Difference between revisions of "The Anatomy of a Physical Expression"

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|title=The Anatomy of a Physical Expression @ The Function Conjunction [Sho Drives Wiki]
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|keywords=electricity,magnetism,motor,generator
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|description=Our own way to chop down a physical equation into building blocks.
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{{DISPLAYTITLE:Function Conjunction → The Anatomy of a Physical Expression}}
 
{{DISPLAYTITLE:Function Conjunction → The Anatomy of a Physical Expression}}
 
Factors serve as '''The Anatomy of a Physical Expression'''. They come in several types as listed below, each characterized as having a distinct role in defining a property of a physical system. The following list items are partially underlined to make memorization easy:
 
Factors serve as '''The Anatomy of a Physical Expression'''. They come in several types as listed below, each characterized as having a distinct role in defining a property of a physical system. The following list items are partially underlined to make memorization easy:
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* <math>\mathbf{v}</math> = velocity
 
* <math>\mathbf{v}</math> = velocity
 
* <math>\mathbf{a}</math> = acceleration
 
* <math>\mathbf{a}</math> = acceleration
−
====Dislocations according to an observer co-moving with <math>q</math> at time <math>t</math>====
+
====Dislocations according to an inertial observer at time <math>t</math>====
 
* <math>\mathbf{r}</math> = position of a charge <math>q</math> at time <math>t</math>, when it receives a light signal from <math>q'</math> that was emitted earlier at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\mathbf{r}</math> = position of a charge <math>q</math> at time <math>t</math>, when it receives a light signal from <math>q'</math> that was emitted earlier at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\frac{∂\mathbf{r}}{∂t}</math> = <math>\mathbf{\dot{r}}</math> = velocity of a charge <math>q</math> at time <math>t</math>, when it receives a light signal from <math>q'</math> that was emitted earlier at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\frac{∂\mathbf{r}}{∂t}</math> = <math>\mathbf{\dot{r}}</math> = velocity of a charge <math>q</math> at time <math>t</math>, when it receives a light signal from <math>q'</math> that was emitted earlier at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\frac{∂^2\mathbf{r}}{∂t^2}</math> = <math>\mathbf{\ddot{r}}</math> = acceleration of a charge <math>q</math> at time <math>t</math>, when it receives a light signal from <math>q'</math> that was emitted earlier at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\frac{∂^2\mathbf{r}}{∂t^2}</math> = <math>\mathbf{\ddot{r}}</math> = acceleration of a charge <math>q</math> at time <math>t</math>, when it receives a light signal from <math>q'</math> that was emitted earlier at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
−
* <math>\mathbf{r'}</math> = position a charge <math>q'</math> was 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>\mathbf{r'}</math> = position 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{r'}}{∂t}</math> = <math>\mathbf{\dot{r}'}</math> = velocity a charge <math>q'</math> was 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>
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* <math>\frac{∂\mathbf{r'}}{∂t}</math> = <math>\mathbf{\dot{r}'}</math> = 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{∂^2\mathbf{r'}}{∂t^2}</math> = <math>\mathbf{\ddot{r}'}</math> = acceleration a charge <math>q'</math> was 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{∂^2\mathbf{r'}}{∂t^2}</math> = <math>\mathbf{\ddot{r}'}</math> = 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>
  
 
===Directions===
 
===Directions===
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* <math>\mathbf{\hat{v}}</math> = velocity unit vector
 
* <math>\mathbf{\hat{v}}</math> = velocity unit vector
 
* <math>\mathbf{\hat{a}}</math> = acceleration unit vector
 
* <math>\mathbf{\hat{a}}</math> = acceleration unit vector
−
====Directions according to an observer co-moving with <math>q</math> at time <math>t</math>====
+
====Directions according to an inertial observer at time <math>t</math>====
 
* <math>\mathbf{\hat{r}}</math> = position unit vector of <math>q</math> at time <math>t</math>
 
* <math>\mathbf{\hat{r}}</math> = position unit vector of <math>q</math> at time <math>t</math>
 
* <math>\mathbf{\hat{\dot{r}}}</math> = velocity unit vector of <math>q</math> at time <math>t</math>
 
* <math>\mathbf{\hat{\dot{r}}}</math> = velocity unit vector of <math>q</math> at time <math>t</math>
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==See also==
 
==See also==
  
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* [[Magnetic Energy]]
 
