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Thread: Classical Electromagnetism

  1. #1 Classical Electromagnetism 
    Moderator Moderator Markus Hanke's Avatar
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    While I have never been an expert in electrical engineering ( frankly, circuit schematics confuse the crap out of me !! ), I am nonetheless fascinated with the sheer beauty and simplicity of the underlying physical principles, namely electromagnetism and its mathematical formulation, Maxwell's equations. Thus I thought I'd give a quick overview of those equations and what they entail, also because these equations get mentioned frequently on various threads. So ladies and gentlemen, allow me to present the reason why the computer, in front of which you are sitting, works the way it does !
    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.


    Last edited by Markus Hanke; January 13th, 2013 at 05:00 AM.
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    A remarkable thing about Maxwell's equations is that they are invariant under Lorentz transformations. Special Relativity could have started with Maxwell, although he wouldn't have known it..


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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by JonG View Post
    A remarkable thing about Maxwell's equations is that they are invariant under Lorentz transformations. Special Relativity could have started with Maxwell, although he wouldn't have known it..
    True, even though in relativistic scenarios I would use the tensor formulation of the equations :



    and



    This is no longer classical though, and harder to understand for someone who is not used to thinking in tensor language...
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    Forum Bachelors Degree Kerling's Avatar
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    Actually, Einstein did in fact start with Maxwell's equations. "Zur Elektrodynamik bewegter Körper"
    History of special relativity - Wikipedia, the free encyclopedia
    In the information age ignorance is a choice.
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    Quote Originally Posted by JonG View Post
    A remarkable thing about Maxwell's equations is that they are invariant under Lorentz transformations. Special Relativity could have started with Maxwell, although he wouldn't have known it..
    Actually Einstein derived a transformation which would (1) leave c invariant and (2) leave the form of Maxwell's equations unchanged. His resulting transformation turned out to be just the Lorentz transformation, as it had to be.
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