Difference between revisions of "Functions composed of Physical Expressions"

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(→‎Lorentz Force for (q,q'))
(→‎Lorentz Force for (q,q'))
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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} \quad = \quad q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
+
: <math>\mathbf{F} \quad = \quad q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]</math>
  
 
The electric field <math>\mathbf{E}</math> is:
 
The electric field <math>\mathbf{E}</math> is:
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The magnetic field <math>\mathbf{B}</math> is:
 
The magnetic field <math>\mathbf{B}</math> is:
  
−
:<math>\mathbf{B} \quad = \quad \nabla \times \mathbf{A}</math>
+
: <math>\mathbf{B} \quad = \quad \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} \quad = \quad q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \right)\right]</math>
+
: <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:
 
Where:
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To restate from a previous section, the magnetic vector potential of a charge <math>q</math> experienced by a charge <math>q</math> is:
 
To restate from a previous section, the magnetic vector potential of a charge <math>q</math> experienced by a charge <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>
  
−
The partial derivative of this with respect to time <math>t</math> is:
+
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{∂\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{d}{dt}(xy) = \dfrac{dx}{dt}y + x\dfrac{dy}{dt}</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>
+
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:
  
−
Where:
+
: <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{∂r'}{∂t}</math> = the velocity of <math>q'</math> at the retarded time <math>t'</math> according to an observer at time <math>t</math> sharing the same inertial frame as <math>q</math>.
+
 
−
* <math>\frac{∂^2r'}{∂t^2}</math> = the acceleration of <math>q'</math> at the retarded time <math>t'</math> according to an observer at time <math>t</math> sharing the same inertial frame as <math>q</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>
−
* <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}</math> = 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> (i.e. the inverse of the norm of the vector difference between the two position vectors <math>r</math> and <math>r'</math>).
+
 
−
* <math>\frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}</math> = the partial derivative of this proximity with respect to time <math>t</math>.
+
{| class="wikitable" width=480
 +
|+ 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{∂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>
 +
|}
 +
 
 +
{| class="wikitable" width=480
 +
|+ 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{∂^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>
 +
|}
  
 
==See also==
 
==See also==

Revision as of 00:52, 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 of a charge [math]q[/math] experienced by a charge [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]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]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]. 		[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]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]. 		[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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