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I know that an electric field is created by a particle with a charge and that a magnetic field is created by a moving charge but how do they combine to make a electromagnetic field?

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What makes an electromagnetic field? How is it created?

There are two perspectives with two different raw materials. Neither perspective claims that charges create electric fields and moving charges create magnetic fields. Because that's an oversimplification that simply isn't true.

An example solution to Maxwell can be provided if both the electric and magnetic fields are each computed as the electric and magnetic parts of the electromagnetic field given by Jefimenko's equations:

$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2}\; \mathrm{d}^3\vec{r}' -\frac{1}{4\pi\epsilon_0c^2}\int\frac{1}{|\vec r-\vec r'|}\frac{\partial \vec J(\vec r',t_r)}{\partial t}\mathbb{d}^3\vec r'$$ and $$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$ where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$

These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field. Note that both the electric and the magnetic part of the electromagnetic field have parts that depend on the time variation of current. And note that when the charges and current both fail to change in time, then it reduces to a simple story like, electric fields are caused by charges and magnetic fields are caused by currents.

So you could use Jefimenko to show how both fields here and now are determined by charge and current (and their time variations) in the past. The advantage is that the fields now are clearly determined by the charge and current (and their tine variations) in the past. But this only picks out one particular solution to Maxwell out of the many possible solutions.

So let's look at the second perspective. Instructors and textbooks that want to oversimplify sometimes try to claim that a changing magnetic field causes an electric field and that changing electric fields cause magnetic fields. That doesn't even make sense as a thing. You can't have a velocity without already having a position.

To have a time changing magnetic field you need to already have a magnetic field. To have a time changing electric field you need to already have an electric field. So you need to have fields in order for them to change in time. Just like you have to have particles with positions in order to have velocities. But when you have a particle with a certain position you are free to then give it any velocity you want. And this isn't true for electromagnetic fields.

Once you specify the electromagnetic field everywhere then the time variations are fixed by the evolution laws (just like when you specify the position and velocity of a particle the acceleration is fixed by Newton's evolution laws $\vec F=m\vec a$). Technically you need the fields now and the current now, and then evolution is fixed by Maxwell. For instance the magnetic part of the electromagnetic field changes according to:

$$\frac{\partial \vec B}{\partial t}=-\vec\nabla \times \vec E.$$ So the change of the magnetic field here and now is caused by the spatial variation of the electric field now.

And $$\frac{\partial \vec E}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec J+\frac{1}{\mu_0}\vec\nabla \times \vec B \right)$$ can tell you how the change of the electric part of the elwctromagnetic field here and now is caused by the current here and now and the spatial variation of the magnetic field now.

Finally the electromagnetic field is a unified object that can be broken into electric and magnetic parts (if you want) but different observers will break it into different parts. They will also compute different charge densities, different currents, different time rates, and so on. But their predictions about physical actions such as whether a particle goes straight or curves, will be the same.

You can combine the electric field and the magnetic field into a single rank two object, the electromagnetic field, but all the rules I have above apply to how how each of the six components are determined. And it's best to think of them as just parts of the unified electromagnetic field from the get go.

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  • $\begingroup$ I have a query to have a more insight on this. It's true that Jefimenko's equations agree with causality; but they are the solutions to the wave-equation which has a source term in the right hand side. But what about the solutions of the source-free wave-equations - Maxwell's equations in vacuum i.e. regions far, far away from the source? The fields of such equations don't contain any source-term. How to apply the law of causality then? You can't use Jefimenko then. $\endgroup$ – user36790 Jun 15 '16 at 16:04
  • $\begingroup$ @MAFIA36790 They are a particular solution, one solution for every right hand side. Including a zero right hand side which produces a zero solution electromagnetic field. But this isn't the field far from sources (as you claim in your comment). Far from sources you still depend on the sources, you just depend on what they were like long long ago. For instance far from the earth you can pick up radio broadcasts, they will be based on what charges did long ago. But it's still based on charges. $\endgroup$ – Timaeus Jun 16 '16 at 0:27
  • $\begingroup$ "But this isn't the field far from sources " - so are you saying the source-free Maxwell's equations are not for far, far away from the source? Also, take the sinusoidal function as the solution of the source-free equation, $E_0\sin(kx-\omega t)_;$ there is no source term in the solution. How could I then say they depend on charges long, long, ago? I really am not getting this as there is no source term in the solution after all the equations are source-free. $\endgroup$ – user36790 Jun 16 '16 at 3:32
  • $\begingroup$ @MAFIA36790 Maxwell's equations are local so you can ask whether a field is a local solution. But there are many possible solutions to Maxwell given some source. Even many possible solutions when the source is zero. Jefimenko provides just one global solution for one global specification of the sources. Maxwell allows an electromagnetic plane wave in a universe that has no sources never did and never will. Jefimenko doesn't allow that. Jefimenko has fewer solutions. Jefimenko allows nonzeros fields far from sources. They just depend on the sources long ago. I've said this all before $\endgroup$ – Timaeus Jun 16 '16 at 3:36
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    $\begingroup$ @MAFIA36790 If there ever was, is, or will be sources, then the actual global fields are not source free. But in a region far from sources the actual field might locally look similar to a globally source free solution. For instance if your source was a dipole oscillating at a fixed frequency and you take off far away in the right direction it might look (locally) very similar to a plane wave. It isn't a plane wave, and the actual global solution isn't source free. The actual solution is global, but we only ever look at a small local region experimentally. $\endgroup$ – Timaeus Jun 16 '16 at 3:52
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I know that an electric field is created by a particle with a charge and that a magnetic field is created by a moving charge but how do they combine to make a electromagnetic field

Electromagnetic fields, i.e. electromagnetic radiation are one step further .

