Jefimenko's equations

Electric and magnetic fields

Jefimenko's equations describe the electric field ${\displaystyle \mathbf {E} }$ and magnetic field ${\displaystyle \mathbf {B} }$ produced by an arbitrary, time-dependent distribution of charge density ${\displaystyle \rho }$ and current density ${\displaystyle \mathbf {J} }$:

${\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\,\iiint \left({\frac {[\rho ]\,\mathbf {n} }{r^{2}}}+{\frac {[{\dot {\rho }}]\,\mathbf {n} }{c\,r}}-{\frac {\left[{\dot {\mathbf {J} }}\right]}{c^{2}r}}\right)\,dV'}$

${\displaystyle \mathbf {B} ={\frac {1}{4\pi \varepsilon _{0}}}\,\displaystyle \iiint \left({\frac {\left[\mathbf {J} \right]\times \mathbf {n} }{c^{2}r^{2}}}+{\frac {\left[{\dot {\mathbf {J} }}\right]\times \mathbf {n} }{c^{3}r}}\right)\,dV'}$

Here, ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity, ${\displaystyle r=|\mathbf {r} -\mathbf {r} '|}$ is the distance between the source point ${\displaystyle \mathbf {r} '}$ and the observation point ${\displaystyle \mathbf {r} }$, and ${\displaystyle \mathbf {n} }$ is the unit vector pointing from the source toward the observer.

Because electromagnetic disturbances propagate at the speed of light ${\displaystyle c}$, the fields observed at a point ${\displaystyle \mathbf {r} }$ at time ${\displaystyle t}$ are determined by the state of the sources at an earlier retarded time, ${\textstyle t_{r}=(t-r/c)}$. The square brackets ${\displaystyle [\ldots ]}$ indicate that the enclosed quantities are evaluated at retarded time:

${\displaystyle [\rho ]=\rho \!\left(t-\displaystyle {\frac {r}{c}}\right)}$ , ${\displaystyle [{\dot {\rho }}]={\dot {\rho }}\!\left(t-\displaystyle {\frac {r}{c}}\right)}$, ${\displaystyle [\mathbf {J} ]=\mathbf {J} \!\left(t-\displaystyle {\frac {r}{c}}\right)}$ and ${\displaystyle \left[{\dot {\mathbf {J} }}\right]={\dot {\mathbf {J} }}\!\left(t-\displaystyle {\frac {r}{c}}\right)}$

In the electrostatic situation, the first term of the electric field equation reduces to the Coulomb's law. Similarly, in the magnetostatic limit, the first term of the magnetic field equation reduces to the Biot–Savart law. Jefimenko's equations are the time-dependent generalizations of these two laws.

Derivation from retarded potentials

Jefimenko's equations can be derived from the retarded potentials V and A:

${\displaystyle V={\frac {1}{4\pi \varepsilon _{0}}}\,\displaystyle \iiint \left({\frac {[\rho ]}{r}}\right)\,dV'}$

${\displaystyle \mathbf {A} ={\frac {1}{4\pi \varepsilon _{0}c^{2}}}\,\displaystyle \iiint \left({\frac {\left[\mathbf {J} \right]}{r}}\right)\,dV'}$

These potentials are the solutions to Maxwell's equations in the potential formulation (see the Helmholtz theorem). To obtain the fields, these expressions are substituted into the definitions of the electromagnetic potentials :

${\displaystyle \mathbf {E} =-{\boldsymbol {\operatorname {grad} }}\!\left(V\right)-{\frac {\partial \mathbf {A} }{\partial t}}}$

${\displaystyle \mathbf {B} ={\boldsymbol {\operatorname {rot} }}\!\left(\mathbf {A} \right)}$

By performing the differentiations with respect to the observer's coordinates ${\displaystyle \mathbf {r} }$ and time ${\displaystyle t}$, the potentials are eliminated in favor of the direct functional dependence on the sources ${\displaystyle \rho }$ and ${\displaystyle \mathbf {J} }$. The fields have an implicit dependence on the position through ${\textstyle t_{r}}$, which must be taken into account in the differentiation.

