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Electromagnetism (2018 Edition)


Contents

EM (ElectroMagnetism) Basic
Coulomb's Law  --  Point Charge, Prefect Conductor, and Capacitor,    Multipole Expansion
Ampere's Law  --  Magnetic Field by Line of Current,   Magnetic Field by Current Loop,   Displacement Current (EM Wave)
Gauss's Law (Absence of Magnetic Monopole)
Faraday's Law  --  Electric Inductor,   LC Circuit,   RLC Circuit,   Electric Generator,   Transformer,   Power Transmission,   Electric Motor,
                            Electrical Safety

Electromagnetism

Electromagnetism is such a mature science, there is not much novel idea that can be added into the subject matter. In this so called "2018 Edition", special attention is devoted to consistent units (in cgs) and formulas throughout - no more troublesome 1/40 and the dimension of the equations can be checked out easily.

Electromagnetism is about electricity, magnetism and light (Figure 01). The electric field E and magnetic field B are described by the Coulomb's
Electromagnetism EM Equations Law (about charge and electric field), the Ampere's Law (about current and magnetic field), and the Faraday's Law (a relationship between E and B), see Figure 02. Faraday's observation has inspired J. C. Maxwell to assemble these laws into a consistent set of equations in 1865 and is now known as Maxwell's equations. The disturbance of the electromagnetic fields was subsequently identified as the light wave in optics (see wave equations below).

Figure 01 Electro-magnetism

Figure 02 Maxwell's Equations

Electromagnetic Wave Equations :
In Figure 02, E and B are both in unit of /cm2, I is the current (/sec), J the current density (/sec-cm2), and c the velocity of light; the in all those electromagnetic units is the statcoulomb - the cgs unit of charge. The dimension for the three basic electric elements are : resistance R in sec/cm, capacitance C in cm, and inductance L in sec.

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EM (ElectroMagnetism) Basic

Gravitational Constant The origin of EM basic can be traced back to Newton's law of Universal Gravitation in 1686. It states that the force :

FG = GmM/r2,
where m and M are the two point masses, r the distance between them, G is a proportional constant known as gravitational constant, which was determined by the Cavendish Experiment (Figure 03) in 1797 to have a value of 6.74x10-8 cm3/gm-sec2 (vs 6.67x10-8 cm3/gm-sec2 in 2014).

Similarly, the force between two electric charges was postulated to have the form :

Fe = keqQ/r2,

Figure 03 Gravitational Constant [view large image]

where q and Q are the two point charges, r the distance between them. The proportional constant ke was measured by torsion balance similar to the Cavendish Experiment.
Currents Interaction The same kind of mathematical scheme has also been applied to the case of force per unit length between two infinitely long, parallel wires separated by a distance r and carrying currents I and I' :
dFm/dl = 2km(I I')/r (see Figure 05),
where I and I' are in unit of statampere (amp) = 1 /sec (one unit of charge passing a given point per second).

Figure 05 Currents Interaction [view large image]

By comparing the magnitude of Fe and dFm/dl for known charges and currents, it is found that ke/km = c2 (velocity of light c = 3x1010 cm/sec).
In term of the permeability (for the effect of magnetic flux on a substance), km = /4, with 0 = 1.4x10-20 gm-cm/2 in vacuum, and 1/00 = c2. Figure 06 lists the relative permeability r = /0 for some substances. It shows most of them do not respond to magnet flux = BS (S = surface area), except iron and two of its neighbors in the periodic table with permeability thousand times weaker than the iron's ferro-magnetization of 280000. The magnetic susceptibility m = r - 1 is for measuring the degree of magnetization of a material in response to an applied magnetic flux.
Permeability Field and Potential The electric field E of a point charge Q can be defined by re-writing the force Fe = qE,
in which, E = Q/r2 is the electric field by the charge Q in unit of /cm2.
The electric potential is defined by : .
A test particle with charge q would have a potential energy EP = qV (see Figure 07).
Voltage is the same as V = EP/q, i.e., potential energy per unit charge. It is always measured across 2 points in the form of difference, i.e., V = Vab = Va - Vb = I R :

