Dimensional Analysis - Part 3
Of oranges, nukes, and virgins,
In part two we saw how kinetic units of measure form an ascending ladder.  Here we shall discuss electrodynamic units of measure and shall discover how these units form a circular chain with no beginning or end.  However, having said that, we must still have some relation that ties kinetic units of measure to electromagnetic units of measure.  The most convenient way to tie these two groups together is through the definition of force on an electric charge, produced by the electric field.

The electric field:
An electric field produces a force that acts upon any electrically charged object, as shown in Eq. 1a.

[Eq. 1a]    

F = Force in Newtons.
Q = Electric charge in Coulombs.
E = Electric field.
Note: In part 3 of this paper we will redefine E as the electric field instead of energy, and use J "joules" for our energy variable.

Therefore, the electric field is defined by Eq. 1b.

[Eq. 1b]    

And since the fundamental units of force are "meter-kilograms per second squared" (2.2.5), the dimensional equivalent of the electric field in fundamental units is:

[Eq. 1c]    
    (dimensional equivalent of electric field in fundamental units)

The unit of measure for a uniform electric field is "volts per meter" or Newtons per Coulomb.  By defining the electric field as volts per meter, we also fix the dimensional equivalent of "volt" in fundamental units as shown in Eq. 1d.

[Eq. 1d]    
    (dimensional equivalent of volt in fundamental units)

e = Volts = Joules per Coulomb.

Force between electric charges:
The force between two electric charges is proportional to the product of charge, and inversely proportional to the square of their separation distance, as shown by Eq. 2a.

[Eq. 2a]    

 = Dielectric constant of free space in Farads per meter.

The dielectric constant is required in order to make Eq. 2a dimensionally balanced.  In other words, from a dimensional analysis view point: Q1*Q2 / L2, does not equal force (L*M / T2).  Therefore the definition of the dielectric constant in terms of fundamental units is derived as follows:

[Eq. 2b]    

The dielectric constant is:

[Eq. 2c]    
    (dimensional equivalent of dielectric const. in fundamental units)

The unit of measure for the dielectric constant is Farads per meter, or in fundamental units, seconds squared-coulombs squared per meter cubed-kilograms.

Generating an electric field:
Now that we have defined the dielectric constant in terms of fundamental units, we can use that definition to examine the process that creates the electric field.  We know from 3.2.1 (above) that an electric field, when multiplied by an electric charge, is equal to a force, and therefore the electric field has the dimensional units of Newtons per coulomb, and from 3.2.2 that electric charges exert a mutual force upon each other.  From these relationships, we surmise that the dielectric constant multiplied by the electric field is equal to the source of the electric field, as shown in eq. 3.

[Eq. 3]    
    (dimensional equivalent of electric field source)

D = Electric flux density in Coulombs per meter squared.

This result is somewhat surprising, since it implies there is no such thing as "naked electric charge", only electric charge per unit area (L2).  In other words, all electric charge is confined to surfaces (a sphere for example).

The magnetic field:
The force experienced by an electric charge moving through a magnetic field (B) is proportional to the quantity of charge, the magnetic field strength, and the velocity of movement.  Furthermore this force manifests at right angles to both the direction of movement and magnetic field vector.  Eq. 4a shows this relationship in mathematical form.

[Eq. 4a]    

Ignoring the vector cross product, we get Eq. 4b.

[Eq. 4b]    

Therefore, we can define the magnetic field in terms of force, charge, and velocity as shown in Eq. 4c.

[Eq. 4c]    
    (dimensional equivalent of magnetic field)

B = Magnetic field in Teslas or kilograms per coulomb-seconds

Another way to view this result is that a magnetic field is a velocity deficient electric field, as shown in Eq. 4d (velocity transforms a magnetic field into an electric field).

[Eq. 4d]    

Electric charge in motion:
If we take our electric flux (D) from 3.2.3 (above) and give it a proper motion, the result is a magnetic flux (H) shown in Eq. 5a.

[Eq. 5a]    

And from the dimensional analysis view point, we have:

[Eq. 5b]    
    (dimensional equivalent of magnetic flux)

H = Magnetic flux in Webbers or coulombs per meter-seconds.

Since electric current (amps) is defined as Coulombs per second, we can also view magnetic flux (H) as "amps per meter".

Generating a magnetic field.
In 3.2.4 we defined the magnetic field, and in 3.2.5 we defined the magnetic flux.  All that is left is to define the relationship between the magnetic field, and the magnetic flux that creates it. This relationship is shown in Eq. 6a.

[Eq. 6a]    

 = Paramagnetic constant of free space in Henries per meter.

The paramagnetic constant is required in order to make the flux and the field it generates dimensionally balanced.  Therefore we define the paramagnetic constant as the ratio of magnetic field strength to magnetic flux, as shown in Eq. 6b.

[Eq. 6b]    
    (dimensional equivalent of paramagnetic constant)

The endless loop:

Sec. 3.2.1
Multiplied by
Sec. 3.2.2
Sec. 3.2.3
Multiplied by
Sec. 2.2.3
shown in
Sec. 2.2.3
Multiplied by
Sec. 3.2.4
Sec. 3.2.6
Multiplied by
Sec. 3.2.5

As mentioned in the introduction (3.1), the electromagnetic units of measure form a circular chain or endless loop (shown above).  At first this result may seem surprising but it is precisely this set of relationships that lead to electromagnetic wave propagation.

Other derivatives:
While the above set of electromagnetic units of measure are arguably the most widely used, many other units of measure may be derived from the electromagnetic units chain.  Among the more useful are:

The speed of light in free space is calculated from the paramagnetic (also known as the permeability of free space) and dielectric constants of free space.  The local speed of light in any media, may be calculated by substituting the local values of paramagnetic and dielectric polarization.  This relationship is shown in Eq. 7.

[Eq. 7]    

The Poynting vector "P" is calculated as a cross product of the electric field and magnetic flux as shown in Eq. 8.

[Eq. 8]    
    (dimensional equivalent of Poynting vector)

In Eq. 9 we discover that mass itself results from electromagnetic rotation in 3 spatial dimensions.

[Eq. 9]    

    (radians per second)

Finally, since both mass and the speed of light can be derived from electromagnetic phenomena, we could choose to rewrite Einstein's famous equation by combining Eq. 7 and Eq. 9, as shown in Eq. 10.

[Eq. 10]    

As we have seen, dimensional analysis is a tool of astounding power.  It provides the physicist with a reliable method to understand the many and varied relationships of physical mathematics, as well as giving us a glimpse directly into the inner workings of the universe.  While this tutorial is finished, the journey is far from over...

Dimensional Analysis - Part 3