The Electric and the Magnetic Fields

The electric and the magnetic fields, Eqs.(6.43) and (6.44), are obtained for each type of e.m. field from the three respective vector potentials,

$$\left[ \begin{array}{c} \phi\\ A_z\\ A_x\\ A_y \end{array} \right... ...{c} \textrm{Transverse}\\ \textrm{Electric}\\ \textrm{Magnetic} \end{array}} ~.$$ (675)

It needs to be reemphasized that for the TEM field it is unnecessary (and, in fact, uncalled for) to calculate the scalars $\Phi$ and $\Psi$ individually. All that is necessary (and sufficient) is the difference $\Phi -\Psi$ .

From the perspective of the principle of conceptual unit-economy⁶⁹ the e.m. potential is superior to the electric and magnetic field. This is because the mathematical characterization of the e.m. potential is simpler than that of the e.m. field. However, in the hierarchy of concepts, the e.m. field is much closer to the electromagnetism's foundation, namely that which is directly accessible to measurements. Thus, in order to comply with this hierarchy and prevent the e.m. potential from being a floating abstraction disconnected from reality, it is mandatory that one explicitly exhibit the e.m. field. We shall do this for TE, TM, and TEM fields relative to cartesian coordinates, and later extend the result to cylindrical and spherical coordinates.

The TE Field:

The result of deriving the e.m. field, Eqs.()-(6.44), from the $TE$ potential in Eq.(6.75), together with the corresponding $TE$ source, have been consolidated into Table 6.1.

Table: The $TE$ system: The components of a $TE$ e.m. field $(\vec E,\vec B)$ are derived from a four-vector $TE$ potential $(\vec A, \phi)$ , a solution the inhomogeneous Maxwell wave Eq.(6.51) on page . Its source is the divergenceless $TE$ charge density-flux four-vector $(\vec J,\rho)$ .

$TE$ Potential				$TE$ Electric Field
$A_x$	$A_y$	$A_z$	$\phi$	$E_x$	$E_y$	$E_z$
$\frac{\partial \Phi^{TE}}{\partial y}$	$-\frac{\partial \Phi^{TE}}{\partial x}$	0	0	$-\frac{\partial }{\partial y}\frac{\partial \Phi^{TE}}{\partial t }$	$\frac{\partial }{\partial x}\frac{\partial \Phi^{TE}}{\partial t}$	0
$TE$ Source				$TE$ Magnetic Field
$J_x$	$J_y$	$J_z$	$\rho$	$B_x$	$B_y$	$B_z$
$\frac{\partial S^{TE}}{\partial y}$	$-\frac{\partial S^{TE}}{\partial x}$	0	0	$\frac{\partial }{\partial x} \frac{\partial \Phi^{TE} }{\partial z}$	$\frac{\partial }{\partial y} \frac{\partial \Phi^{TE}}{\partial z}$	$-\left( \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} \right) \Phi^{TE}$

Any e.m. field of the type exhibited in this table is called purely transverse electric ($TE$ ). This is because the electric field vector $\vec E$ is nonzero only in the transverse $(x,y)$ -plane, the plane perpendicular to the longitudinal direction, the $z$ -axis. The longitudinal electric field component,

$$E_{long.}\equiv E_{z}=0~,$$

of a $TE$ electromagnetic field vanishes!

The TM Field:

The result of deriving the e.m. field, Eqs.(6.43)-(6.44), from the $TM$ potential in Eq.(6.75), together with the corresponding $TM$ source, have been consolidated into Table 6.2.

Table: The $TM$ system: The components of a $TM$ e.m. field $(\vec E,\vec B)$ are derived from a four-vector $TM$ potential $(\vec A, \phi)$ , a solution the inhomogeneous Maxwell wave Eq.(6.51) on page . Its source is the divergenceless $TM$ charge density-flux four-vector $(\vec J,\rho)$ .

