Difference between revisions of "Functions composed of Physical Expressions"

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(Lorentz Force for (q,q'))
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{{#seo:
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|title=Functions composed of Physical Expressions @ The Function Conjunction [Sho Drives Wiki]
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|titlemode=replace
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|keywords=electricity,magnetism,motor,generator
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|description=Visit http://sho.wiki/ for tips, hints, and whys for all things Sho Drives related!
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{{DISPLAYTITLE:Function Conjunction → Functions composed of Physical Expressions}}
 
{{DISPLAYTITLE:Function Conjunction → Functions composed of Physical Expressions}}
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Math can be hard and physics can be even harder. But if that is your thing, continue on for more about '''Functions composed of Physical Expressions'''. For experts!
 +
 +
:Or, if you want to the see the S.H.O. Drive [[Presentation]], check it out '''[[Presentation|here]]'''.
 +
==Preface==
 +
 +
Physics can be a challenging subject, especially in more difficult areas such as in quantum mechanics and string theory. However, the physics involved in classical electromagnetism is an intermediate level of difficulty. It's harder than Newtonian dynamics, but all of it exists in three dimensions of space + one dimension of time. No wormholes or hyper-dimensions here! Just classical electromagnetism. ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 17:09, 15 May 2016 (PDT)
 +
 
==Functions for a point charge <math>q'</math>==
 
==Functions for a point charge <math>q'</math>==
  
 
The electric scalar potential <math>\mathbf{\varphi}</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> is:
 
The electric scalar potential <math>\mathbf{\varphi}</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> is:
  
<math>\mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}</math>
+
<math>\mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}</math>
  
The magnetic vector potential <math>A</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> which had a velocity <math>\frac{d\mathbf{r'}}{dt}</math> at <math>\left(\mathbf{r'},t'\right)</math> is:
+
The magnetic vector potential <math>A</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> which had a velocity <math>\frac{\mathbf{r'}}{∂t}</math> at <math>\left(\mathbf{r'},t'\right)</math> is:
  
<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{1}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>
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<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \quad \underset{constant}{\frac{1}{c^2}} \quad \underset{dislocation}{\frac{\mathbf{r'}}{∂t}}</math>
  
<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>
+
<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{\mathbf{r'}}{∂t}}</math>
  
 
==Functions for an ordered pair of point charges <math>(q,q')</math>==
 
==Functions for an ordered pair of point charges <math>(q,q')</math>==
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A charge <math>q</math> subject to an electric scalar potential <math>\varphi</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> has an electric potential energy of:
 
A charge <math>q</math> subject to an electric scalar potential <math>\varphi</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> has an electric potential energy of:
  
:<math>q\varphi\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{qq'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}</math>
+
:<math>q\varphi\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{qq'}{4\pi\ \epsilon_0}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}</math>
  
A charge <math>q</math> subject to a magnetic vector potential <math>A</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> which had a velocity <math>\frac{d\mathbf{r'}}{dt}</math> at <math>\left(\mathbf{r'},t'\right)</math> has a potential momentum of:
+
A charge <math>q</math> subject to a magnetic vector potential <math>A</math> at <math>\left(\mathbf{r},t\right)</math> due to a point charge <math>q'</math> which had a velocity <math>\frac{\mathbf{r'}}{∂t}</math> at <math>\left(\mathbf{r'},t'\right)</math> has a potential momentum of:
  
:<math>q\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \varphi\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{q}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>
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:<math>q\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \varphi\left(\mathbf{r},\mathbf{r'}\right) \quad \underset{constant}{\frac{q}{c^2}} \quad \underset{dislocation}{\frac{\mathbf{r'}}{∂t}}</math>
  
:<math>q\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{\mu_0\ qq'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>
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:<math>q\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ qq'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{\mathbf{r'}}{∂t}}</math>
  
 
=== Lorentz Force for <math>(q,q')</math> ===
 
=== Lorentz Force for <math>(q,q')</math> ===
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A charge <math>q</math> which has a velocity of <math>\mathbf{v}</math> at <math>\left(\mathbf{r},t\right)</math> will experience a Lorentz force due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> of:
 
A charge <math>q</math> which has a velocity of <math>\mathbf{v}</math> at <math>\left(\mathbf{r},t\right)</math> will experience a Lorentz force due to a point charge <math>q'</math> at <math>\left(\mathbf{r'},t'\right)</math> of:
  
:<math>\mathbf{F} = q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
+
: <math>\mathbf{F} \quad = \quad q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
  
 
The electric field <math>\mathbf{E}</math> is:
 
The electric field <math>\mathbf{E}</math> is:
  
:<math>\mathbf{E} = - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}</math>
+
:<math>\mathbf{E} \quad = \quad - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}</math>
  
 
The magnetic field <math>\mathbf{B}</math> is:
 
The magnetic field <math>\mathbf{B}</math> is:
  
:<math>\mathbf{B} = \nabla \times \mathbf{A}</math>
+
: <math>\mathbf{B} \quad = \quad \nabla \times \mathbf{A}</math>
  
 
The Lorentz Force can be expressed directly in terms of the potentials:
 
The Lorentz Force can be expressed directly in terms of the potentials:
  
:<math>\mathbf{F} = q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right]</math>
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: <math>\mathbf{F} \quad = \quad q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right]</math>
  
 
Where:
 
Where:
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* <math>\mathbf{v} \times \left( \nabla \times \mathbf{A} \right)</math> = the cross product of the velocity <math>\mathbf{v}</math> of the charge <math>q</math> and the curl of the magnetic vector potential <math>\nabla \times \mathbf{A} = \mathbf{B}</math> due to charge <math>q'</math>.
 
