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Maxwell's Equations

[last updated: 2024-11-12]
Still fumbling...
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  • Electrodynamics is the study of electromagnetism.
    Electrodynamics describes the behavior of the forces between electrically charged particles.
    These forces, Electro-magnet-ism are the result of the electric and magnetic fields
    that are created by charged particles as a result of their charge or movement.
  • Electrostatics is the subset (and starting point) of electrodynamics that deals with stationary or slow-moving electrically charged particles.
    • The first property/behavior to consider in electrostatics is the force between charged particles as described by Coulomb's Law:
        While Coulomb's Law is not strictly part of "Maxwell's equations," it regardless is the foundation of all electromagnetic phenomena,
        dealing as it does with the electric force between charged particles.
          F = (1 / 4 π ε0) (q1 q2 / r2)
          F = force in Newtons
          r = separation distance in meters
            ε0 is the "electric constant" or "vacuum permittivity of free space", or "distributed capacitance of a vacuum".
              ε0 = 8.8541878188(14)×10−12 F⋅m−1

            Note the units of ε0 are "F⋅m−1", Farads/meter, ie. "distributed capacitance."

          q1 and q2 are the electric charge on the particles being considered, and are expressed in coulombs.

            A coulomb is defined as the amount of charge carried by 1 amp flowing for 1 second.
            The charge on a single electron is 1.602 x 10-19 coulombs.
            A coulomb is approx equal to the charge on 6.24 x 1018 electrons
      • The principle of Superposition
        states that the force between any two charged particles is independent and unaffected
        by any other charged particles in the system.
        Therefore, to find the net force on the target charge Q,
        find the force contributed by the first particle q1,
        then repeat for q2, and continue through all particles in the system,
        then vector sum the forces from all the individual qn particles to find the net total force on Q.
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      • The Electric Field:
        • Electric fields originate from electric charges and time-varying electric currents.
        • The Electric Field is defined as a vector field (ie. with magnitude and direction) that associates every point in space surrounding a charge (or current) with a force that would be exerted on a unit (ie. 1 coulomb) test charge located at that point.
        • Units of E are volts/meter, or Newtons/coulomb.

        • If you rearrange the components of Coulomb's Law like this:
            F = ( q1 / 4 π ε0 ) * ( 1 / r2 ) * q2

          then, since F is really a force vector, replace it with F⃗,
              and to capture the direction, add the unit vector r from q1 to q2: r^1,2

            F⃗ = ( q1 / 4 π ε0 ) * ( 1 / r2 ) * q2 * r^1,2

          then, since a unit vector = the vector divided by its magnitude:

            r^1,2 = r⃗ / | r⃗ |

          it follows that: | r⃗ | = r⃗ / r^1,2

          and r2 = the magnitude of r, squared = | r⃗ |2 = ( r⃗ / r^1,2 )2

            F⃗ = ( q1 / 4 π ε0 ) * ( 1 / ( r⃗ / r^1,2 )2 ) * q2 * r^1,2

          or:
          ... this isn't quite right yet ...

            F⃗ = ( q1 / 4 π ε0 ) * ( r^1,22 / r⃗2 ) * q2 * r^1,2


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      from wikipedia:

      https://en.wikipedia.org/wiki/Maxwell's_equations
      are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism,

      https://en.wikipedia.org/wiki/Electromagnetism
      In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electrostatics and magnetism, which are distinct but closely intertwined phenomena. Electromagnetic forces occur between any two charged particles. Electric forces cause an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs between charged particles in relative motion. These two forces are described in terms of electromagnetic fields. Macroscopic charged objects are described in terms of Coulomb's law for electricity and Ampère's force law for magnetism; the Lorentz force describes microscopic charged particles.
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      from Google genAI:
      Maxwell's equations tell us how electric and magnetic fields are generated by charges and currents, and how they interact with each other, essentially describing the relationship between changing electric fields creating magnetic fields and vice versa, which ultimately leads to the understanding of electromagnetic waves like light as a propagating phenomenon through space; they are a set of four fundamental equations that provide a complete description of electromagnetism.
      Key points about Maxwell's equations:

      Generating fields:
      They explain how electric charges create electric fields and how moving charges (currents) generate magnetic fields.

