Function Conjunction → 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 $\frac{d\mathbf{r'}}{dt}$ 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:

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

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:

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

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

Lorentz Force for $(q,q')$

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

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:

$\mathbf{F}_{(q,q')} = q\left[\mathbf{E}_{(q,q')} + \mathbf{v} \times \mathbf{B}_{(q,q')}\right]$

The electric field $\mathbf{E}_{(q,q')}$ is:

$\mathbf{E}_{(q,q')} = - \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t}$

The magnetic field $\mathbf{B}_{(q,q')}$ is:

$\mathbf{B}_{(q,q')} = \nabla \times \mathbf{A}_{(q,q')}$

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

$\mathbf{F}_{(q,q')} = q\left[- \nabla \varphi_{(q,q')} - \frac{∂\mathbf{A}_{(q,q')}}{∂t} + \mathbf{v} \times \left( \nabla \times \mathbf{A}_{(q,q')} \right)\right]$

See also

• The Anatomy of a Physical Expression

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