利用者:Glayhours/sandbox/リエナール・ヴィーヘルト・ポテンシャル

• Liénard–Wiechert potential の翻訳の残り。

Universal Speed Limit

与えられた位置 r と時刻 t において粒子に加わる力は、その源となる粒子たちの遅延時間t_r における位置に依存し、 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 (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 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.

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 $v_{M}<c$, 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

$f'(t')={\frac {\mathbf {r} -\mathbf {r} _{s}(t_{r})}{|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}}\cdot (-\mathbf {v} _{s}(t'))+c\geq c-\left|{\frac {\mathbf {r} -\mathbf {r} _{s}(t_{r})}{|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}}\right|\,|\mathbf {v} _{s}(t')|=c-|\mathbf {v} _{s}(t')|\geq c-v_{M}>0$

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 $v<c$ 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

For a given point $(\mathbf {r} ,t)$ and trajectory of the point source $\mathbf {r} _{s}(t')$, there is at most one value of the retarded time $t_{r}$, i.e., one value $t_{r}$ such that $|\mathbf {r} -\mathbf {r} _{s}(t_{r})|=c(t-t_{r})$. To see this, suppose there are two retarded times $t_{1}$ and $t_{2}$, with $t_{1}\leq t_{2}$. Then, $|\mathbf {r} -\mathbf {r} _{s}(t_{1})|=c(t-t_{1})$ and $|\mathbf {r} -\mathbf {r} _{s}(t_{2})|=c(t-t_{2})$. Subtracting gives $c(t_{2}-t_{1})=|\mathbf {r} -\mathbf {r} _{s}(t_{1})|-|\mathbf {r} -\mathbf {r} _{s}(t_{2})|\leq |\mathbf {r} _{s}(t_{2})-\mathbf {r} _{s}(t_{1})|$ by the triangle inequality. Unless $t_{2}=t_{1}$, this then implies that the average velocity of the charge between $t_{1}$ and $t_{2}$ is $|\mathbf {r} _{s}(t_{2})-\mathbf {r} _{s}(t_{1})|/(t_{2}-t_{1})\geq c$, which is impossible. The intuitive interpretation is that we can only ever "see" the point source at one location/time at once unless it travels at least at the speed of light to another location. As the source moves forward in time, the cross section of its light cone at present time contracts faster than the source can approach, so it can never intersect the point $\mathbf {r}$ again.

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

Lorentz 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

${\begin{aligned}{\frac {d\varphi }{dt}}=&-{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{2}}}{\frac {d}{dt}}\left[(|\mathbf {r} -\mathbf {r_{s}} |(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})\right]\\=&-{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{2}}}{\frac {d}{dt}}\left[|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}\right]\\=&-{\frac {qc}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}+{\beta _{s}}^{2}-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right]\end{aligned}}$

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {A} =&-{\frac {q}{4\pi \epsilon _{0}c}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{2}}}{\big (}{\boldsymbol {\nabla }}\left[\left(|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}\right)\right]\cdot {\boldsymbol {\beta }}_{s}-\left[\left(|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}\right)\right]{\boldsymbol {\nabla }}\cdot {\boldsymbol {\beta }}_{s}{\big )}\\=&-{\frac {q}{4\pi \epsilon _{0}c}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\cdot \\&\left[(\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})-{\beta }_{s}^{2}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})-{\beta }_{s}^{2}\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}+\left((\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right)(\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})+{\big (}|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}{\big )}(\mathbf {n} \cdot {\dot {\boldsymbol {\beta }}}_{s}/c)\right]\\=&{\frac {q}{4\pi \epsilon _{0}c}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[\beta _{s}^{2}-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right]\end{aligned}}$

These show that the Lorentz gauge is satisfied, namely that ${\frac {d\varphi }{dt}}+c^{2}{\boldsymbol {\nabla }}\cdot \mathbf {A} =0$.

