Difference between revisions of "Function Conjunction"

From S.H.O.
Jump to: navigation, search
(→‎The Anatomy of a Physical Expression)
Line 17: Line 17:
 
|-
 
|-
 
|valign=top align=center| '''Examples:'''<br><math>\mu_0, \epsilon_0</math><br><math>k_B, \alpha, c</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>\mu_0, \epsilon_0</math><br><math>k_B, \alpha, c</math><br>or<br><math>1</math>
−
|valign=top align=center| '''Examples:'''<br><math>\mu_r, \epsilon_r</math><br><math>N</math><br>or<br><math>1</math>
+
|valign=top align=center| '''Examples:'''<br><math>\mu_r, \epsilon_r</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math><br>or<br><math>1</math>
−
|valign=top align=center| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}</math><br>or<br><math>1</math>
+
|valign=top align=center| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}, \frac{1}{|\mathbf{r}-\mathbf{r'}|^2}</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}</math><br><math>\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}</math><br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}</math><br><math>\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}</math><br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}</math><br><math>\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}</math><br><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math><br>or<br><math>1</math>
 
|valign=top align=center| '''Examples:'''<br><math>\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}</math><br><math>\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}</math><br><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math><br>or<br><math>1</math>
Line 33: Line 33:
  
 
===Coefficients===
 
===Coefficients===
 +
* <math>\mu_r</math> = Relative Magnetic Permeability of Free Space
 +
* <math>\epsilon_r</math> = Relative Electric Permittivity of Free Space
  
 
===Quantities===
 
===Quantities===
Line 43: Line 45:
  
 
===Proximities===
 
===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===
 
===Dislocations===
 +
* <math>\mathbf{\hat{x}}</math> = position
 +
* <math>\mathbf{\hat{v}}</math> = velocity
 +
* <math>\mathbf{\hat{a}}</math> = acceleration
 
* <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 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 time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\frac{d\mathbf{r}}{dt}</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 time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 
* <math>\frac{d\mathbf{r}}{dt}</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 time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
Line 50: Line 57:
 
* <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> 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{d\mathbf{r'}}{dt}</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>
 
* <math>\frac{d\mathbf{r'}}{dt}</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>
−
* <math>\frac{d^2\mathbf{r'}}{dt^2}</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{d^2\mathbf{r'}}{dt^2}</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>  
  
 
===Directions===
 
===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
 +
* <math>\mathbf{\hat{r}}</math> = position unit vector of <math>q</math>
 +
* <math>\mathbf{\hat{\dot{r}}}</math> = velocity unit vector of <math>q</math>
 +
* <math>\mathbf{\hat{\ddot{r}}}</math> = acceleration unit vector of <math>q</math>
 +
* <math>\mathbf{\hat{r'}}</math> = position unit vector of <math>q'</math>
 +
* <math>\mathbf{\hat{\dot{r'}}}</math> = velocity unit vector of <math>q'</math>
 +
* <math>\mathbf{\hat{\ddot{r'}}}</math> = acceleration unit vector of <math>q'</math>
  
 
==Functions Composed of Physical Expressions==
 
==Functions Composed of Physical Expressions==
Line 64: Line 80:
 
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>\mathbf{v'}</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>\mathbf{v'}</math> at <math>\left(\mathbf{r'},t'\right)</math> is:
  
−
<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'},\mathbf{v}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}/c^2}</math>
+
<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>
  
−
<math>\mathbf{A}\left(\mathbf{r},\mathbf{r'},\mathbf{v}\right) =  \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}}</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>

Revision as of 20:14, 23 April 2016

The Anatomy of a Physical Expression

Constant × Coefficient × Quantity × Proximity × Dislocation × Direction
Examples:
μ0,ϵ0
kB,α,c
or
1 		Examples:
μr,ϵr
or
1 		Examples:
q,λq,σq,ρq
m,ρ
or
1 		Examples:
1|r−r′|,1|r−r′|2
or
1 		Examples:
r,drdt,d2rdt2
r′,dr′dt,d2r′dt2
x,v,a,β
or
1 		Examples:
ˆr,ˆ˙r,ˆ¨r
^r′,^˙r′,^¨r′
ˆx,ˆv,ˆa
or
1

Constants

  • μ0 = Magnetic Permeability of Free Space
  • ϵ0 = Electric Permittivity of Free Space
  • kB = Boltzmann's constant
  • α = Fine Structure Constant
  • c = Speed of Light

Coefficients

  • μr = Relative Magnetic Permeability of Free Space
  • ϵr = Relative Electric Permittivity of Free Space

Quantities

  • q = point charge
  • λq = linear charge density (for continuous charge)
  • σq = surface charge density (for continuous charge)
  • ρq = volume charge density (for continuous charge)
  • m = mass
  • ρ = volume mass density

Proximities

  • 1|r−r′| = Inverse of the magnitude of the separation between positions r and r′
  • 1|r−r′|2 = Inverse square of the magnitude of the separation between positions r and r′

Dislocations

  • ˆx = position
  • ˆv = velocity
  • ˆa = acceleration
  • r = position of a charge q at time t, when it receives a light signal from q′ that was emitted earlier at time t′=t−|r−r′|/c
  • drdt = velocity of a charge q at time t, when it receives a light signal from q′ that was emitted earlier at time t′=t−|r−r′|/c
  • d2rdt2 = acceleration of a charge q at time t, when it receives a light signal from q′ that was emitted earlier at time t′=t−|r−r′|/c
  • r′ = position a charge q′ was at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t
  • dr′dt = velocity a charge q′ was at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t
  • d2r′dt2 = acceleration a charge q′ was at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t

Directions

  • ˆx = position unit vector
  • ˆv = velocity unit vector
  • ˆa = acceleration unit vector
  • ˆr = position unit vector of q
  • ˆ˙r = velocity unit vector of q
  • ˆ¨r = acceleration unit vector of q
  • ^r′ = position unit vector of q′
  • ^˙r′ = velocity unit vector of q′
  • ^¨r′ = acceleration unit vector of q′

Functions Composed of Physical Expressions

Functions for a point charge q′

The electric scalar potential φ at (r,t) due to a point charge q′ at (r′,t′) is:

φ(r,r′)=q′4π ϵ0constant×1|r−r′|proximity

The magnetic vector potential A at (r,t) due to a point charge q′ which had a velocity v′ at (r′,t′) is:

A(r,r′)=φ(r,r′)×1c2constant×dr′dtdislocation

A(r,r′)=μ0 q′4πconstant×1|r−r′|proximity×dr′dtdislocation

Retrieved from "http://www.sho.wiki/index.php?title=Function_Conjunction&oldid=466"