* <math>\mathbf{v} \times \left( \nabla \times \mathbf{A} \right)</math> = the cross product of the velocity <math>\mathbf{v}</math> of the charge <math>q</math> and the curl of the magnetic vector potential <math>\nabla \times \mathbf{A} = \mathbf{B}</math> due to charge <math>q'</math>.
  
To restate from a previous section, the magnetic vector potential of a charge <math>q</math> experienced by a charge <math>q</math> is:
+
To restate from a previous section, the magnetic vector potential from a charge <math>q'</math> at the position <math>\mathbf{r}</math> at <math>q</math> is:
 +
 
 +
: <math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}</math>
 +
 
 +
Using the product rule, the partial derivative of this with respect to time <math>t</math> can be found. For example, the derivative of a product of two variables <math>x</math> and <math>y</math> with respect to time <math>t</math> is:
 +
 
 +
: <math>\frac{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}</math>.
 +
 
 +
Therefore, the partial derivative of the magnetic vector potential at <math>\mathbf{r}</math> due to <math>q'</math> with respect to time <math>t</math> is:
 +
 
 +
: <math>\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{ \frac{ ∂\left[ \frac{∂\mathbf{r'}}{∂t} \right] }{∂t}} \right]</math>
  
<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}</math>
+
: <math>\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{ \frac{∂^2r'}{∂t^2} } \right]</math>
  
The partial derivative of this with respect to time <math>t</math> is:
+
{| class="wikitable" width=480
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|+ First term in the brackets
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|-
 +
|width=240 valign=top align=center| <math>\frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}</math><br>The partial derivative, with respect to time <math>t</math>, of the proximity of the position <math>\mathbf{r}</math> of <math>q</math> at time <math>t</math> to the position <math>r'</math> of <math>q'</math> at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>.
 +
|width=240 valign=top align=center| <math>\frac{∂r'}{∂t}</math><br>According to an observer at time <math>t</math>: the velocity a charge <math>q'</math> had at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math>
 +
|}
  
<math>\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \underset{dislocation}{ \frac{ ∂\left[ \frac{∂\mathbf{r'}}{∂t} \right] }{∂t}} \right]</math>
+
{| class="wikitable" width=480
 +
|+ Second term in the brackets
 +
|-
 +
|width=240 valign=top align=center| <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}</math><br>The proximity of the position <math>\mathbf{r}</math> of <math>q</math> at time <math>t</math> to the position <math>r'</math> of <math>q'</math> at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>.
 +
|width=240 valign=top align=center| <math>\frac{∂^2r'}{∂t^2}</math><br>According to an observer at time <math>t</math>: the acceleration a charge <math>q'</math> had at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math>
 +
|}
  
 
==See also==
 
==See also==
  
 +
* [[Magnetic Energy]]
 
* [[The Anatomy of a Physical Expression]]
 
* [[The Anatomy of a Physical Expression]]
 +
* [[Electromagnetic Potentials]]
  
 
{{Site map}}
 
{{Site map}}
  
 
[[Category:Function Conjunction]]
 
[[Category:Function Conjunction]]

Latest revision as of 19:26, 14 July 2017

Math can be hard and physics can be even harder. But if that is your thing, continue on for more about Functions composed of Physical Expressions. For experts!

Or, if you want to the see the S.H.O. Drive Presentation, check it out here.

Preface

Physics can be a challenging subject, especially in more difficult areas such as in quantum mechanics and string theory. However, the physics involved in classical electromagnetism is an intermediate level of difficulty. It's harder than Newtonian dynamics, but all of it exists in three dimensions of space + one dimension of time. No wormholes or hyper-dimensions here! Just classical electromagnetism. S.H.O. talk 17:09, 15 May 2016 (PDT)

Functions for a point charge [math]q'[/math]

The electric scalar potential [math]\mathbf{\varphi}[/math] at [math]\left(\mathbf{r},t\right)[/math] due to a point charge [math]q'[/math] at [math]\left(\mathbf{r'},t'\right)[/math] is:

[math]\mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}[/math]

The magnetic vector potential [math]A[/math] at [math]\left(\mathbf{r},t\right)[/math] due to a point charge [math]q'[/math] which had a velocity [math]\frac{∂\mathbf{r'}}{∂t}[/math] at [math]\left(\mathbf{r'},t'\right)[/math] is:

