Function Conjunction

The Anatomy of an Physical Expression

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

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

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

Dislocations

• $\mathbf{r}$ = 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 $t$

Functions Composed of Physical Expressions

Functions for a point charge $q'$

Electric scalar potential $\mathbf{\varphi}$

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

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

Magnetic vector potential $A$

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

$\mathbf{A}\left(\mathbf{r},\mathbf{r'},\mathbf{v}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}/c^2} = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}}$