Similarly one calculates:

${\boldsymbol {\nabla }}\varphi =-{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[\mathbf {n} \left(1-{\beta _{s}}^{2}+(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right)-{\boldsymbol {\beta }}_{s}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})\right]$

${\frac {d\mathbf {A} }{dt}}={\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\left[{\boldsymbol {\beta }}_{s}\left(\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}-{\beta _{s}}^{2}+(\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right)+|\mathbf {r} -\mathbf {r_{s}} |{\dot {\boldsymbol {\beta }}}_{s}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})/c\right]$

By noting that for any vectors $\mathbf {u}$, $\mathbf {v}$, $\mathbf {w}$:

$\mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\mathbf {w}$

The expression for the electric field mentioned above becomes

${\begin{aligned}\mathbf {E} (\mathbf {r} ,t)=&{\frac {q}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {r} -\mathbf {r} _{s}|^{2}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})^{3}}}\cdot \\&\left[\left(\mathbf {n} -{\boldsymbol {\beta }}_{s}\right)(1-{\beta _{s}}^{2})+|\mathbf {r} -\mathbf {r} _{s}|(\mathbf {n} \cdot {\dot {\boldsymbol {\beta }}}_{s}/c)(\mathbf {n} -{\boldsymbol {\beta }}_{s})-|\mathbf {r} -\mathbf {r} _{s}|{\big (}\mathbf {n} \cdot (\mathbf {n} -{\boldsymbol {\beta }}_{s}){\big )}{\dot {\boldsymbol {\beta }}}_{s}/c\right]\end{aligned}}$

which is easily seen to be equal to $-{\boldsymbol {\nabla }}\varphi -{\frac {d\mathbf {A} }{dt}}$

Similarly ${\boldsymbol {\nabla }}\times \mathbf {A}$ gives the expression of the magnetic field mentioned above:

${\begin{aligned}{\mathbf {B} }=&{\boldsymbol {\nabla }}\times \mathbf {A} =-{\frac {q}{4\pi \epsilon _{0}c}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{2}}}{\big (}{\boldsymbol {\nabla }}\left[\left(|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}\right)\right]\times {\boldsymbol {\beta }}_{s}-\left[\left(|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}\right)\right]{\boldsymbol {\nabla }}\times {\boldsymbol {\beta }}_{s}{\big )}\\=&-{\frac {q}{4\pi \epsilon _{0}c}}{\frac {1}{|\mathbf {r} -\mathbf {r_{s}} |^{2}\left(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s}\right)^{3}}}\cdot \\&\left[(\mathbf {n} \times {\boldsymbol {\beta }}_{s})-({\boldsymbol {\beta }}_{s}\times {\boldsymbol {\beta }}_{s})(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})-{\beta }_{s}^{2}\mathbf {n} \times {\boldsymbol {\beta }}_{s}+\left((\mathbf {r} -\mathbf {r_{s}} )\cdot {\dot {\boldsymbol {\beta }}}_{s}/c\right)(\mathbf {n} \times {\boldsymbol {\beta }}_{s})+{\big (}|\mathbf {r} -\mathbf {r_{s}} |-(\mathbf {r} -\mathbf {r_{s}} )\cdot {\boldsymbol {\beta }}_{s}{\big )}(\mathbf {n} \times {\dot {\boldsymbol {\beta }}}_{s}/c)\right]\\=&-{\frac {q}{4\pi \epsilon _{0}c}}{\frac {1}{|\mathbf {r} -\mathbf {r} _{s}|^{2}(1-\mathbf {n} \cdot {\boldsymbol {\beta }}_{s})^{3}}}\cdot \\&\left[\left(\mathbf {n} \times {\boldsymbol {\beta }}_{s}\right)(1-{\beta _{s}}^{2})+|\mathbf {r} -\mathbf {r} _{s}|(\mathbf {n} \cdot {\dot {\boldsymbol {\beta }}}_{s}/c)(\mathbf {n} \times {\boldsymbol {\beta }}_{s})+|\mathbf {r} -\mathbf {r} _{s}|{\big (}\mathbf {n} \cdot (\mathbf {n} -{\boldsymbol {\beta }}_{s}){\big )}\mathbf {n} \times {\dot {\boldsymbol {\beta }}}_{s}/c\right]={\frac {\mathbf {n} }{c}}\times \mathbf {E} \end{aligned}}$