Liénard–Wiechert potential

{{#invoke: Sidebar | collapsible }} Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Built directly from Maxwell's equations, these potentials describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials.

These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900 and continued into the early 1900s. It has been shown in 2008  that the formulas found have to be reviewedTemplate:What (revision is not mentioned in this page) to be totally correct in the case of accelerated charges.

Implications

The study of classical electrodynamics was instrumental in Einstein's development of the theory of relativity. Analysis of the motion and propagation of electromagnetic waves led to the special relativity description of space and time. The Liénard–Wiechert formulation is an important launchpad into more complex analysis of relativistic moving particles.

The Liénard–Wiechert description is accurate for a large (ie, not quantum mechanical)Template:Why, independent (ie, free of external influence) Template:Clarify moving particle.The Liénard–Wiechert formulation provides always two sets of solutions: Advanced fields are absorbed by the charges and retarded field are emitted. Schwarzschild and Fokker considered the advanced field of a system of moving charges, and the retarded field of a system of changes having the same geometry and opposite charges. Linearity of Maxwell's equations in vacuum allows to add both systems, so that charges disappear: This trick allows Maxwell's equations become linear in matter. Multiplying electric parameters of both problems by arbitrary real constants produces a coherent interaction of light with matter which generalizes Einstein's theory (A. Einstein, “Zur Quantentheorie der Strahlung.” Phys. Z. 18 121-128, 1917) which is now considered as founding theory of lasers: it is not necessary to study a large set of identical molecules to get coherent amplification in the mode obtained by arbitrary multiplications of advanced and retarded fields. To compute energy, it is necessary to use the absolute fields which includes zero point field. Else, an error appears, for instance in photon counting.

It is important to take into account the zero point field discovered by Planck (M. Planck, Deutsche Physikalische Gesellschaft, Vol. 13, 1911, pp. 138-175.). It replaces Einstein's "A" coefficient and explains that classical electron is stable on Rydberg's classical orbits. More, introducing the fluctuations of zero point field produces Willis E. Lamb's correction of levels of H atom.

Quantum electrodynamics helped bring together the radiative behavior with the quantum constraints. It introduces quantization of normal modes of electromagnetic field in assumed perfect optical resonators.

Universal Speed Limit

The force on a particle at a given location r and time t depends in a complicated way on the position of the source particles at an earlier time tr due to the finite speed, c, at which electromagnetic information travels. A particle on Earth 'sees' a charged particle accelerate on the Moon as this acceleration happened 1.5 seconds ago, and a charged particle's acceleration on the Sun as happened 500 seconds ago. This earlier time in which an event happens such that a particle at location r 'sees' this event at a later time t is called the retarded time, tr. The retarded time varies with position; for example the retarded time at the Moon is 1.5 seconds before the current time and the retarded time on the Sun is 500 s before the current time. The retarded time tr=tr(r,t) is defined implicitly by

$t_{r}=t-{\frac {R(t_{r})}{c}}$ where $R(t_{r})$ is the distance of the particle from the source at the retarded time. Only electromagnetic wave effects depend fully on the retarded time.

A novel feature in the Liénard–Wiechert potential is seen in the breakup of its terms into two types of field terms (see below), only one of which depends fully on the retarded time. The first of these is the static electric (or magnetic) field term that depends only on the distance to the moving charge, and does not depend on the retarded time at all, if the velocity of the source is constant. The other term is dynamic, in that it requires that the moving charge be accelerating with a component perpendicular to the line connecting the charge and the observer and does not appear unless the source changes velocity. This second term is connected with electromagnetic radiation.

The first term describes near field effects from the charge, and its direction in space is updated with a term that corrects for any constant-velocity motion of the charge on its distant static field, so that the distant static field appears at distance from the charge, with no aberration of light or light-time correction. This term, which corrects for time-retardation delays in the direction of the static field, is required by Lorentz invariance. A charge moving with a constant velocity must appear to a distant observer in exactly the same way as a static charge appears to a moving observer, and in the latter case, the direction of the static field must change instantaneously, with no time-delay. Thus, static fields (the first term) point exactly at the true instantaneous (non-retarded) position of the charged object if its velocity has not changed over the retarded time delay. This is true over any distance separating objects.

The second term, however, which contains information about the acceleration and other unique behavior of the charge that cannot be removed by changing the Lorentz frame (inertial reference frame of the observer), is fully dependent for direction on the time-retarded position of the source. Thus, electromagnetic radiation (described by the second term) always appears to come from the direction of the position of the emitting charge at the retarded time. Only this second term describes information transfer about the behavior of the charge, which transfer occurs (radiates from the charge) at the speed of light. At "far" distances (longer than several wavelengths of radiation), the 1/R dependence of this term makes electromagnetic field effects (the value of this field term) more powerful than "static" field effects, which are described by the 1/R2 potential of the first (static) term and thus decay more rapidly with distance from the charge.

