Function Conjunction

The Anatomy of a Physical Expression

Constant	$\times$	Coefficient	$\times$	Quantity	$\times$	Proximity	$\times$	Dislocation	$\times$	Direction
Examples: $\mu_0, \epsilon_0$ $k_B, \alpha, c$ or $1$		Examples: $\mu_r, \epsilon_r$ or $1$		Examples: $q,\lambda_q,\sigma_q,\rho_q$ $m,\rho$ or $1$		Examples: $\frac{1}{|\mathbf{r}-\mathbf{r'}|}, \frac{1}{|\mathbf{r}-\mathbf{r'}|^2}$ or $1$		Examples: $\mathbf{x}, \mathbf{v}, \mathbf{a}$ $\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}$ $\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}$ or $1$		Examples: $\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}$ $\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}$ $\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}$ or $1$

Constants

• $\mu_0$ = Magnetic Permeability of Free Space
• $\epsilon_0$ = Electric Permittivity of Free Space
• $k_B$ = Boltzmann's constant
• $\alpha$ = Fine Structure Constant
• $c$ = Speed of Light

Coefficients

• $\mu_r$ = Relative Magnetic Permeability of Free Space
• $\epsilon_r$ = Relative Electric Permittivity of Free Space

Quantities

• $q$ = point charge
• $\lambda_q$ = linear charge density (for continuous charge)
• $\sigma_q$ = surface charge density (for continuous charge)
• $\rho_q$ = volume charge density (for continuous charge)
• $m$ = mass
• $\rho$ = volume mass density

Proximities

• $\frac{1}{|\mathbf{r}-\mathbf{r'}|}$ = inverse of the magnitude of the separation between positions $\mathbf{r}$ and $\mathbf{r'}$
• $\frac{1}{|\mathbf{r}-\mathbf{r'}|^2}$ = inverse square of the magnitude of the separation between positions $\mathbf{r}$ and $\mathbf{r'}$

Dislocations

• $\mathbf{\hat{x}}$ = position
• $\mathbf{\hat{v}}$ = velocity
• $\mathbf{\hat{a}}$ = acceleration
• $\mathbf{r}$ = position of a charge $q$ at time $t$, when it receives a light signal from $q'$ that was emitted earlier at time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$
• $\frac{d\mathbf{r}}{dt}$ = velocity of a charge $q$ at time $t$, when it receives a light signal from $q'$ that was emitted earlier at time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$
• $\frac{d^2\mathbf{r}}{dt^2}$ = acceleration of a charge $q$ at time $t$, when it receives a light signal from $q'$ that was emitted earlier at time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$
• $\mathbf{r'}$ = position a charge $q'$ was at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$, when it emitted a light signal which has now reached $q$ at position $\mathbf{r}$ and time $t$
• $\frac{d\mathbf{r'}}{dt}$ = velocity a charge $q'$ was at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$, when it emitted a light signal which has now reached $q$ at position $\mathbf{r}$ and time $t$
• $\frac{d^2\mathbf{r'}}{dt^2}$ = acceleration a charge $q'$ was at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$, when it emitted a light signal which has now reached $q$ at position $\mathbf{r}$ and time $t$

Directions

• $\mathbf{\hat{x}}$ = position unit vector
• $\mathbf{\hat{v}}$ = velocity unit vector
• $\mathbf{\hat{a}}$ = acceleration unit vector
• $\mathbf{\hat{r}}$ = position unit vector of $q$
• $\mathbf{\hat{\dot{r}}}$ = velocity unit vector of $q$
• $\mathbf{\hat{\ddot{r}}}$ = acceleration unit vector of $q$
• $\mathbf{\hat{r'}}$ = position unit vector of $q'$
• $\mathbf{\hat{\dot{r'}}}$ = velocity unit vector of $q'$
• $\mathbf{\hat{\ddot{r'}}}$ = acceleration unit vector of $q'$

Functions Composed of Physical Expressions

Functions for a point charge $q'$

The electric scalar potential $\mathbf{\varphi}$ at $\left(\mathbf{r},t\right)$ due to a point charge $q'$ at $\left(\mathbf{r'},t'\right)$ is:

$\mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}$

The magnetic vector potential $A$ at $\left(\mathbf{r},t\right)$ due to a point charge $q'$ which had a velocity $\mathbf{v'}$ at $\left(\mathbf{r'},t'\right)$ is:

$\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{1}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}$

$\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}$