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| − | ==The Anatomy of a Physical Expression==
| + | '''[[The Anatomy of a Physical Expression]]''' |
| | | | |
| − | <div style="overflow-x: auto">
| + | : A physical expression is a product of factors, each with their own distinct role in defining a property of a physical system. Types include constants, coefficients, quantities, proximities, dislocations, and directions. |
| − | {| class="wikitable"
| + | |
| − | |-
| + | |
| − | ! Constant
| + | |
| − | !rowspan=2|<math>\times</math>
| + | |
| − | ! Coefficient
| + | |
| − | !rowspan=2|<math>\times</math>
| + | |
| − | ! Quantity
| + | |
| − | !rowspan=2|<math>\times</math>
| + | |
| − | ! Proximity
| + | |
| − | !rowspan=2|<math>\times</math>
| + | |
| − | ! Dislocation
| + | |
| − | !rowspan=2|<math>\times</math>
| + | |
| − | ! Direction
| + | |
| − | |-
| + | |
| − | |valign=top align=center| '''Examples:'''<br><math>\mu_0, \epsilon_0</math><br><math>k_B, \alpha, c</math><br>or<br><math>1</math>
| + | |
| − | |valign=top align=center| '''Examples:'''<br><math>\mu_r, \epsilon_r</math><br>or<br><math>1</math>
| + | |
| − | |valign=top align=center| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math><br>or<br><math>1</math>
| + | |
| − | |valign=top align=center| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}, \frac{1}{|\mathbf{r}-\mathbf{r'}|^2}</math><br>or<br><math>1</math>
| + | |
| − | |valign=top align=center| '''Examples:'''<br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}</math><br><math>\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}</math><br><math>\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}</math><br>or<br><math>1</math>
| + | |
| − | |valign=top align=center| '''Examples:'''<br><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math><br><math>\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}</math><br><math>\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}</math><br>or<br><math>1</math>
| + | |
| − | |}
| + | |
| − | </div>
| + | |
| − | | + | |
| − | ===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>
| + | |
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| | ==Functions Composed of Physical Expressions== | | ==Functions Composed of Physical Expressions== |
