Function Conjunction

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The Anatomy of a Physical Expression

Constant [math]\times[/math] Coefficient [math]\times[/math] Quantity [math]\times[/math] Proximity [math]\times[/math] Dislocation [math]\times[/math] Direction
Examples:
[math]\mu_0, \epsilon_0[/math]
[math]k_B, \alpha, c[/math]
or
[math]1[/math]
Examples:
[math]\mu_r, \epsilon_r[/math]
[math]N[/math]
or
[math]1[/math]
Examples:
[math]q,\lambda_q,\sigma_q,\rho_q[/math]
[math]m,\rho[/math]
or
[math]1[/math]
Examples:
[math]\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}[/math]
or
[math]1[/math]
Examples:
[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]
[math]\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta[/math]
or
[math]1[/math]
Examples:
[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]
[math]\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}[/math]
or
[math]1[/math]

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

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

Dislocations

  • [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

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'},\mathbf{v}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}/c^2}[/math]

[math]\mathbf{A}\left(\mathbf{r},\mathbf{r'},\mathbf{v}\right) = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}}[/math]