From S.H.O.
Math can be hard and physics can be even harder. But if that is your thing, continue on for more about Functions composed of Physical Expressions. For experts!
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Preface
Physics can be a challenging subject, especially in more difficult areas such as in quantum mechanics and string theory. However, the physics involved in classical electromagnetism is an intermediate level of difficulty. It's harder than Newtonian dynamics, but all of it exists in three dimensions of space + one dimension of time. No wormholes or hyper-dimensions here! Just classical electromagnetism. S.H.O. talk 17:09, 15 May 2016 (PDT)
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].
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[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]
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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].
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[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]
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