From S.H.O.
The Anatomy of a Physical Expression
Constant
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[math]\times[/math]
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Coefficient
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[math]\times[/math]
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Quantity
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[math]\times[/math]
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Proximity
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[math]\times[/math]
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Dislocation
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[math]\times[/math]
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Direction
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Examples: [math]\mu_0, \epsilon_0[/math] [math]k_B, \alpha, c[/math] or [math]1[/math]
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Examples: [math]\mu_r, \epsilon_r[/math] or [math]1[/math]
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Examples: [math]q,\lambda_q,\sigma_q,\rho_q[/math] [math]m,\rho[/math] or [math]1[/math]
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Examples: [math]\frac{1}{|\mathbf{r}-\mathbf{r'}|}, \frac{1}{|\mathbf{r}-\mathbf{r'}|^2}[/math] or [math]1[/math]
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Examples: [math]\mathbf{x}, \mathbf{v}, \mathbf{a}[/math] [math]\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}[/math] [math]\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}[/math] or [math]1[/math]
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Examples: [math]\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}[/math] [math]\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}[/math] [math]\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}[/math] or [math]1[/math]
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Constants
- [math]\mu_0[/math] = Magnetic Permeability of Free Space
- [math]\epsilon_0[/math] = Electric Permittivity of Free Space
- [math]k_B[/math] = Boltzmann's constant
- [math]\alpha[/math] = Fine Structure Constant
- [math]c[/math] = Speed of Light
Coefficients
- [math]\mu_r[/math] = Relative Magnetic Permeability of Free Space
- [math]\epsilon_r[/math] = Relative Electric Permittivity of Free Space
Quantities
- [math]q[/math] = point charge
- [math]\lambda_q[/math] = linear charge density (for continuous charge)
- [math]\sigma_q[/math] = surface charge density (for continuous charge)
- [math]\rho_q[/math] = volume charge density (for continuous charge)
- [math]m[/math] = mass
- [math]\rho[/math] = volume mass density
Proximities
- [math]\frac{1}{|\mathbf{r}-\mathbf{r'}|}[/math] = inverse of the magnitude of the separation between positions [math]\mathbf{r}[/math] and [math]\mathbf{r'}[/math]
- [math]\frac{1}{|\mathbf{r}-\mathbf{r'}|^2}[/math] = inverse square of the magnitude of the separation between positions [math]\mathbf{r}[/math] and [math]\mathbf{r'}[/math]
Dislocations
- [math]\mathbf{\hat{x}}[/math] = position
- [math]\mathbf{\hat{v}}[/math] = velocity
- [math]\mathbf{\hat{a}}[/math] = acceleration
- [math]\mathbf{r}[/math] = position of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
- [math]\frac{d\mathbf{r}}{dt}[/math] = velocity of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
- [math]\frac{d^2\mathbf{r}}{dt^2}[/math] = acceleration of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
- [math]\mathbf{r'}[/math] = position a charge [math]q'[/math] was 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]
- [math]\frac{d\mathbf{r'}}{dt}[/math] = velocity a charge [math]q'[/math] was 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]
- [math]\frac{d^2\mathbf{r'}}{dt^2}[/math] = acceleration a charge [math]q'[/math] was 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]
Directions
- [math]\mathbf{\hat{x}}[/math] = position unit vector
- [math]\mathbf{\hat{v}}[/math] = velocity unit vector
- [math]\mathbf{\hat{a}}[/math] = acceleration unit vector
- [math]\mathbf{\hat{r}}[/math] = position unit vector of [math]q[/math]
- [math]\mathbf{\hat{\dot{r}}}[/math] = velocity unit vector of [math]q[/math]
- [math]\mathbf{\hat{\ddot{r}}}[/math] = acceleration unit vector of [math]q[/math]
- [math]\mathbf{\hat{r'}}[/math] = position unit vector of [math]q'[/math]
- [math]\mathbf{\hat{\dot{r'}}}[/math] = velocity unit vector of [math]q'[/math]
- [math]\mathbf{\hat{\ddot{r'}}}[/math] = acceleration unit vector of [math]q'[/math]
Functions Composed of Physical Expressions
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]\mathbf{v'}[/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]