}{\partial t} + 4\pi\mathbf{J} \right)\end{align}</math>

In one sense this does not simplify anything. The four equations are still there. In a different sense it has done something worthwhile. It has brought out that magnetic and electromagnetic fields are two aspects of a field, which is best thought of as being represented with complex numbers.

{{collapse bottom}}

==To Dig Deeper==

{{Dig|http://cachestocaches.com/2016/3/what-are-maxwells-equations/|Caches-to-Caches}} - A nuanced description of Maxwell's equations that aim to make the physics more understandable. The use of the more symmetric version of the equations<ref>{{cite book|title=Jackson 1962}}</ref> is inspired by that article.

{{Dig|WP|Maxwell's_Equations}} - As ever Wikipedia aims to be encyclopedic, making the subject matter more than a bit impenetrable.

Maxwell's equations

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JamesCrook: collapse

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JamesCrook: Template.

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JamesCrook: template.

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JamesCrook: New page.

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Maxwell's equations

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JamesCrook: subscripts better explained.

File:CambridgeMap.png

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JamesCrook: Derivative work from Open Street Maps.

== Summary ==
Derivative work from Open Street Maps.
== Licensing ==
{{subst:No license from license selector|Don't know}}

File:Chronogram1.png

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JamesCrook: Based on info in a wikipedia chronogram.

== Summary ==
Based on info in a wikipedia chronogram.
== Licensing ==
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File:WikidiagramsComparisons2.png

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JamesCrook:

== Licensing ==
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JamesCrook:

== Licensing ==
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