      Time-varying fields:
      A crucial aspect is that Maxwell's equations account for changing electric and magnetic fields, meaning a changing electric field can induce a magnetic field and vice versa.
      Electromagnetic waves:
      By combining the equations, it can be shown that fluctuations in electromagnetic fields propagate as waves at a constant speed, which is the speed of light in a vacuum.
      Unified theory:
      Maxwell's equations unify electricity and magnetism, demonstrating that they are two aspects of the same phenomenon.
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    • Four laws of electromagnetism:
      • Faraday's laws:
        • Faraday published his first law, the law of electromagnetic induction, in 1831.
          It states:
            If a conductor is placed in a varying magnetic field, an EMF (voltage) is induced in it.
            If the conductor is in a closed circuit, then a current will flow.
        • Faraday's second law states:
            "The induced emf in a coil is equal to the rate of change of flux linkage."

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      • Ampere's circuital law:
        • This law was not discovered by Ampere, regardless of its name, but in fact is one of Maxwell's equations published in 1860.
        • This law specifies the strength of the magnetic field that is created by a current in a conductor.

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      • Lenz' law:
          “When emf is generated by a change in magnetic flux, the polarity of the induced emf is such that it generates a current whose magnetic field is in a direction that opposes the change that produced it (the original magnetic field).”

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      • Lorentz force law:
          "The Lorentz force is the force that a particle experiences due to electric and magnetic fields. Electric fields exert a force on a particle whether it is moving or not, while magnetic fields exert a force only when the particle is in motion."

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    • Maxwell's equations:
      • James Clarke Maxwell published his initial paper (his Treatise...) in 1873.
        In fact the 4 equations now recognized as "Maxwell's equations" were never identified nor presented by Maxwell.
        They in fact were developed and defined in subsequent years by George Francis FitzGerald, Oliver Lodge, and Oliver Heaviside,
        with significant contributions from Heinrich Hertz and J. H. Poynting.

      • (from McGrawHill...)
        Four laws known collectively as Maxwell’s equations:
        • An electrically charged particle creates an electric field. This is known as Gauss's law.
        • Magnetically charged particles do not exist and therefore certain types of magnetic fields do not exist. This is known as Gauss's law for magnetism.
        • A changing magnetic field creates an electric field. This is Faraday's law of induction.
        • A moving or spinning electric charge creates a magnetic field. This is Ampère’s law.
          Also, a changing electric field creates a magnetic field. When both principles are combined, it is known as the Ampère-Maxwell law.

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    • Links/Refs:

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    • Addendum:
      • "electrically charged particles have an intrinsic magnetic moment"
        "Magnetic moment" is a vector quantity, a measure of the strength and direction of the magnetic field produced by an (stationary?) object.
        It is proportional to the amount of torque the object would experience when placed in an external magnetic field.
        Units: measured in Ampere-meter squared (A·m²).
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      • ...The first two laws of electrodynamics (Faraday and Ampere) are incorporated into Maxwell's equations, which describe the electromagnetic behavior mathematically (quantitatively).
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      • From Hunt (p.71):
        "[Heaviside's articles] ... produced a ... clear introduction to the main points of field theory. [Their] centerpiece was a discussion of how the vector operators "divergence" and "curl" could be used to derive the strengths of charges and currents from a knowledge of the forces and fluxes in the surrounding field. ... [He was] a strong supporter or Faraday's and Maxwell's doctrine that the field was the real seat of electromagnetic phenomena..."
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      • Phonomena to comprehend/explain:
        "Displacement current": Is this "back-EMF" from building and collapsing a magnetic field around a current-carrying conductor?
        "Faraday Effect": When a polarized light is put through a glass (an "optically transparent dielectric material") in a magnetic field, its plane of polarization rotates.
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      • from Google GenAI:
        The equation for the electric field around a charged particle is E = k * Q / r² where:
          E: is the electric field strength (in Newtons per Coulomb, N/C)
          k: is Coulomb's constant (approximately 8.99 x 10⁹ N·m²/C²)
          Q: is the magnitude of the charge of the particle (in Coulombs, C)
          r: is the distance from the particle to the point where the electric field is being measured (in meters, m)


          Key points:
          The electric field is a vector quantity, having both magnitude and direction.
          This equation is based on the concept of a "test charge" - a small positive charge placed at a point r distance from the charged particle.
          The equation gives the electric field strength at that test point.
          Examining the equation shows that the electric field strength decreases with the square of the distance from the charged particle.

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