# EM wave/radiation from which equations?

• xxxyyy
In summary, it is said that radiation comes from the last term, the one with acceleration in it. This formula is a particular case of the Jefimenko equations, the ones derived from Lienard-Wiechert potentials. These potentials are particular solutions of the inhomogeneus wave equations for the potentials (in Lorenz gauge). I should add the general solution of the homogeneus equestions, i.e. wave solutions, to get the true general solution. Wave solutions for the potentials translate into wave solutions for the fields.
xxxyyy
Hi there,
in Feynmann's lectures, (Vol 2 eq (21.1) Heaviside-Feynman forumla)
https://www.feynmanlectures.caltech.edu/II_21.html

it is said that radiation comes from the last term, the one with acceleration in it. This formula is a particular case of the Jefimenko equations, the ones derived from Lienard-Wiechert potentials.
These potentials are particular solutions of the inhomogeneus wave equations for the potentials (in Lorenz gauge). I should add the general solution of the homogeneus equestions, i.e. wave solutions, to get the true general solution.

Does wave solutions for the potentials translate into wave solutions for the fields?

And where does exactly the wave phenomenon come from? from the homogeneus solution or from the second time derivative Feynman was talking about in his (21.1) formula?
Thank you!

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xxxyyy said:
Does wave solutions for the potentials translate into wave solutions for the fields?
Absolutely yes. If you have solutions ##V,\vec{A}## for the scalar potential and vector potential that satisfies the homogeneous or inhomogeneous wave equation, then the fields defined by $$\vec{E}=-\nabla V-\frac{\partial \vec{A}}{\partial t}$$ and $$\vec{B}=\nabla\times\vec{A}$$ satisfy the wave equation as well. To prove it is a not so hard vector calculus excercise, you will utilize Maxwell's equation (in differential form) and some vector calculus identities and the Lorentz gauge condition.

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Set ##\phi = 0## such that ##\mathbf{E} = - \dfrac{\partial \mathbf{A}}{\partial t}## and ##\mathbf{B} = \nabla \times \mathbf{A}##. Set also ##\nabla \cdot \mathbf{A} = 0## then from Maxwell\begin{align*}
\Delta \mathbf{A} - \frac{\partial^2 \mathbf{A}}{\partial t^2} &= 0 \ \ \ (1) \\
\end{align*}Take the curl of ##(1)##,\begin{align*}
\Delta ( \nabla \times \mathbf{A} )- \frac{\partial^2}{\partial t^2}\left( \nabla \times \mathbf{A} \right) &= 0 \\

\Delta \mathbf{B} - \frac{\partial^2 \mathbf{B}}{\partial t^2} &= 0

\end{align*}Instead take the time derivative of ##(1)##,\begin{align*}
\Delta \left( \frac{\partial \mathbf{A}}{\partial t}\right) - \frac{\partial^2}{\partial t^2} \left( \frac{\partial \mathbf{A}}{\partial t} \right) &= 0\\

\Delta \mathbf{E} - \frac{\partial^2 \mathbf{E}}{\partial t^2} &= 0

\end{align*}The wave equations holds on ##\mathbf{E}## and ##\mathbf{B}##. Let ##\xi## be any component of either of these vectors, i.e. ##\Delta \xi - \dfrac{\partial^2 \xi}{\partial t^2} = 0## and let ##\xi(\boldsymbol{x},t) = f(p)## where ##p = k_1 x + k_2 y + k_3 z + k_4 t##. Then ##\dfrac{\partial^2 \xi}{\partial x_i^2} = k_i^2 \dfrac{d^2 f}{dp^2}## hence ##k_1^2 + k_2^2 + k_3^2 - k_4^2 = 0##. You may simply let ##k_4 = \pm 1## so that ##\hat{\mathbf{k}} = (k_1, k_2, k_3)## form the components of a unit vector. Then ##p_{\pm} = \hat{\mathbf{k}} \cdot \boldsymbol{x} \pm t## and for each component the general solution is ##\xi(\boldsymbol{x}, t) = f(p_+) + g(p_{-})##

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Delta2
Does wave phenomena arise from the third term in Feynman equation (21.1)?
At the end of 21-3 paragraph, Feyman boldy says that the complete theory of (classical) EM is in that box... but why doesn't he include the solutions of the homogeneous equations (the wave solutions)? Is really everything, that's a wave, in that third term of (21.1)?
I have here a (very respectable) book that includes them and it even says that, for bounded charge distributions, the Lienard-Wiechert potential are negligible far away from the sources, because they fall of faster then 1/r.
I'm confused.

## 1. What is an EM wave/radiation?

An EM wave/radiation is a type of energy that is propagated through space in the form of electric and magnetic fields. It is also known as electromagnetic radiation and is responsible for many natural phenomena such as light, radio waves, and X-rays.

## 2. What are the equations that describe EM waves/radiation?

The equations that describe EM waves/radiation are Maxwell's equations. They are a set of four equations that describe the behavior of electric and magnetic fields and their relationship to each other. These equations were developed by James Clerk Maxwell in the 19th century and are considered one of the most important contributions to the field of electromagnetism.

## 3. How are the electric and magnetic fields related in an EM wave/radiation?

In an EM wave/radiation, the electric and magnetic fields are perpendicular to each other and also to the direction in which the wave is propagating. This means that as the electric field changes, the magnetic field also changes, and vice versa. This relationship is described by Maxwell's equations.

## 4. What is the speed of an EM wave/radiation?

The speed of an EM wave/radiation is a constant value, known as the speed of light. In a vacuum, this speed is approximately 299,792,458 meters per second. This value is denoted by the letter "c" in equations and is an important constant in physics.

## 5. How are EM waves/radiation produced?

EM waves/radiation can be produced in a variety of ways, such as by accelerating charged particles, through changes in electric and magnetic fields, and through nuclear reactions. Some common sources of EM waves include light bulbs, radio antennas, and X-ray machines.

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