Magnetic Potential

Terms: Magnetic Potential (40,500), magnetic vector potential (18,700), magnetic scalar potential (1,030), Electromagnetic four-potential (266), four potential (343,000),

Boolean: "magnetic potential" AND weber (295),

Terms: weber/m (320,000), wb/m (46,800), weber/m2 (127), Tesla meter (1,890), volt second/meter (93), volt/m/s (4), volt second (2,330),

\vec{E} = -\vec{\nabla} \phi - \frac{\partial \vec{A}}{\partial t} \qquad \left( -\vec{\nabla} \phi - \frac{1}{c} \frac{\partial \vec{A}}{\partial t} \right)

\vec{B} = \vec{\nabla} \times \vec{A}

Electromagnetic four-potential

For a given charge and current distribution, \rho(\vec{x},t) and \vec{j}(\vec{x},t), the solutions to these equations in SI units are

\phi (\vec{x}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}
\vec A (\vec{x}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\vec{j}( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|},

where \tau = t - \frac{\left|\vec{x}-\vec{x}'\right|}{c} is the retarded time. This is sometimes also expressed with \rho(\vec{x}',\tau)=[\rho(\vec{x}',t)], where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.

When the integrals above are evaluated for typical cases, eg of an oscillating current (or charge), they are found to give both a magnetic field component varying as r - 2 (the induction field) and a component decreasing as r - 1 (the radiation field).

Terms: Jefimenko's equations (111), retarded time (3,360),

Jefimenko's equations

In the vacuum, the electric field \vec{E} and the magnetic field \vec{B} are given in terms of the charge density \rho\, and the current density \vec{J} as:


\vec{E}(\vec{r},t) = \frac{1}{4\pi\epsilon_0}\int{\left(\frac{\rho(\vec{r'},t_r)\,\vec{R}}{R^3}+\frac{\vec{R}}{R^2c}\frac{\partial\rho(\vec{r'},t_r)}{\partial t} - \frac{1}{Rc^2}\frac{\partial \vec{J}(\vec{r'},t_r)}{\partial t}\right)\mathrm{d}^3\vec{r'}}

\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi}\int{\left(\frac{\vec{J}(\vec{r'},t_r)\times\vec{R}}{R^3}+\frac{1}{R^2c}\frac{\partial \vec{J}(\vec{r'},t_r)}{\partial t}\times\vec{R}\right)\mathrm{d}^3\vec{r'}}

where \vec{R} = \vec{r} - \vec{r'}, and t_r = t - R/c \, (the retarded time).