Figure 06 Permeability List [view large image]

Figure 07 Field and Potential


Ferromagnetization Inside dielectric medium, the electric field is replaced by the displacement field D = KE to take care the unknown composition within the material (the K here is the dimensionless dielectric constant, but denoted by most textbooks as - a rather confusing practice). Similarly for the magnetic field B inside diamagnetic (r < 1, repulsive) and paramagnetic (r > 1, attractive) materials, it is replace by the magnetizing field H = rB (most textbooks would use instead of r). Ferromagnetic material retains magnetization (induced magnetic dipole) after the external field is removed (Fogire 08).

Figure 08 Ferromagnetization [view large image]

Electromagnetic wave is the propagation of the vibrational E and B fields : .

The above formulation seems to be rather ad hoc and specific for mainly point charges and parallel currents. The followings are more general in the form of the four laws of the Maxwell's equations involving more complicated mathematics.

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Coulomb's Law

The "Divergence" of a vector function D is a general mathematic concept about an incoming vector function at one side of a volume and its variation on the emergent side (Figure 09a). The change could be induced by internal process such as contraction/expansion, or via external source/sink. There could be no change at all, in that case D = 0. The Coulomb's Law identifies D as the electric field E and there is a source/sink in the form of + / -
Coulomb's Law Vector Formulas Gauss' Theorm charge density , i.e. E = (Figure 09b and see Figure 10 for some formulas of vector analysis in three different coordinate systems). Sometimes the solution can be obtained readily by integrating the divergence over a volume. As shown in Figure 11, the Gauss' Theorem transforms the volume integral to a surface integral, from which some examples for its

Figure 09 Coulomb's Law
[view large image]

Figure 10 Vector Formulas
[view large image]

Figure 11 Gauss' Theoerm

application are shown in the followings. The surface enclosing the volume is called Gaussian Surface.

  • Point Charge, Prefect Conductor, and Capacitor -
    Gaussian Surface Capacitance

    Figure 12 Gaussian Surface
    [view large image]

    The factor of 40 disappears in the spherical surface integration because we adopt the statcoulomb as the unit of charge (see comments on Fe).

    Figure 13 Capacitance
    [view large image]


  • Multipole Expansion -
    The general solution for the electric field E as shown in Figure 09 seems to be intractable for arbitrary distribution of charges. The method of multipole expansion for the potential defined by E = - enables extraction of some information depending on the level of detail. Since
    Multipole Expansion

    Figure 13a Multipole Expansion [view large image]

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    Ampere's Law

    Curl of Vector Stokes' Theorem The curl of a vector function A, i.e., (A) is about going around a looped path as shown in Figure 14, which illustrates the process with the four sides of a rectangle on the y-z plane. A little bit more imagination would be able to generalize the concept to 3-D space (see Figure 10 for the expressions in three different coordinate systems). The Ampere's Law identifies the vector function to be the magnetic field and the change is related to the source/sink of electric current passing through the surface enclosed by the loop, i.e., B = J/c, where J is the current density in unit of C/cm2-sec. This is in analogy to the "divergence" in which the vector function goes through a closed surface and the source/sink is the charge within the enclosed volume. So the Ampere's Law is prescribed in a formulation one dimension lower and its corresponding integral form is performed on a surface enclosed by the loop (see Figure 02) and derivation below.

    Figure 14 Curl of Vector [view large image]

    Figure 15 Stokes' Theorem [view large image]


    The transformation of the surface integral of B to line integral of B as shown in Figure 15 is known as Stokes' Theorem. By constructing small closed loops inside the enclosed surface, it is noted that all the two vectors sharing the path are always in opposite direction and hence canceled out except those at the edge where the path is not shared. Thus, we can express the Ampere's Law in line integral form :
    .