Existence and uniqueness of the retarded time

Existence

The retarded time is not guaranteed to exist in general. For example, if, in a given frame of reference, an electron has just been created, then at this very moment another electron does not yet feel its electromagnetic force at all. However, under certain conditions, there always exists a retarded time. For example, if the source charge has existed for an unlimited amount of time, during which it has always travelled at a speed not exceeding $v_{M} , then there exists a valid retarded time $t_{r}$ . To see this, consider the function $f(t')=|{\mathbf {r} }-{\mathbf {r} }_{s}(t')|-c(t-t')$ . At the present time, $t'=t$ , we have $f(t')=|\mathbf {r} -\mathbf {r} _{s}(t')|-c(t-t')=|\mathbf {r} -\mathbf {r} _{s}(t')|\geq 0$ . The derivative $f'(t')$ is given by

By the mean value theorem, $f(t-\Delta t)\leq f(t)-f'(t)\Delta t\leq f(t)-(c-v_{M})\Delta t$ . By making $\Delta t$ sufficiently large, we can force this to be negative, i.e., at some point in the past, $f(t')<0$ . By the intermediate value theorem, there exists an intermediate $t_{r}$ with $f(t_{r})=0$ , the defining equation of the retarded time. Intuitively, as the source charge moves back in time, the cross section of its light cone at present time expands faster than it can recede, so eventually it must reach the point ${\mathbf {r} }$ . Note that this is not necessarily true if the source charge's speed is allowed to be arbitrarily close to $c$ , i.e., if for any given speed $v there was some time in the past when the charge was moving at this speed. In this case the cross section of the light cone at present time approaches the point ${\mathbf {r} }$ as we travel back in time but does not necessarily ever reach it.

Uniqueness

We conclude that, under certain conditions, the retarded time exists and is unique.

Equations

Definition of Liénard–Wiechert potentials

$\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q}{(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }}_{s})|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\right)_{t_{r}}$ and

${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }}_{s})|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\right)_{t_{r}}={\frac {{\boldsymbol {\beta }}_{s}(t_{r})}{c}}\varphi ({\mathbf {r} },t)$ Corresponding values of electric and magnetic fields

We can calculate the electric and magnetic fields directly from the potentials using the definitions:

${\mathbf {E} }=-\nabla \varphi -{\dfrac {\partial {\mathbf {A} }}{\partial t}}$ and ${\mathbf {B} }=\nabla \times {\mathbf {A} }$ The calculation is nontrivial and requires a number of steps. The electric and magnetic fields are (in non-covariant form):

${\mathbf {E} }({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q({\mathbf {n} }-{\boldsymbol {\beta }})}{\gamma ^{2}(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }})^{3}|{\mathbf {r} }-{\mathbf {r} }_{s}|^{2}}}+{\frac {q{\mathbf {n} }\times {\big (}({\mathbf {n} }-{\boldsymbol {\beta }})\times {\dot {\boldsymbol {\beta }}}{\big )}}{c(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }})^{3}|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\right)_{t_{r}}$ and

${\mathbf {B} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\left({\frac {qc({\boldsymbol {\beta }}\times {\mathbf {n} })}{\gamma ^{2}(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }})^{3}|{\mathbf {r} }-{\mathbf {r} }_{s}|^{2}}}+{\frac {q{\mathbf {n} }\times {\Big (}{\mathbf {n} }\times {\big (}({\mathbf {n} }-{\boldsymbol {\beta }})\times {\dot {\boldsymbol {\beta }}}{\big )}{\Big )}}{(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }})^{3}|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\right)_{t_{r}}={\frac {{\mathbf {n} }(t_{r})}{c}}\times {\mathbf {E} }({\mathbf {r} },t)$ Note that the ${\mathbf {n} }-{\boldsymbol {\beta }}$ part of the first term updates the direction of the field toward the instantaneous position of the charge, if it continues to move with constant velocity $c{\boldsymbol {\beta }}$ . This term is connected with the "static" part of the electromagnetic field of the charge.

The second term, which is connected with electromagnetic radiation by the moving charge, requires charge acceleration ${\dot {\boldsymbol {\beta }}}$ and if this is zero, the value of this term is zero, and the charge does not radiate (emit electromagnetic radiation). This term requires additionally that a component of the charge acceleration be in a direction transverse to the line which connects the charge $q$ and the observer of the field ${\mathbf {E} }({\mathbf {r} },t)$ . The direction of the field associated with this radiative term is toward the fully time-retarded position of the charge (i.e. where the charge was when it was accelerated).

