Difference between revisions of "Functions composed of Physical Expressions"

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(→‎Lorentz Force for (q,q'))
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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 from a charge <math>q'</math> at the position <math>r</math> at <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>
 
: <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>
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: <math>\frac{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}</math>.
 
: <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>r</math> due to <math>q'</math> with respect to time <math>t</math> is:
+
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{ ∂\left[ \frac{∂\mathbf{r'}}{∂t} \right] }{∂t}} \right]</math>
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|+ First term in the brackets
 
|+ First term in the brackets
 
|-
 
|-
−
|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>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'</math>.
+
|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>
 
|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>
 
|}
 
|}
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|+ Second term in the brackets
 
|+ 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>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'</math>.
+
|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>
 
|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>
 
|}
 
|}

Revision as of 13:47, 15 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) \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

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HQ ● Glossary ● April 2016 Presentation

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