Electromagnetic waves are produced whenever charged particles are accelerated, and these waves can subsequently interact with any charged particles. EM waves carry energy, momentum and angular momentum away from their source particle and can impart those quantities to matter with which they interact.

That is the observational fact. An antenna, for example, radiates electromagnetic waves because charges move sinusoidally continually accelerating and decelerating over its length.

The theoretical model that fits all the data and can predict situations for electromagnetic waves is Maxwell's equations.

At the quantum level, the electromagnetic wave is composed of photons , and visible light is generated in the quantum framework by atoms and molecules. There is mathematical continuity between the two frameworks, but that is another story.

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Static electric fields were described by Coulomb's Law in 1785, with a similar law for magnets; https://en.wikipedia.org/wiki/Coulomb%27s_law#History

When Oersted discovered the magnetic field generated by an electric current (thanks to the invention of the battery by Volta), it became clear that the electric and magnetic fields were interconnected somehow: http://www-spof.gsfc.nasa.gov/Education/whmfield.html

Ampere conducted a careful series of experiments which, along with the work of many others, established the basic form of the laws covering currents; significant additions were made by Faraday, and the important work of Maxwell completed the unification of the electric and magnetic fields into a single theory of electromagnetic fields.

The combined field is made by combining the electric and magnetic fields; this can be done conveniently in the form of the Maxwell's electromagnetic tensor, an antisymmetric tensor: https://en.wikipedia.org/wiki/Electromagnetic_tensor

For many applications this is not required; then you just use the Lorentz force law, $F=q [V + v \times B]$; see https://en.wikipedia.org/wiki/Lorentz_force

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how do they combine to make a electromagnetic field?

They do not.

If you ignore the sources of those fields (i.e., electrically or magnetically charges particles), you can happily think about those fields as just separate properties in space.

You then can slap the gravitational field on top of it, as well as the field describing air temperature at each point. Add the field of motion vectors describing the fluid movement of the air, and the field of air density, etc. etc.

I hope you see what I mean. There's nothing special about an electric and magnetic field overlapping per se, it only gets interesting when they affect the particles that created them in the first place.

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an electric field is created by a particle with a charge

The electric field from charge is not created, it exists. Physics talks about intrinsic properties of particles which exist independent from our observation. Despite an intrinsic property is axiomatically (because without observation we can't be sure that the electric field exist and any observation of an electric field does change its behavior), the being of electric fields around particles, which we than call charges, is a foundation of physics. from Wikipedia https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/LineasCampo.svg/320px-LineasCampo.svg.png

a magnetic field is created by a moving charge

A moving charge is not the only phenomenon for a magnetic field. The main reason for the existence of a magnetic field is an other intrinsic property of particles. Electrons, protons, neutrons as well as their antiparticles have the intrinsic property of a magnetic dipole moment. In difference to the electric field this field is not symmetrical in all directions. So to talk about magnetic dipole moment helps to recapitulate this fact all the time.

https://upload.wikimedia.org/wikipedia/commons/thumb/0/0c/VFPt_cylindrical_magnet_thumb.svg/220px-VFPt_cylindrical_magnet_thumb.svg.png

The outer effects from this two properties are quite different. A collection of equally charged particles produce a common field. This is easily reproducible and observable for example with an electroscope. To show the magnetic properties one need an external magnetic field. All materials show magnetic properties. Some of them, once magnetized, are stable magnets. Some materials are showing magnetic properties only under the influence of an external field (you could hold a pice of metal with an magnet, but two pieces of metal do not show any magnetic force to each over). Some materials show magnetic properties under the influence only of strong magnetic field and by low temperatures. All this has to do with the more or less chaotic distribution of the magnetic dipole moments of the involved particles.

A magnetic field, induced by accelerated charges has to do with the alignment of the magnetic dipole moments of this charges. In detail see this answer or this publication (I'm against to mark it explicitely with "my answer" and "publication of mine" but it is wished to do so in this forum).

but how do they combine to make a electromagnetic field?

They do not combine in your sense. Particles emit and absorb (receive) photons. They get disturbed by photons and the came again to a lower state by emitting photons. This is used by us in an electric bulb as well as for radio waves or lasers. When a stream of photons is produced we are talking about electromagnetic radiation. This could be a naturally process like from stars or a made by us process like a radio wave. To be precise, every body with temperature below 0 Kelvin emits electromagnetic radiation, means, all bodies are radiators of photons.

As long as the stream of photons is not coherent (coherence = same starting point and same wavelength for all involved photons) the observation of the electromagnetic property of a radiation is not possible. To talk about a electromagnetic wave is possible for radio waves. It is interesting, that we can deduce from radio waves some phenomenons: A radio wave has an electric und a magnetic field component, this components (in vacuum) are aligned to each over perpendicular and both perpendicular to the direction of propagation. The propagation happens with the velocity of light.

from Wikipedia https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Onde_electromagnetique.svg/320px-Onde_electromagnetique.svg.png

Since radio waves are composed of a hugh number of photons (see my answer here) we can deduce that each photon has the same properties as the whole radio wave. So every photons is an electromagnetic wave.

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