Difference between revisions of "Functions composed of Physical Expressions"

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(→‎Functions for an ordered pair of point charges (q,q'))
(→‎Functions for an ordered pair of point charges (q,q'))
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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_{(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\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}\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>
  
−
:<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}\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> ===
 
=== 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>.
+
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:
 
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>
+
:<math>\mathbf{F} = q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
  
−
The electric field <math>\mathbf{E}_{(q,q')}</math> is:
+
The electric field <math>\mathbf{E}</math> is:
  
−
:<math>\mathbf{E}_{(q,q')} = - \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t}</math>
+
:<math>\mathbf{E} = - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}</math>
  
−
The magnetic field <math>\mathbf{B}_{(q,q')}</math> is:
+
The magnetic field <math>\mathbf{B}</math> is:
  
−
:<math>\mathbf{B}_{(q,q')} = \nabla \times \mathbf{A}_{(q,q')}</math>
+
:<math>\mathbf{B} = \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,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>
+
:<math>\mathbf{F} = q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right]</math>
  
 
==See also==
 
==See also==

Revision as of 23:48, 14 May 2016

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 dr′dt 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

Functions for an ordered pair of point charges (q,q′)

A charge q subject to an electric scalar potential φ at (r,t) due to a point charge q′ at (r′,t′) has an electric potential energy of:

qφ(r,r′)=qq′4π ϵ0constant×1|r−r′|proximity

A charge q subject to a magnetic vector potential A at (r,t) due to a point charge q′ which had a velocity dr′dt at (r′,t′) has a potential momentum of:

qA(r,r′)=φ(r,r′)×qc2constant×dr′dtdislocation
qA(r,r′)=μ0 qq′4πconstant×1|r−r′|proximity×dr′dtdislocation

Lorentz Force for (q,q′)

The Lorentz Force between charges (q,q′) can be derived from the scalar potential φ and the vector potential A.

A charge q which has a velocity of v at (r,t) will experience a Lorentz force due to a point charge q′ at (r′,t′) of:

F=q[E+v×B]

The electric field E is:

E=−∇φ−∂A∂t

The magnetic field B is:

B=∇×A

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

F=q[−∇φ−∂A∂t+v×(∇×A)]

See also

Site map

HQ ● Glossary ● April 2016 Presentation

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