* [[Functions composed of Physical Expressions]]
 
* [[Functions composed of Physical Expressions]]
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* [[Electromagnetic Potentials]]
  
 
{{Site map}}
 
{{Site map}}
  
 
[[Category:Function Conjunction]]
 
[[Category:Function Conjunction]]

Latest revision as of 20:18, 4 July 2016

Factors serve as The Anatomy of a Physical Expression. They come in several types as listed below, each characterized as having a distinct role in defining a property of a physical system. The following list items are partially underlined to make memorization easy:

  1. Constants
  2. Coefficients
  3. Quantities
  4. Proximities
  5. Dislocations
  6. Directions


Definition

Constant (or 1) [math]\times[/math]
Coefficient (or 1) [math]\times[/math]
Quantity (or 1) [math]\times[/math]
Proximity (or 1) [math]\times[/math]
Dislocation (or 1) [math]\times[/math]
Direction (or 1) [math]=[/math]
A Physical Expression

Constants

  • [math]c[/math] = Speed of Light
  • [math]G[/math] = Gravitational constant
  • [math]k_B[/math] = Boltzmann's constant
  • [math]\alpha[/math] = Fine Structure constant
  • [math]\mu_0[/math] = Magnetic Permeability of Free Space
  • [math]\epsilon_0[/math] = Electric Permittivity of Free Space

Coefficients

  • [math]\mu_r[/math] = Relative Magnetic Permeability
  • [math]\epsilon_r[/math] = Relative Electric Permittivity

Quantities

  • [math]q[/math] = point charge
  • [math]\lambda_q[/math] = linear charge density (for continuous charge)
  • [math]\sigma_q[/math] = surface charge density (for continuous charge)
  • [math]\rho_q[/math] = volume charge density (for continuous charge)
  • [math]m[/math] = mass
  • [math]\rho[/math] = volume mass density

Proximities

  • [math]\frac{1}{|\mathbf{r}-\mathbf{r'}|}[/math] = inverse of the magnitude of the separation between positions [math]\mathbf{r}[/math] and [math]\mathbf{r'}[/math]
  • [math]\frac{1}{|\mathbf{r}-\mathbf{r'}|^2}[/math] = inverse square of the magnitude of the separation between positions [math]\mathbf{r}[/math] and [math]\mathbf{r'}[/math]

Dislocations

  • [math]\mathbf{x}[/math] = position
  • [math]\mathbf{v}[/math] = velocity
  • [math]\mathbf{a}[/math] = acceleration

Dislocations according to an inertial observer at time [math]t[/math]

  • [math]\mathbf{r}[/math] = position of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\frac{∂\mathbf{r}}{∂t}[/math] = [math]\mathbf{\dot{r}}[/math] = velocity of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\frac{∂^2\mathbf{r}}{∂t^2}[/math] = [math]\mathbf{\ddot{r}}[/math] = acceleration of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\mathbf{r'}[/math] = position 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{r'}}{∂t}[/math] = [math]\mathbf{\dot{r}'}[/math] = 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{∂^2\mathbf{r'}}{∂t^2}[/math] = [math]\mathbf{\ddot{r}'}[/math] = 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]

Directions

  • [math]\mathbf{\hat{x}}[/math] = position unit vector
  • [math]\mathbf{\hat{v}}[/math] = velocity unit vector
  • [math]\mathbf{\hat{a}}[/math] = acceleration unit vector

Directions according to an inertial observer at time [math]t[/math]

  • [math]\mathbf{\hat{r}}[/math] = position unit vector of [math]q[/math] at time [math]t[/math]
  • [math]\mathbf{\hat{\dot{r}}}[/math] = velocity unit vector of [math]q[/math] at time [math]t[/math]
  • [math]\mathbf{\hat{\ddot{r}}}[/math] = acceleration unit vector of [math]q[/math] at time [math]t[/math]
  • [math]\mathbf{\hat{r'}}[/math] = position unit vector of [math]q'[/math] at retarded time [math]t'[/math]
  • [math]\mathbf{\hat{\dot{r'}}}[/math] = velocity unit vector of [math]q'[/math] at retarded time [math]t'[/math]
  • [math]\mathbf{\hat{\ddot{r'}}}[/math] = acceleration unit vector of [math]q'[/math] at retarded time [math]t'[/math]

See also

Site map

HQ ● Glossary ● April 2016 Presentation

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