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    (Edit Source - Edit) : Is The Differential Vector Element of Path Length Tangential To The Path/curve, As Well

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    senthilanview
    The document discusses electromagnetic fields and their relationship to line integrals and curls. It explains that the curl of a vector field represents the circulation of that field around a closed curve, and how this relates to Stokes' theorem. It also describes how the time evolution of electromagnetic fields is represented by the partial derivatives of the fields with respect to time, and how this predicts the propagation of electromagnetic waves.

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    (Edit Source - Edit) : Is The Differential Vector Element of Path Length Tangential To The Path/curve, As Well

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    senthilanview
    15 views2 pages
    The document discusses electromagnetic fields and their relationship to line integrals and curls. It explains that the curl of a vector field represents the circulation of that field around a closed curve, and how this relates to Stokes' theorem. It also describes how the time evolution of electromagnetic fields is represented by the partial derivatives of the fields with respect to time, and how this predicts the propagation of electromagnetic waves.

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    Curl

    Original Title

    Curl

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    The document discusses electromagnetic fields and their relationship to line integrals and curls. It explains that the curl of a vector field represents the circulation of that field around a closed curve, and how this relates to Stokes' theorem. It also describes how the time evolution of electromagnetic fields is represented by the partial derivatives of the fields with respect to time, and how this predicts the propagation of electromagnetic waves.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    15 views2 pages

    (Edit Source - Edit) : Is The Differential Vector Element of Path Length Tangential To The Path/curve, As Well

    Uploaded by

    senthilanview
    The document discusses electromagnetic fields and their relationship to line integrals and curls. It explains that the curl of a vector field represents the circulation of that field around a closed curve, and how this relates to Stokes' theorem. It also describes how the time evolution of electromagnetic fields is represented by the partial derivatives of the fields with respect to time, and how this predicts the propagation of electromagnetic waves.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    curl[edit source | editbeta]

    Open surface and boundary . F could be the E or B fields. Again, n is the unit normal. (The curl of a vector field doesn't literally look like the "circulations", this is a heuristic depiction).

    The "circulation of the fields" can be interpreted from the line integrals of the fields around the closed curve :

    where d is the differential vector element of path length tangential to the path/curve, as well as their curls:

    These line integrals and curls are connected by Stokes' theorem, and are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.

    Time evolution[edit source | editbeta]


    The "dynamics" or "time evolution of the fields" is due to the partial derivatives of the fields with respect to time:

    These derivatives are crucial for the prediction of field propagation in the form of electromagnetic waves. Since the surface is taken to be time-independent, we can make the following transition in Faraday's law:

    see differentiation under the integral sign for more on this result.

    Conceptual descriptions[edit source | editbeta]


    Gauss's law[edit source | editbeta]
    Gauss's law describes the relationship between a static electric field and the electric charges that cause it: The static electric field points away from positive charges and towards negative charges. In the field line description, electric field lines begin only at positive electric charges and end only at negative electric charges. 'Counting' the number of field lines passing though a closed surface, therefore, yields the total charge (including bound charge due to polarization of material) enclosed by that surface divided by dielectricity of free space (the vacuum permittivity). More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the enclosed electric charge.

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