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| − | ==The Anatomy of a Physical Expression== | + | {{#seo: |
| | + | |title=Function Conjunction @ Sho Drives Wiki |
| | + | |titlemode=replace |
| | + | |keywords=electricity,magnetism,motor,generator |
| | + | |description=To facilitate understanding of the S.H.O. Drive, this place catalogs the ideas, creations, and techniques underlying its invention. |
| | + | }} |
| | | | |
| − | <div style="overflow-x: auto">
| + | To facilitate understanding of the S.H.O. Drive, this '''Function Conjunction''' will catalog the ideas, creations, and techniques underlying its invention. |
| − | {| 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{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><math>\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta</math><br>or<br><math>1</math>
| + | |
| − | |valign=top align=center| '''Examples:'''<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><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math><br>or<br><math>1</math>
| + | |
| − | |}
| + | |
| − | </div>
| + | |
| | | | |
| − | ===Constants===
| + | '''[[Magnetic Energy]]''' |
| − | * <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===
| + | : The Magnetic Energy is the energy existing in magnetic fields. |
| − | * <math>\mu_r</math> = Relative Magnetic Permeability of Free Space
| + | |
| − | * <math>\epsilon_r</math> = Relative Electric Permittivity of Free Space
| + | |
| | | | |
| − | ===Quantities===
| + | '''[[The Anatomy of a Physical Expression]]''' |
| − | * <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===
| + | : 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. |
| − | * <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===
| + | '''[[Functions composed of Physical Expressions]]''' |
| − | * <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===
| + | : A function composed of physical expressions is simply the result of the summations, differences, exponentiations, logarithms, or distributed multiplications or divisions of these physical expressions, or in the simplest case, a function is simply equal to an expression, such as <math>E(m) = mc^2</math>, where <math>E</math> is a function of <math>m</math>. |
| − | * <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==
| + | '''[[Electromagnetic Potentials]]''' |
| | | | |
| − | ===Functions for a point charge <math>q'</math>===
| + | : The basic idea here is that the electromagnetic potentials <math>\phi</math> and <math>A</math> and their derivatives can be used to derive all electromagnetism. |
| | | | |
| − | 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:
| + | {{Site map}} |
| | | | |
| − | <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>
| + | [[Category:Function Conjunction| ]] |
| − | | + | |
| − | 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>
| + | |
To facilitate understanding of the S.H.O. Drive, this Function Conjunction will catalog the ideas, creations, and techniques underlying its invention.