For the electric field, the detailed calculation gives:

${\displaystyle -{\boldsymbol {\operatorname {grad} }}\!\left(V\right)={\frac {1}{4\pi \varepsilon _{0}}}\,\iiint \left({\frac {[\rho ]\,\mathbf {n} }{r^{2}}}+{\frac {[{\dot {\rho }}]\,\mathbf {n} }{c\,r}}\right)\,dV'}$

${\displaystyle -{\frac {\partial \mathbf {A} }{\partial t}}=-{\frac {1}{4\pi \varepsilon _{0}c^{2}}}\,\iiint \left({\frac {\left[{\dot {\mathbf {J} }}\right]}{r}}\right)\,dV'}$

Distinction between near field and radiated field

The terms of the Panofsky–Phillips equations can be separated into two categories: The "near field" terms have a ${\displaystyle 1/r^{2}}$ dependence, and are significant only in the immediate vicinity of the sources. The "far field" terms have a ${\displaystyle 1/r}$ dependence. Due to their slower decrease with distance, they dominate as one moves away from the sources (see near field and far field).

${\displaystyle \definecolor {vert}{rgb}{0,0.6,0}\mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\,\displaystyle \iiint \left({\color {vert}{\frac {[\rho ]\mathbf {n} }{r^{2}}}+{\frac {\left(\left[\mathbf {J} \right]\cdot \mathbf {n} \right)\mathbf {n} +\left(\left[\mathbf {J} \right]\times \mathbf {n} \right)\times \mathbf {n} }{c\,r^{2}}}}\mathbin {\color {red}{+}} {\color {red}{\frac {\left(\left[{\dot {\mathbf {J} }}\right]\times \mathbf {n} \right)\times \mathbf {n} }{c^{2}r}}}\right)\,dV'}$

${\displaystyle \definecolor {vert}{rgb}{0,0.6,0}\mathbf {B} ={\frac {1}{4\pi \varepsilon _{0}}}\,\displaystyle \iiint \left({\color {vert}{\frac {\left[\mathbf {J} \right]\times \mathbf {n} }{c^{2}r^{2}}}}\mathbin {\color {red}{+}} {\color {red}{\frac {\left[{\dot {\mathbf {J} }}\right]\times \mathbf {n} }{c^{3}r}}}\right)\,dV'}$

• In green are the near fields (decaying as 1/r²).
• In red are the radiated fields (decaying as 1/r).

In the far-field region, near fields can be neglected and the expressions of the emitted fields reduce to:

${\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}c^{2}}}\,\displaystyle \iiint \left({\frac {\left(\left[{\dot {\mathbf {J} }}\right]\times \mathbf {n} \right)\times \mathbf {n} }{r}}\right)\,dV'}$

${\displaystyle \mathbf {B} ={\frac {1}{4\pi \varepsilon _{0}c^{3}}}\,\displaystyle \iiint \left({\frac {\left[{\dot {\mathbf {J} }}\right]\times \mathbf {n} }{r}}\right)\,dV'}$

One therefore has ${\displaystyle \mathbf {E} \perp \mathbf {B} \perp \mathbf {n} }$ , with E and B related by:

The Panofsky–Phillips equation allows one to directly distinguish between near fields and radiated fields for the electric field—something that is not apparent in Jefimenko’s formulation. In Jefimenko's electric field equation, both the second and third terms appear to contribute to the far field because they scale with ${\displaystyle 1/r}$. However, this is illusory; the longitudinal component of the third term cancels the ${\displaystyle 1/r}$ dependence of the second term, leaving only the transverse components to propagate as radiation. Despite this complication, Jefimenko’s electric field equation is more compact and sometimes easier to use for calculations. One of the exceptions is the case of electric dipole radiation.