The Two forms of Maxwell's equations

In summary, the divergence and curl vectors are different ways of looking at the same thing - the integral of the E dot dA.
  • #1
DeepSeeded
113
1
Hello,

I was tempted to put this in the math section but it is more of a visualization problem though it is most likley due to my lack of understanding the divergence and curl operators fully.

I am comfortable with the closed loop integral of E dot dA and can visualize it as a solid closed surface. However when I think of the divergence of the E vector I think of a tiny little piece of the E vector.

How can these be the same thing?

Same goes for the Curl vector being the same as the closed loop integral of a line.

In 8.02 they used the integral form, in 8.03 they are using the differential form and now I am confused.
 
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  • #2
The integral of the E dot dA is not a surface but the flux through that surface. You can visualize it as the "amount of field" crossing the surface. This is not the same as the divergence but same as the integral of the divergence over the volume included in the surface.
The divergence is the flux through a tiny, infinitesimal, closed surface (divided by the volume inside the surface). By integration of the divergence over the volume you just add all these fluxes inside the volume included by the surface and end up with the "total" (net) flux through the surface.

You can write Maxwell equations in differential or integral form. It does not follow that the flux is equal to the divergence.
 
  • #3
OK, my misunderstanding was that the definition of the divergence is an infinitsmal sphere which is closed and not an open point. I see that it says this now on wikipedia but it was not so clear.

Looks like the definition of the curl is an infinitsmal closed line integral which also works to explain Ampere's laws.

I think I am getting it now, thank you.
 
  • #4
Those ideas are subtle and require some attention and patience when your firstencounter them. Try looking for other interpretations and explanations and reread the ones you like several times over a few days...took awhile to for those ideas to sink in for me...

I'm always amazed at the people who first dreamed these kinds of things up...pretty classy!
 

What are the two forms of Maxwell's equations?

The two forms of Maxwell's equations are the differential form and the integral form. The differential form uses vector calculus to describe the behavior of electromagnetic fields, while the integral form uses Gauss's law and Ampere's law to relate the fields to sources of electric charge and current.

What is the importance of Maxwell's equations?

Maxwell's equations are fundamental laws in electromagnetism that describe the relationship between electric and magnetic fields and their sources. They have been crucial in understanding and predicting the behavior of electromagnetic waves, which has led to advancements in technology such as radio, television, and wireless communication.

How do Maxwell's equations relate to light?

Maxwell's equations are the foundation of the electromagnetic theory of light. They show that light is an electromagnetic wave, with an electric field perpendicular to a magnetic field, and that the speed of light is determined by the electric permittivity and magnetic permeability of space.

What are the units used in Maxwell's equations?

The units used in Maxwell's equations depend on the system of measurement being used. In the SI system, the electric and magnetic fields are measured in volts per meter and teslas, respectively. The sources of electric charge and current are measured in coulombs and amperes, respectively.

What are some applications of Maxwell's equations?

Maxwell's equations have a wide range of applications, including the development of technologies such as radio and television broadcasting, radar, and wireless communication. They are also used in medical imaging techniques such as magnetic resonance imaging (MRI) and in understanding the behavior of particles in particle accelerators.

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