$TM$ Potential				$TM$ Electric Field
$A_x$	$A_y$	$A_z$	$\phi$	$E_x$	$E_y$	$E_z$
0	0	$\frac{\partial \Phi^{TM}}{\partial t}$	$-\frac{\partial \Phi^{TM}}{\partial z}$	$\frac{\partial }{\partial x} \frac{\partial \Phi^{TM} }{\partial z}$	$\frac{\partial }{\partial y} \frac{\partial \Phi^{TM}}{\partial z}$	$\left( \frac{\partial^2}{\partial z^2} -\frac{\partial^2}{\partial t^2} \right) \Phi^{TM}$
$TM$ Source				$TM$ Magnetic Field
$J_x$	$J_y$	$J_z$	$\rho$	$B_x$	$B_y$	$B_z$
0	0	$\frac{\partial S^{TM}}{\partial t}$	$-\frac{\partial S^{TM}}{\partial z}$	$\frac{\partial }{\partial y}\frac{\partial \Phi^{TM}}{\partial t }$	$-\frac{\partial }{\partial x}\frac{\partial \Phi^{TM}}{\partial t}$	0

Any e.m. field of the type exhibited in this table is called purely transverse magnetic ($TM$ ). This is becausethe magnetic field vector $\vec B$ is nonzero only in the transverse $(x,y)$ -plane, the plane perpendicular to the longitudinal direction, the $z$ -axis: the longitudinal magnetic field component,

$$B_{long.}\equiv B_{z}=0~,$$

of a $TM$ electromagnetic field vanishes!

Remark. Note that the $TM$ field is the same as the $TE$ field except that the roles of $\vec E$ and $\vec B$ are essentially reversed:

$$\vec E \longrightarrow -\vec B$$

$$\vec B \longrightarrow ~~\vec E\quad\textrm{(whenever $\Phi^{TM}$ satisfies the sourceless wave equation)}$$

Table: The $TEM$ system: The components of a $TEM$ e.m. field $(\vec E,\vec B)$ are derived from a four-vector $TEM$ potential $(\vec A, \phi)$ , a solution the inhomogeneous Maxwell wave Eq.(6.51) on page . Its source is the divergenceless $TEM$ charge flux-density four-vector $(\vec J,\rho)$ .

$TEM$ Potential				$TEM$ Electric Field
$A_x$	$A_y$	$A_z$	$\phi$	$E_x$	$E_y$	$E_z$
$\frac{\partial \Psi}{\partial x}$	$\frac{\partial \Psi}{\partial y}$	$\frac{\partial \Phi}{\partial z}$	$- \frac{\partial \Phi}{\partial t}$	$\frac{\partial }{\partial x}\frac{\partial (\Phi-\Psi)}{\partial t }$	$\frac{\partial }{\partial y}\frac{\partial (\Phi-\Psi)}{\partial t}$	$0$
$TEM$ Source				$TEM$ Magnetic Field
$J_x$	$J_y$	$J_z$	$\rho$	$B_x$	$B_y$	$B_z$
$\frac{\partial I^{TEM}}{\partial x}$	$\frac{\partial I^{TEM}}{\partial y}$	$\frac{\partial J^{~}}{\partial z}$	$- \frac{\partial J^{~}}{\partial t}$	$\frac{\partial }{\partial y} \frac{\partial (\Phi-\Psi) }{\partial z}$	$-\frac{\partial }{\partial x} \frac{\partial (\Phi-\Psi)}{\partial z}$	0

The TEM Field:

The result of deriving the e.m. field, Eqs.(6.43)- (6.44), from the $TEM$ potential in Eq.(6.75), together with the corresponding $TEM$ source, have been consolidated into Table 6.3.

Any e.m. field of the type exhibited in the table is called purely transverse electric and magnetic ($TEM$ ). This is because both the $\vec E$ field and the $\vec B$ field lie strictly in the trasverse $(x,y)$ -plane. There are no longitudinal components:

$$E_{long.}\equiv E_z=0\quad \textrm{and} \quad B_{long.}\equiv B_z=0~.$$ (676)

Footnotes

... unit-economy⁶⁹

As identified in the footnote on Page