[math]\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \quad \underset{constant}{\frac{1}{c^2}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}[/math]

[math]\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}[/math]

Functions for an ordered pair of point charges [math](q,q')[/math]

A charge [math]q[/math] subject to an electric scalar potential [math]\varphi[/math] at [math]\left(\mathbf{r},t\right)[/math] due to a point charge [math]q'[/math] at [math]\left(\mathbf{r'},t'\right)[/math] has an electric potential energy of:

[math]q\varphi\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{qq'}{4\pi\ \epsilon_0}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}[/math]

A charge [math]q[/math] subject to a magnetic vector potential [math]A[/math] at [math]\left(\mathbf{r},t\right)[/math] due to a point charge [math]q'[/math] which had a velocity [math]\frac{∂\mathbf{r'}}{∂t}[/math] at [math]\left(\mathbf{r'},t'\right)[/math] has a potential momentum of:

[math]q\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \varphi\left(\mathbf{r},\mathbf{r'}\right) \quad \underset{constant}{\frac{q}{c^2}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}[/math]
[math]q\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ qq'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}[/math]

Lorentz Force for [math](q,q')[/math]

The Lorentz Force between charges [math](q,q')[/math] can be derived from the scalar potential [math]\varphi[/math] and the vector potential [math]\mathbf{A}[/math].

A charge [math]q[/math] which has a velocity of [math]\mathbf{v}[/math] at [math]\left(\mathbf{r},t\right)[/math] will experience a Lorentz force due to a point charge [math]q'[/math] at [math]\left(\mathbf{r'},t'\right)[/math] of:

[math]\mathbf{F} \quad = \quad q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right][/math]

The electric field [math]\mathbf{E}[/math] is:

[math]\mathbf{E} \quad = \quad - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}[/math]

The magnetic field [math]\mathbf{B}[/math] is:

[math]\mathbf{B} \quad = \quad \nabla \times \mathbf{A}[/math]

The Lorentz Force can be expressed directly in terms of the potentials:

[math]\mathbf{F} \quad = \quad q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right][/math]

Where:

  • [math]-\nabla \varphi[/math] = negative the gradient of the scalar potential [math]\varphi[/math].
  • [math]-\frac{∂\mathbf{A}}{∂t}[/math] = negative the partial derivative of the magnetic vector potential [math]\mathbf{A}[/math] with respect to time [math]t[/math].
  • [math]\mathbf{v} \times \left( \nabla \times \mathbf{A} \right)[/math] = the cross product of the velocity [math]\mathbf{v}[/math] of the charge [math]q[/math] and the curl of the magnetic vector potential [math]\nabla \times \mathbf{A} = \mathbf{B}[/math] due to charge [math]q'[/math].

To restate from a previous section, the magnetic vector potential from a charge [math]q'[/math] at the position [math]\mathbf{r}[/math] at [math]q[/math] is:

[math]\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \quad \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}}[/math]

Using the product rule, the partial derivative of this with respect to time [math]t[/math] can be found. For example, the derivative of a product of two variables [math]x[/math] and [math]y[/math] with respect to time [math]t[/math] is:

[math]\frac{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}[/math].

Therefore, the partial derivative of the magnetic vector potential at [math]\mathbf{r}[/math] due to [math]q'[/math] with respect to time [math]t[/math] is:

[math]\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{ \frac{ ∂\left[ \frac{∂\mathbf{r'}}{∂t} \right] }{∂t}} \right][/math]
[math]\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} \quad = \quad \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \left[ \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \quad \underset{dislocation}{\frac{∂\mathbf{r'}}{∂t}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \quad \underset{dislocation}{ \frac{∂^2r'}{∂t^2} } \right][/math]
First term in the brackets
[math]\frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}[/math]
The partial derivative, with respect to time [math]t[/math], of the proximity of the position [math]\mathbf{r}[/math] of [math]q[/math] at time [math]t[/math] to the position [math]r'[/math] of [math]q'[/math] at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math].
[math]\frac{∂r'}{∂t}[/math]
According to an observer at time [math]t[/math]: the velocity a charge [math]q'[/math] had at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math], when it emitted a light signal which has now reached [math]q[/math] at position [math]\mathbf{r}[/math] and time [math]t[/math]
Second term in the brackets
[math]\frac{1}{|\mathbf{r}-\mathbf{r'}|}[/math]
The proximity of the position [math]\mathbf{r}[/math] of [math]q[/math] at time [math]t[/math] to the position [math]r'[/math] of [math]q'[/math] at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math].
[math]\frac{∂^2r'}{∂t^2}[/math]
According to an observer at time [math]t[/math]: the acceleration a charge [math]q'[/math] had at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math], when it emitted a light signal which has now reached [math]q[/math] at position [math]\mathbf{r}[/math] and time [math]t[/math]

See also

Site map

HQGlossaryApril 2016 Presentation