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| | :<math>\mathbf{F} = q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A} \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> |
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| | + | Where: |
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| | + | * <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>. |
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| | ==See also== | | ==See also== |
Revision as of 00:00, 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) = \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\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}\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}\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[/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} = q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right][/math]
The electric field [math]\mathbf{E}[/math] is:
- [math]\mathbf{E} = - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}[/math]
The magnetic field [math]\mathbf{B}[/math] is:
- [math]\mathbf{B} = \nabla \times \mathbf{A}[/math]
The Lorentz Force can be expressed directly in terms of the potentials:
- [math]\mathbf{F} = 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].
See also
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