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  1. Definition of Electromagnetic field?

  2. What is the Electromagnetic field at a given point in space due to a point charge being accelerated non-uniformly?

  3. Is there a single equation that can give me the electromagnetic force experienced by a body of unit charge (and should I include unit velocity/momentum/acceleration)?

  4. And how do I determine from which direction an electromagnetic wave is coming from?

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  • $\begingroup$ Electic field is calculated using this equation: $E = \frac{q_1q_2}{4\pi\epsilon_0 r^2}$ where $q_1$ and $q_2$ are charges, $\epsilon_0$ is Vacuum Permittivity and $r$ is distance between two charges and force is calculated using: $F=q(E+v\times B)$ (Lorentz Force) where $v$ is velocity, $E$ is electric field, $q$ is charge and $B$ is the magnetic field $\endgroup$ – Gigi Butbaia Jun 8 '14 at 10:35
  • $\begingroup$ @GigiButbaia, in your comment above, v is the velocity of what? The source or the one which is experiencing the force? $\endgroup$ – The Light Spark Jun 8 '14 at 11:28
  • $\begingroup$ @GigiButbaia Wouldn't you have to take into account radiation? $\endgroup$ – jinawee Jun 8 '14 at 13:13
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    $\begingroup$ You need the retarded potential for that. Calculate the charge and current density of your moving charge (with a $\delta$-distibution) and that will give you $\phi$ and $\vec A$. From that, you can apply the gradient and curl to get $E$ and $B$. $\endgroup$ – Martin Ueding Jul 19 '14 at 15:03
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The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of the magnetic vector potential and the scalar electric 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, but note that this is not obvious just by looking at the equations for the potentials. The wikipedia article does however also give explicit formulae for the fields, and these do contain terms in the acceleration of the charge, which is what gives rise to radiation.

These potentials generalize the standard formulae for the fields due to stationary charges. The field at a point at a given time depends on the position of the source particle at an earlier time, the retarded time, due to the finite speed, $c$, at which electromagnetic information travels. This is exactly what you would expect naively.

The electromagnetic force experienced by a body of given charge and velocity can then, of course, be calculated from the standard formula $F=q(E+v×B)$ as mentioned by Pu Zhang.

In other words, Light Spark, you do not have to solve Maxwell's equations if you know the movements of all the charges - this has been done for you.

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http://www.feynmanlectures.caltech.edu/II_21.html

The formula you are looking for is (21.1). The whole chapter linked contains a derivation (that omits some mathematical details) of the equation from Maxwell Equations.

You might also want to give a look at http://www.feynmanlectures.caltech.edu/II_26.html, that explains how the fields can be obtained by the retarded potentials.

Finally, if you want to see some applications of (21.1) and its links with electromagnetic radiation, you can start to read from http://www.feynmanlectures.caltech.edu/I_28.html .

Note that (21.1) is not a “definition” of the electromagnetic field at a point; the electromagnetic field is operatively defined via the law of force $F=q(E+v\times B)$. Instead, it is a theorem that follows from Maxwell's equations, that are the fundamental laws. Clearly, since it provides the fields for a particle with arbitrary motion, it also provides the fields for an arbitrary distribution of charges, so it can itself be regarded as a fundamental law.

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  • $\begingroup$ In all its splendor: $$\mathbf E = \dfrac{q}{4\pi \varepsilon _0} (\dfrac{\mathbf e '_r}{r'^2}+\dfrac{r'}{c}\dfrac{\text d}{\text d t}(\dfrac{\mathbf e '_r}{r'^2})+\dfrac{1}{c^2}\dfrac{\text d ^2}{\text d t^2}(\mathbf e ' _r))$$ $\endgroup$ – pppqqq Aug 25 '14 at 21:27
  • $\begingroup$ How come we both saw this question from June 8 just a few minutes apart? $\endgroup$ – akrasia Aug 25 '14 at 21:47
  • $\begingroup$ @akrasia I think that old questions without an accepted answer automatically pop-up after some time ! $\endgroup$ – pppqqq Aug 25 '14 at 21:56
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Your questions can be all answered by Maxwell's equations. Electric current and charge are the sources of electromagnetic fields. (These two are actually related to each other, also implied by Maxwell's equations.) Once you have the source, electromagnetic fields can be found by solving the equations. It doesn't matter how the charges are moving.

As for the force, $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ consists of contributions from both electric and magnetic fields.

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