Difference between revisions of "Functions composed of Physical Expressions"

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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 $\frac{d\mathbf{r'}}{dt}$ at $\left(\mathbf{r'},t'\right)$ is:

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

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

Functions for an ordered pair of point charges $(q,q')$

A charge $q$ subject to an electric scalar potential $\varphi$ at $\left(\mathbf{r},t\right)$ due to a point charge $q'$ at $\left(\mathbf{r'},t'\right)$ has an electric potential energy of:

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

A charge $q$ subject to a magnetic vector potential $A$ at $\left(\mathbf{r},t\right)$ due to a point charge $q'$ which had a velocity $\frac{d\mathbf{r'}}{dt}$ at $\left(\mathbf{r'},t'\right)$ has a potential momentum of:

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

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

Lorentz Force for $(q,q')$

The Lorentz Force between charges $(q,q')$ can be derived from the scalar potential $\varphi$ and the vector potential $\mathbf{A}$.

A charge $q$ which has a velocity of $\mathbf{v}$ at $\left(\mathbf{r},t\right)$ will experience a Lorentz force due to a point charge $q'$ at $\left(\mathbf{r'},t'\right)$ of:

$\mathbf{F} = q\left[\mathbf{E} + \mathbf{v} \times \mathbf{B}\right]$

The electric field $\mathbf{E}$ is:

$\mathbf{E} = - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}$

The magnetic field $\mathbf{B}$ is:

$\mathbf{B} = \nabla \times \mathbf{A}$

The Lorentz Force can be expressed directly in terms of the potentials:

$\mathbf{F} = q\left[- \nabla \varphi - \frac{∂\mathbf{A}}{∂t} + \mathbf{v} \left( \nabla \times \mathbf{A} \right)\right]$

Where:

• $-\nabla \varphi$ = negative the gradient of the scalar potential $\varphi$.
• $-\frac{∂\mathbf{A}}{∂t}$ = negative the partial derivative of the magnetic vector potential $\mathbf{A}$ with respect to time $t$.
• $\mathbf{v} \left( \nabla \times \mathbf{A} \right)$ = the cross product of the velocity $\mathbf{v}$ of the charge $q$ and the curl of the magnetic vector potential $\nabla \times \mathbf{A} = \mathbf{B}$ due to charge $q'$.

To restate from a previous section, the magnetic vector potential of a charge $q$ experienced by a charge $q$ is:

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

The partial derivative of this with respect to time $t$ is:

$\frac{∂\mathbf{A}\left(\mathbf{r},\mathbf{r'}\right)}{∂t} = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} + \underset{proximity}{ \frac{ ∂\left[ \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right] }{∂t}} \underset{dislocation}{\frac{d\mathbf{r'}}{dt}} + \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \underset{dislocation}{ \frac{ ∂\left[ \frac{d\mathbf{r'}}{dt} \right] }{∂t}}$

See also

• The Anatomy of a Physical Expression

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