Derivation

In the case that there are no boundaries surrounding the sources, the retarded solutions for the scalar and vector potentials (SI units) of the nonhomogeneous wave equations with sources given by the charge and current densities $\rho ({\mathbf {r} },t)$ and ${\mathbf {J} }({\mathbf {r} },t)$ are in the Lorenz gauge (see Nonhomogeneous electromagnetic wave equation)

$\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho ({\mathbf {r} }',t_{r}')}{|{\mathbf {r} }-{\mathbf {r} }'|}}d^{3}{\mathbf {r} }'$ and

${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\int {\frac {{\mathbf {J} }({\mathbf {r} }',t_{r}')}{|{\mathbf {r} }-{\mathbf {r} }'|}}d^{3}{\mathbf {r} }'$ For a moving point charge whose trajectory is given as a function of time by ${\mathbf {r} }_{s}(t')$ , the charge and current densities are as follows:

$\rho ({\mathbf {r} }',t')=q\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t'))$ ${\mathbf {J} }({\mathbf {r} }',t')=q{\mathbf {v} }_{s}(t')\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t'))$ Substituting into the expressions for the potential gives

$\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {q\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t_{r}'))}{|{\mathbf {r} }-{\mathbf {r} }'|}}d^{3}{\mathbf {r} }'$ ${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\int {\frac {q{\mathbf {v} }_{s}(t_{r}')\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t_{r}'))}{|{\mathbf {r} }-{\mathbf {r} }'|}}d^{3}{\mathbf {r} }'$ These integrals are difficult to evaluate in their present form, so we will rewrite them by replacing $t_{r}'$ with $t'$ and integrating over the delta distribution $\delta (t'-t_{r}')$ :

$\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\iint {\frac {q\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t'))}{|{\mathbf {r} }-{\mathbf {r} }'|}}\delta (t'-t_{r}')\,dt'\,d^{3}{\mathbf {r} }'$ ${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\iint {\frac {q{\mathbf {v} }_{s}(t')\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t'))}{|{\mathbf {r} }-{\mathbf {r} }'|}}\delta (t'-t_{r}')\,dt'\,d^{3}{\mathbf {r} }'$ We exchange the order of integration:

$\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\iint {\frac {\delta (t'-t_{r}')}{|{\mathbf {r} }-{\mathbf {r} }'|}}q\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t'))\,d^{3}{\mathbf {r} }'dt'$ ${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\iint {\frac {\delta (t'-t_{r}')}{|{\mathbf {r} }-{\mathbf {r} }'|}}q{\mathbf {v} }_{s}(t')\delta ^{3}({\mathbf {r'} }-{\mathbf {r} }_{s}(t'))\,d^{3}{\mathbf {r} }'dt'$ $\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\int q{\frac {\delta (t'-t_{r}')}{|{\mathbf {r} }-{\mathbf {r} }_{s}(t')|}}dt'$ ${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\int q{\mathbf {v} }_{s}(t'){\frac {\delta (t'-t_{r}')}{|{\mathbf {r} }-{\mathbf {r} }_{s}(t')|}}\,dt'$ $\delta (f(t'))=\sum _{i}{\frac {\delta (t'-t_{i})}{|f'(t_{i})|}}$ {\begin{aligned}\delta (t'-t_{r}')={\frac {\delta (t'-t_{r})}{{\frac {\partial }{\partial t'}}(t'-t_{r}')|_{t'=t_{r}}}}=&{\frac {\delta (t'-t_{r})}{{\frac {\partial }{\partial t'}}(t'-(t-{\frac {1}{c}}|{\mathbf {r} }-{\mathbf {r} }_{s}(t')|))|_{t'=t_{r}}}}={\frac {\delta (t'-t_{r})}{1+{\frac {1}{c}}({\mathbf {r} }-{\mathbf {r} }_{s}(t'))/|{\mathbf {r} }-{\mathbf {r} }_{s}(t')|\cdot (-{\mathbf {v} }_{s}(t'))|_{t'=t_{r}}}}=\\&={\frac {\delta (t'-t_{r})}{1-{\boldsymbol {\beta }}_{s}\cdot ({\mathbf {r} }-{\mathbf {r} }_{s})/|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\end{aligned}} $\varphi ({\mathbf {r} },t)={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q}{|{\mathbf {r} }-{\mathbf {r} }_{s}|(1-{\boldsymbol {\beta }}_{s}\cdot ({\mathbf {r} }-{\mathbf {r} }_{s})/|{\mathbf {r} }-{\mathbf {r} }_{s}|)}}\right)_{t_{r}}={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q}{(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }}_{s})|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\right)_{t_{r}}$ ${\mathbf {A} }({\mathbf {r} },t)={\frac {\mu _{0}}{4\pi }}\left({\frac {q{\mathbf {v} }}{|{\mathbf {r} }-{\mathbf {r} }_{s}|(1-{\boldsymbol {\beta }}_{s}\cdot ({\mathbf {r} }-{\mathbf {r} }_{s})/|{\mathbf {r} }-{\mathbf {r} }_{s}|)}}\right)_{t_{r}}={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-{\mathbf {n} }\cdot {\boldsymbol {\beta }}_{s})|{\mathbf {r} }-{\mathbf {r} }_{s}|}}\right)_{t_{r}}$ which are the Liénard–Wiechert potentials.

