Liénard–Wiechert potential

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Liénard-Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential. 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^[1] and continued into the early 1900s.

The Liénard-Wiechert potentials can be generalized according to gauge theory.

The Liénard-Wiechert potentials are the initial terms in an expansion of retarded potential solutions of the nonhomogeneous wave equations (the retarded Lorentz-gauge potentials) in terms of co-moving moments of localized, time-dependent, moving charges and currents; and the following terms give explicit expressions for retarded potential solutions related to moving dipoles and quadrupoles.^[2]

1 Implications
2 Universal Speed Limit
3 Equations
- 3.1 Definition of Liénard-Wiechert potentials
- 3.2 Corresponding values of electric and magnetic fields
4 Derivation
- 4.1 Retarded potential solutions
5 See also
6 References

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, independent moving particle, but breaks down at the quantum level.

Quantum mechanics sets important constraints on the ability of a particle to emit radiation. The classical formulation, as laboriously described by these equations, expressly violates experimentally observed phenomena. For example, an electron around an atom does not emit radiation in the pattern predicted by these classical equations. Instead, it is governed by quantized principles regarding its energy state. In the later decades of the twentieth century, quantum electrodynamics helped bring together the radiative behavior with the quantum constraints.

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 $t r$ 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, $t r$ . 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 can be calculated as:

$t_r=t-\frac{R(t_r)}{c}$

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 field term, and depends only on the distance to the moving charge; 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. 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 position of the object, if its velocity has not changed over the retarded time delay.

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 to 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.

Equations

Definition of Liénard-Wiechert potentials

The Liénard-Wiechert potentials $\varphi$ (scalar potential field) and $\mathbf{A}$ (vector potential field) are for a source point charge $q$ at position $\mathbf{r}_s$ traveling with velocity $\mathbf{v}_s$ :

$\varphi(\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \left(\frac{q}{(1 - \mathbf{n} \cdot \boldsymbol{\beta})|\mathbf{r} - \mathbf{r}_s|} \right)_{t_r}$

and

$\mathbf{A}(\mathbf{r},t) = \frac{\mu_0c}{4 \pi} \left(\frac{q \boldsymbol{\beta}}{(1 - \mathbf{n} \cdot \boldsymbol{\beta})|\mathbf{r} - \mathbf{r}_s|} \right)_{t_r} = \frac{\boldsymbol{\beta}(t_r)}{c} \varphi(\mathbf{r}, t)$

where $\boldsymbol{\beta}(t) = \frac{\mathbf{v}_s(t)}{c}$ .

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 non trivial and requires a number of steps. The electric and magnetic fields are (in non-covariant form):

$\mathbf{E}(\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \left(\frac{q(\mathbf{n} - \boldsymbol{\beta})}{\gamma^2 (1 - \mathbf{n} \cdot \boldsymbol{\beta})^3 |\mathbf{r} - \mathbf{r}_s|^2} %2B \frac{q \mathbf{n} \times \big((\mathbf{n} - \boldsymbol{\beta}) \times \dot{\boldsymbol{\beta}}\big)}{c(1 - \mathbf{n} \cdot \boldsymbol{\beta})^3 |\mathbf{r} - \mathbf{r}_s|} \right)_{t_r}$

and

$\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \left(\frac{q c(\boldsymbol{\beta} \times \mathbf{n})}{\gamma^2 (1-\mathbf{n} \cdot \boldsymbol{\beta})^3 |\mathbf{r} - \mathbf{r}_s|^2} %2B \frac{q \mathbf{n} \times \Big(\mathbf{n} \times \big((\mathbf{n} - \boldsymbol{\beta}) \times \dot{\boldsymbol{\beta}}\big) \Big)}{(1 - \mathbf{n} \cdot \boldsymbol{\beta})^3 |\mathbf{r} - \mathbf{r}_s|} \right)_{t_r} = \frac{\mathbf{n}(t_r)}{c} \times \mathbf{E}(\mathbf{r}, t)$

where $\boldsymbol{\beta}(t) = \frac{\mathbf{v}_s(t)}{c}$ , $\mathbf{n}(t) = \frac{\mathbf{r} - \mathbf{r}_s(t)}{|\mathbf{r} - \mathbf{r}_s(t)|}$ and $\gamma(t) = \frac{1}{\sqrt{1 - |\boldsymbol{\beta}(t)|^2}}$ (the Lorentz factor).

Note that the $\mathbf{n} - \boldsymbol{\beta}$ part of the first term updates the direction of the field toward the instantantaneous position of the charge, if it continues to move with constant velocity $\boldsymbol{c}{\beta}$ .

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. 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

Retarded potential solutions

In the case that there are no boundaries surrounding the sources, the retarded solutions for the scalar and vector potentials (CGS 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 (see Nonhomogeneous electromagnetic wave equation)

$\varphi (\mathbf{r}, t) = \int { { \delta \left ( t' %2B { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } \rho (\mathbf{r}', t') d^3r' dt'$

and

$\mathbf{A} (\mathbf{r}, t) = \int { { \delta \left ( t' %2B { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } { \mathbf{J} (\mathbf{r}', t')\over c} d^3r' dt'$

where

${ \delta \left ( t' %2B { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) }$

is a Dirac delta function. For a moving point charge at $\mathbf{r}_0(t')$ traveling with velocity $\mathbf{v}_0(t')$ , the current and charge densities are

$\mathbf{J} (\mathbf{r}', t') = e \mathbf{v}_0(t') \delta \left ( \mathbf{r}' - \mathbf{r}_0(t') \right )$

$\rho (\mathbf{r}', t') = e \delta \left ( \mathbf{r}' - \mathbf{r}_0 (t') \right )$

and the retarded potential solutions simplify to the Liénard-Wiechert potentials.

References

^ http://verplant.org/history-geophysics/Wiechert.htm
^ Ribarič, M., and L. Šušteršič, Expansion in terms of time-dependent, moving charges and currents, SIAM J. Appl. Math. 55, 593-624.

Griffiths, David. Introduction to Electrodynamics. Prentice Hall, 1999. ISBN 0-13-805326-X.