Maxwell's equations - Revision history

JamesCrook: Permissivity.

2020-10-01T17:05:42Z

Permissivity.

JamesCrook: Clearer?

2020-09-30T13:01:57Z

Clearer?

JamesCrook: /* Using Complex Numbers */

2020-09-30T11:54:11Z

Using Complex Numbers

JamesCrook: Spell it out.

2020-09-30T11:47:40Z

Spell it out.

JamesCrook: Framing statement.

2020-09-30T11:31:00Z

Framing statement.

JamesCrook at 11:09, 30 September 2020

2020-09-30T11:09:10Z

JamesCrook: Teasers.

2020-09-30T11:05:52Z

Teasers.

JamesCrook: Neater.

2020-09-30T11:02:10Z

Neater.

JamesCrook: Complex number version.

2020-09-30T10:40:09Z

Complex number version.

JamesCrook: collapse

2020-09-30T09:58:33Z

collapse

@@ Line 8: / Line 8: @@
 * 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>.
-It is astonishing that electricity, magnetism, light and radio waves are interrelated.
+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==

@@ Line 5: / Line 5: @@
 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>.
+* 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.
 * 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.

@@ Line 88: / Line 88: @@
 \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.
+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}
@@ Line 97: / Line 98: @@
 \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.
+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==

@@ Line 88: / Line 88: @@
 \end{align}</math>
-Then F has combined electromagnetic and magnetic fields into one vector, and the four equations reduce down to two.
+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}
@@ Line 97: / Line 97: @@
 \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.
+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.
 {{collapse bottom}}

@@ Line 3: / Line 3: @@
 '''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 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.
 ==Prerequisities==

@@ Line 21: / Line 21: @@
 ==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 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.
@@ Line 74: / Line 75: @@
 ==Using Complex Numbers==
 {{collapse top|1=Four Equations to two - Using <math>\mathbb{C}</math>omplex numbers}}

@@ Line 19: / Line 19: @@
 To-Be-Written.
 ==Integral Version==
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 The equivalence of the integral and differential forms of the equations follows from [[Stoke's Theorem]] and [[Green's Theorem]].