Difference between revisions of "Functions composed of Physical Expressions"

From S.H.O.
Jump to: navigation, search
(→‎Functions for an ordered pair of point charges (q,q'))
(→‎Functions for an ordered pair of point charges (q,q'))
Line 14: Line 14:
 
==Functions for an ordered pair of point charges <math>(q,q')</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>\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> 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_{(q,q')}\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_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{qq'}{4\pi\ \epsilon_0}} \times \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{d\mathbf{r'}}{dt}</math> at <math>\left(\mathbf{r'},t'\right)</math> has a potential momentum of:
  
−
<math>q\mathbf{A}_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) = q\varphi_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{1}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>
+
:<math>q\mathbf{A}_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) = q\varphi_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{1}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>
  
−
<math>q\mathbf{A}_{(q,q')}\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>
+
:<math>q\mathbf{A}_{(q,q')}\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>
 +
 
 +
=== 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_{(q,q')}</math> and the vector potential <math>\mathbf{A}_{(q,q')}</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}_{(q,q')} = q\left[\mathbf{E}_{(q,q')} + \mathbf{v} \times \mathbf{B}_{(q,q')}\right]</math>
 +
 
 +
The electric field <math>\mathbf{E}_{(q,q')}</math> is:
 +
 
 +
:<math>\mathbf{E}_{(q,q')} = - \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t}</math>
 +
 
 +
The magnetic field <math>\mathbf{B}_{(q,q')}</math> is:
 +
 
 +
:<math>\mathbf{B}_{(q,q')} = \nabla \times \mathbf{A}_{(q,q')}</math>
 +
 
 +
The Lorentz Force can be expressed directly in terms of the potentials:
 +
 
 +
:<math>\mathbf{F}_{(q,q')} = q\left[- \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A}_{(q,q')} \right)\right]</math>
  
 
==See also==
 
==See also==

Revision as of 23:43, 14 May 2016

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) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \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:

[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'}\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]

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_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{qq'}{4\pi\ \epsilon_0}} \times \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:

[math]q\mathbf{A}_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) = q\varphi_{(q,q')}\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{1}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}[/math]
[math]q\mathbf{A}_{(q,q')}\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]

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_{(q,q')}[/math] and the vector potential [math]\mathbf{A}_{(q,q')}[/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}_{(q,q')} = q\left[\mathbf{E}_{(q,q')} + \mathbf{v} \times \mathbf{B}_{(q,q')}\right][/math]

The electric field [math]\mathbf{E}_{(q,q')}[/math] is:

[math]\mathbf{E}_{(q,q')} = - \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t}[/math]

The magnetic field [math]\mathbf{B}_{(q,q')}[/math] is:

[math]\mathbf{B}_{(q,q')} = \nabla \times \mathbf{A}_{(q,q')}[/math]

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

[math]\mathbf{F}_{(q,q')} = q\left[- \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A}_{(q,q')} \right)\right][/math]

See also

Site map

HQ ● Glossary ● April 2016 Presentation

Retrieved from "http://www.sho.wiki/index.php?title=Functions_composed_of_Physical_Expressions&oldid=713"