Lorenz gauge, electric and magnetic fields

In order to calculate the derivatives of $\varphi$ and ${\mathbf {A} }$ it is convenient to first compute the derivatives of the retarded time. Taking the derivatives of both sides of its defining equation (remembering that $\mathbf {r_{s}} =\mathbf {r_{s}} (t_{r})$ ):

$t_{r}+{\frac {1}{c}}|\mathbf {r} -\mathbf {r_{s}} |=t$ Differentiating with respect to t,

${\frac {dt_{r}}{dt}}+{\frac {1}{c}}{\frac {dt_{r}}{dt}}{\frac {d|\mathbf {r} -\mathbf {r_{s}} |}{dt_{r}}}=1$ ${\frac {dt_{r}}{dt}}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)=1$ ${\frac {dt_{r}}{dt}}={\frac {1}{\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)}}$ Similarly, Taking the gradient with respect to ${\mathbf {r} }$ gives

${\boldsymbol {\nabla }}t_{r}+{\frac {1}{c}}{\boldsymbol {\nabla }}|\mathbf {r} -\mathbf {r_{s}} |=0$ ${\boldsymbol {\nabla }}t_{r}+{\frac {1}{c}}\left({\boldsymbol {\nabla }}t_{r}{\frac {d|\mathbf {r} -\mathbf {r_{s}} |}{dt_{r}}}+\mathbf {n} \right)=0$ ${\boldsymbol {\nabla }}t_{r}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)=-\mathbf {n} /c$ ${\boldsymbol {\nabla }}t_{r}=-{\frac {\mathbf {n} /c}{\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)}}$ It follows that

${\frac {d|\mathbf {r} -\mathbf {r_{s}} |}{dt}}={\frac {dt_{r}}{dt}}{\frac {d|\mathbf {r} -\mathbf {r_{s}} |}{dt_{r}}}={\frac {-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}c}{\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)}}$ ${\boldsymbol {\nabla }}|\mathbf {r} -\mathbf {r_{s}} |={\boldsymbol {\nabla }}t_{r}{\frac {d|\mathbf {r} -\mathbf {r_{s}} |}{dt_{r}}}+\mathbf {n} ={\frac {\mathbf {n} }{\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)}}$ These can be used in calculating the derivatives of the vector potential and the resulting expressions are

{\begin{aligned}{\frac {d\varphi }{dt}}=&-{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{2}}}{\frac {d}{dt}}\left[(|\mathbf {r} -\mathbf {r_{s}} |(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})\right]\\=&-{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{2}}}{\frac {d}{dt}}\left[|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}\right]\\=&-{\frac {qc}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}+{\beta _{s}}^{2}-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right]\end{aligned}} These show that the Lorenz gauge is satisfied, namely that ${\frac {d\varphi }{dt}}+c^{2}{\boldsymbol {\nabla }}\cdot \mathbf {A} =0$ .

Similarly one calculates:

${\boldsymbol {\nabla }}\varphi =-{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[\mathbf {n} \left(1-{\beta _{s}}^{2}+(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right)-{\boldsymbol {\beta }}_{s}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})\right]$ ${\frac {d\mathbf {A} }{dt}}={\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[{\boldsymbol {\beta }}_{s}\left(\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}-{\beta _{s}}^{2}+(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right)+|\mathbf {r} -\mathbf {r_{s}} |{\dot {\boldsymbol {\beta }}}_{s}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})/c\right]$ $\mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\mathbf {w}$ The expression for the electric field mentioned above becomes

{\begin{aligned}\mathbf {E} (\mathbf {r} ,t)=&{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r} _{s}|^{2}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})^{3}}}\cdot \\&\left[\left(\mathbf {n} -{\boldsymbol {\beta }}_{s}\right)(1-{\beta _{s}}^{2})+|\mathbf {r} -\mathbf {r} _{s}|(\mathbf {n} \cdot {\dot {\boldsymbol {\beta }}}_{s}/c)(\mathbf {n} -{\boldsymbol {\beta }}_{s})-|\mathbf {r} -\mathbf {r} _{s}|{\big (}\mathbf {n} \cdot (\mathbf {n} -{\boldsymbol {\beta }}_{s}){\big )}{\dot {\boldsymbol {\beta }}}_{s}/c\right]\end{aligned}} Similarly ${\boldsymbol {\nabla }}\times \mathbf {A}$ gives the expression of the magnetic field mentioned above: