# Thread: Maxwell Equations

1. Note : the below is a simplified version of the Maxwell equations in their microscopic form, i.e. in vacuum, as they can be found in an introductory course on electrostatics.

Before we get into it though a quick explanation of the symbols used ( bold face denotes vector quantities ) : : Electric field in [V/m] : Magnetic field in [T] : Total charge enclosed in volume V : Vacuum permittivity : Vacuum permeability : Electric and magnetic flux, respectively : Vector surface element : Vector line element : Boundary of a volume of space, i.e. a surface : Boundary of a surface, i.e. a line

GAUSS's LAW FOR ELECTRIC FIELDS Plain text meaning : The electric field through a closed surface is directly proportional to the amount of electrostatic charge enclosed within that surface. Equivalently stated : all electric field lines begin or end at electric charges.

GAUSS's LAW FOR MAGNETIC FIELDS Plain text meaning : In the presence of a magnetic field, the same amount of magnetic field lines enters a given volume of space than exits it. Equivalently : Magnetic field lines form closed loops, or : magnetic field lines do not end. Or, even more generally : there are no magnetic monopoles.

AMPERE's CIRCUIT LAW Plain text meaning : Both circuit current and displacement current act as sources of the magnetic field.

FARADAY's LAW Plain text meaning : A changing magnetic flux induces an electric field.

LORENTZ FORCE A charge q moving through an electromagnetic field with velocity v experiences a force along electric, but perpendicular to magnetic field lines.

SAMPLE CALCULATION

Let's give a very simple, if not to say trivial, example of a field calculation using Maxwell's equations.
Let's say we want to find the electric field around a stationary, constant, isolated point charge Q in free space. The simplest relation for the electric field is Gauss's law : Since this takes place in free space, and the charge is isolated and stationary, it is intuitive that this problem possesses spherical symmetry - this means that we can choose the uniform surface of a sphere centred on the charge Q as our Gaussian surface : Since the charge is stationary and constant, the electric field will be constant as well, and the integral simply becomes : and thus which is just the familiar Coulomb's law for a single point charge in free space.  2. I must admit I prefer the Differential form, so that in 4-vector form I only need to remember two :P  3. Originally Posted by Kerling I must admit I prefer the Differential form, so that in 4-vector form I only need to remember two :P
I agree that the differential form is mathematically simpler, however, to me at least, the integral formulation is much easier to visualise. I can directly picture the meaning of the equations through surface fluxes. But that's really just a personal preference   4. Originally Posted by Markus Hanke I agree that the differential form is mathematically simpler, however, to me at least, the integral formulation is much easier to visualise. I can directly picture the meaning of the equations through surface fluxes. But that's really just a personal preference That is actually quite interesting that you mention that. I always envision the vector fields, and because of that, the differential form is for me more easy to remember. It is nice to see, the same law being envisioned completely differently.   5. There is something you should check out and that's the naming conentions for B and H. In the simplest case H = uB where u is the permiability of free space.

B = Magnetic Flux Density (page 173 Eq. (5.1) )

H = Magnetic Field (page 192 Eq. (5.81) )

Pages and Equation numbers are from Classical Electrodynamics - 3rd Ed. by J.D. Jackson

Other texts call B the magnetic field. E.g. my old copy of Haliday & Resnick call B the magnetic field.  6. Originally Posted by Popper There is something you should check out and that's the naming conentions for B and H. In the simplest case H = uB where u is the permiability of free space.

B = Magnetic Flux Density (page 173 Eq. (5.1) )

H = Magnetic Field (page 192 Eq. (5.81) )

Pages and Equation numbers are from Classical Electrodynamics - 3rd Ed. by J.D. Jackson

Other texts call B the magnetic field. E.g. my old copy of Haliday & Resnick call B the magnetic field.
Correct, however, the equations in my OP are the microscopic ones, i.e. in vacuum, hence no H field. I did notice that sometimes there seems to be some confusion as to the usage of H and B, but here I follow the conventions presented in Young & Freedman's University Physics, partly because I have not yet gotten around to study other ED textbooks. The Wikipedia article on Maxwell's equations follows a similar distinction between microscopic and macroscopic versions.  7. I don't like the way electromagnetism is taught. What Markus presents is all pretty standard, but it rather separates the electric field from the magnetic field. It's only when we get to Lorentz force is the electromagnetic field mentioned. Take a look at Minkowski's Space and Time on wiki. See this little paragraph towards the end, alongside figure 3:

"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect."

That's my bolding. E and B don't designate fields, they designate forces. Electromagnetic field interactions involve linear force and/or rotational force which results in motion. If we only see linear motion we speak of an electric field, if we only see rotational motion we speak of a magnetic field. But the only field that's there is the electromagnetic field. I don't think this comes out very well.

Kerling: are you familiar with Percy Hammond and The role of potentials in electromagnetism? Or the book he co-authored with D Boldomir: Geometry of Electromagnetic Systems. Nobody seems to have heard of it. And sadly it's eighty quid.  8. Farsight, this is just a basic overview over Maxwell's equations, as taken from an introductory undergraduate text. It is neither a complete nor comprehensive picture of electrodynamics - I deliberately did not go into any of the alternative formulations, like the one using potentials, simply because their mathematical form does not lend itself to easy visualisation. Also, from a purely mathematical point of view, doing any actual calculations with the potentials formalism is a nightmare.

Suffice it to say that I agree with you that potentials are more fundamental than E and B fields, and that these equations can be formulated in terms of potentials. I think the alternative formalisms for electrodynamics warrant a separate thread, and are outside the scope of this sticky. I will address it when I get a chance.

