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Appendices
Appendix I: The Nature of Electromagnetic Fields
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A field is simply a region in which each point in the region is associated with a value of a parameter. This parameter may be a scalar quantity, such as temperature, or a vector quantity, such as force. Field Equations define the value of the parameter at any point in terms of the coordinates of that point.
The Electromagnetic (EM) field is actually a combination of two fields.
An Electric Field is a region in which there would be a Coulomb Force on a small test charge. The Coulomb Force is caused by the distribution of every other charged particle in and around the region. The Electric Field Intensity E is the parameter which defines the net force per unit charge on a small test charge at any point. E represents the sum of all the individual Coulomb forces from every other charged particle at that point. As E defines a force, which has both magnitude and direction, an electric field is a vector field.
Similarly, a Magnetic Field arises as the result of the physical distribution of magnetic materials and electric currents in the vicinity, each of which will have a definable effect on a test object in their vicinity. The Magnetic Flux Density B is a measure of the combined effect of all the individual contributions.
Magnetic Flux Density B also has both magnitude and direction. Therefore, the magnetic field is also a vector field.
However, because the effect of a magnetic field on different test objects is much more variable than that of an electric field on charged objects, it is not possible to equate B with a force in the same way that E is. Instead B represents one value in various field equations from which the force on a test object may be calculated, depending on whether that test object is charged, moving, or ferromagnetic.
Maxwell's Equations and the Lorentz Force Law are the field equations for the electromagnetic field. They define the forces experienced by objects in the EM field and also define the interaction of the electric and magnetic parts of the EM field. Particle motions resulting from the forces, together with the interactions of the electric and magnetic components, change the data values of the EM field itself. Therefore, the EM field is usually time-variable, and the interactions with charged particles in the field behave dynamically.
Field lines are virtual plots of the data which enable visualization of the field at any moment. For example, contour lines are the field lines of a 2-dimensional scalar field of heights above a datum. In a vector field, field lines can be used to indicate the direction of the field diagrammatically, for example, the magnetic field lines that represent the Earth's magnetic field, or they can be used to 'contour' the magnitudes of the data instead of indicating the direction.
Because the EM field is simply a field of data points in the form of vectors, the field lines have no physical reality. They are merely aids to visualization of the data set. That data can then be used to work out what physical effects would be experienced by a test object placed in the field.
To take another example, contour lines on a land survey map do not have a physical reality, but they can be used to work out the height and therefore the potential energy of a test object placed at any point in the real landscape represented by the contour lines.
The EM field is often thought of as "possessing" energy and momentum. The concept is useful in calculations involving transmission of energy and momentum across space, but the concept can lead to confusion about the physical reality of the field. It is all too easy to then think of the field as a physical entity that can store energy rather than as a mathematical data set representing the energies of physical objects. Saying that an EM field itself has energy is just a convenient shorthand meaning the total energy that would be necessary to assemble the physical objects under consideration. A new arrangement may have a different total energy. The new arrangement will certainly have a different field, or set of data values. It is erroneous to think of the field itself as having momentum and energy.
For example, consider the Moon and Earth system. The net gravitational field is a result of the position and mass of both bodies. There is even a point of zero gravitational force somewhere between them, that is, the field at that point is zero. The Moon has a certain potential energy relative to the Earth, and so the system has a certain total potential energy. If the Moon were to be lowered gently to the surface of the Earth, then the Earth-Moon system would lose potential energy. At the same time, the net gravitational field resulting from the two bodies would be modified, and the zero point would disappear.
To say that the Earth-Moon gravitational field itself has lost energy is incorrect. The field has merely been modified as a result of the changes to the arrangement of the physical bodies whose parameters it represents. It is only in the sense that the energy of the field can used as shorthand to represent the sum of the energies of the physical components which the field represents that the field itself can be said to have lost energy.
Why does this matter? Because it is important to keep a perspective on what the EM field is and not let it assume an independent physical reality it does not have. A field cannot exist as a separate entity. A field cannot be directly manipulated. A field merely represents certain parameters of the adjacent physical bodies in the form of a set of data.
In particular, EM field lines are only aids to data visualization. They have no physical reality; neither do contour lines on a map.
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end of Appendix 1
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next: Appendix II: Vector Algebra
previous: Summary & Conclusion
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