Page 31 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 11

Here is the vector of electric field strength, is the integration area, dA is the infinitesimal
element of this area, and is the electrical charge. According to Table 1.5, lines 21, 1, and 9,

2
−1
the units for electric field strength E is [m ∙ kg ∙ −3 ∙ A ], for area element dA is [m ], and
for the electric charge is [A ∙ s]. Treating dimensions as algebraic quantities one can see their

relationship as
−1
(m ∙ kg ∙ −3 ∙ A ) ∙ m 2 1
= =
4
A ∙ s m −3 ∙ kg −1 ∙ ∙ A 2
Examining line 34 of Table 1.5 one can come to the conclusion that the factor k must be the
inverse value of the free space permittivity . Therefore, we can expect that
0

Φ = ∯ ∘ = 0 [V∙ m]

Later we will see that this equation is the exact formulation of the integral form of Gauss’s law.
Following the same path, we can prove that in Gauss’s law for magnetic flux the factor k =
0
Φ = ∯ ∘ = [V ∙ s = Wb]
0

where is the permeability of free space.
0
It is worth to note that the factors in Gauss’s law as in all following electrodynamics equations
depend on the chosen unit system. For example, in Gaussian units, unlike SI units, the
dimensional coefficients and disapper from Gauss’s law formulation.
0
0
Unfortunately, the dimensional analysis does not include the magnitudes of the based units. If
so, it can predict the numerical value of factor k only up to a multiplicative constant. For

example, in the last case, nothing will change in dimensional analysis if = 2/ or =
0
1/ . The additional multiplicative constant usually can be established through the
0
measurements or other means.
These examples demonstrate the significant role that the dimensional analysis, based on having
the same units on both sides of the equation, can play as a prediction and verifying tool. In fact,
we can establish the Unit Law as having different units on the two sides of the equation does
guarantee that the equation is wrong. In conclusion, note that in Gaussian CGS (short for
centimeter-gram-second) unit system the dimensional coefficients and swap to some
0
0
combinations of constant 4 and speed of light c.
1.3.4 Table of Mathematical Operators in Use

The following table provides the meaning of some mathematical operators for use later. We put
it here for the reader’s convenience. See more in Appendix.

Table 1.6
Symbol Meaning Comments

, , is magnitude of vector function
= + + in x, y, z direction, respectively.

0
0
0
= + + Vector differential operator called del or
0
0 0 nubla operator written in Cartesian space.