That's my bolding. E and B don't designate fields, they designate forces.
Well, the E field is closely related to a force ( that's how it is defined, after all ), but the B field isn't - it is a flux density. Remember that the B field does not perform work ( the motion of a particle under the influence of a B field alone is always constant-speed motion ), thus it cannot be a force.

but it rather separates the electric field from the magnetic field
That's true, and it is done this way for a reason, namely computational convenience. Calculating E and B fields is much easier than calculating potentials, or using the full tensor formalism for the combined electromagnetic F field. However, we all understand that E and B cannot be separated, and that they are just components of the same underlying F field.

This is not a bad little article; seeing the author use "curl", "div" and "grad" instead of the del operator evokes memories for me, because these are the forms of the operators as I first learned them all those many years ago ( with "rot" instead of "curl" )   9. All points noted Markus, I look forward to a separate thread sometime.  10. Originally Posted by Markus Hanke Correct, however, the equations in my OP are the microscopic ones, i.e. in vacuum, hence no H field. I did notice that sometimes there seems to be some confusion as to the usage of H and B, but here I follow the conventions presented in Young & Freedman's University Physics, partly because I have not yet gotten around to study other ED textbooks. The Wikipedia article on Maxwell's equations follows a similar distinction between microscopic and macroscopic versions.
All I meant to say was that it would be nice to have a footnote noting that some authors refer to B as the magnetic field while others refer to B it as the magnetic flux density. It would help them to have that in the back of their minds when the go from one text to another.  11. We know that We can see this in the next graph which it is one half period

Attachment 71

Where with , i represent all the electric field vectors in half period to simplify. is the surface where it flows and whit the perimeter of where circulates .

We can see this in the video maxwell equations of mechanical universe Maxwell's Equations - YouTube , (specially in the minute 1:56 and 7:36)

But that can't be in EM waves because in half period the vectors that circulates points in just one direction. The circulations have to be like this

Attachment 72

Where And in that way the flows of in the surface wil generated the circulation of Attachment 73

So, How is really the circulation of the vectors? It's describe a circumference? or rectangle? Because the Maxwell equations do not specify that.

greetings.  12. Originally Posted by Farsight That's my bolding. E and B don't designate fields, they designate forces. Electromagnetic field interactions involve linear force and/or rotational force which results in motion. If we only see linear motion we speak of an electric field, if we only see rotational motion we speak of a magnetic field. But the only field that's there is the electromagnetic field. I don't think this comes out very well.
I don't agree with that, Farsight. As soon as one has a motion of a charged object with relation to any existing electric field, a magnetic field is created, as well for a linear motion as for a rotational motion.  13. You've missed the point of the screw nature of electromagnetism, thierry. Imagine you have no initial motion with respect to an electron. You might say there was an electric field present. Typically people will say that moving that electron creates a magnetic field. But motion is relative. You could keep the electron motionless and you could move instead. You don't alter the electron or its field simply by moving. You merely begin to see a different aspect of that field, that's all.  14. Originally Posted by Farsight You've missed the point of the screw nature of electromagnetism, thierry. Imagine you have no initial motion with respect to an electron. You might say there was an electric field present. Typically people will say that moving that electron creates a magnetic field. But motion is relative. You could keep the electron motionless and you could move instead. You don't alter the electron or its field simply by moving. You merely begin to see a different aspect of that field, that's all.
Yes, indeed, you only see a different aspect of that fied: the field that is perpendicularly to the motion, and that is hidden as long as there is no mutual motion between both electric charges.

But "the screw nature of electromagnetism" is induction, which is the second part of the story. Firstly, a magnetic field must be created by the mutual motion between the charges. Then only another charge can be induced and get the Lorentz force working on it.

As a matter of fact, exactly the same happens in gravitomagnetism (which Markus Hanke says is from GR, hence perfectly valuable), with the difference that mass hasn't to be charged before it can act. So, all masses under a mutual motion will interact through a second gravity field, that we could call gyrotation (the so-called magnetic part of gravitomagnetism). You could also speek of the screw nature of gravitomagnetism, and all the other properties of electromagnetism are directly transportable into gravitomagnetism.  15. Originally Posted by 1thierry which Markus Hanke says is from GR, hence perfectly valuable
I never said that at all. What I said is that it is a weak-field and non-relativistic approximation which is obtained from perturbing the Minkowski metric, in a manner akin to linearised gravity.  16. Well, all the roads lead to Rome...  17. Originally Posted by Popper There is something you should check out and that's the naming conentions for B and H. In the simplest case H = uB where u is the permiability of free space.

B = Magnetic Flux Density (page 173 Eq. (5.1) )

H = Magnetic Field (page 192 Eq. (5.81) )

Pages and Equation numbers are from Classical Electrodynamics - 3rd Ed. by J.D. Jackson

Other texts call B the magnetic field. E.g. my old copy of Haliday & Resnick call B the magnetic field.
Minor correction: No matter what you choose to call B and H, they're related by B = uH, not H = uB.  18. Hi
I'm studying maxwell's equations. I need historical references for Maxwell equations. I've read Maxwell articles and Heaviside articles and  But I need a much deeper references.
What were the changes of these equations in history? Where these equations have been come?
Can you help me?  19. Maxwell (1864) derives Maxwell's equation using only Faraday's induction law (Maxwell, Part III) yet Faraday's induction effect is not luminous. What is going on?  20. Originally Posted by Doctorevil Does this mean that Einstein's SR is invalid?.
No it doesn't, what gives you this idea?  21. Originally Posted by AndrewC No it doesn't, what gives you this idea?
Doctorevil is notorious for making sh*t up. S/he is embarrassingly clueless, but free of doubt.  Posting Permissions
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