Maxwell's 1st equation

gaussianMaxwell’s first law is based on Gauss’ law for electric fields, which was formulated by Carl Friedrich Gauss in 1835, although not published until 1867. This work could in turn be linked to Coulomb’s law defining the force [F] between two charges [q1,q2] published in 1785. At the root of all these equations is the idea of an electric field emanating outwards from a point charge, as illustrated in the diagram. It is worth noting that that while a charge particle can exist in isolation, analogous to an electric monopole, the concept of an electric field can still only be measured between 2 charge particles. However, we can conceptually visualise an electric field emanating out from the charge particle [q] equally in all direction. As such, we can also define the electric field [E] at any point on the spherical Gaussian surface shown as [ds]:

[1]      12

We might recognise that the division of the charge [q] by surface area of a sphere [4πr2] provides the definition of the electric field strength at any point on the surface. We can also quickly explain the notation [ur] as the unit vector reflecting the direction of the electric field vector [E]. However, having highlighted the vector requirements, this aspect will not always be highlighted in subsequent equations.

But what is the physical meaning of [ε0]?

This parameter represents permittivity [ε], which is said to be a measure of how much resistance is encountered when forming an electric field in a vacuum, i.e. it reflects how an electric field affects, and is affected by, a dielectric medium. In the case of a vacuum, which is the primary focus of this discussion, the permittivity of free space is a constant, represented by the symbol [ε0], and has the value: 8.85*10−12F/m. On this basis, we can now proceed to define the total electric flux [φE] through the Gaussian surface:

[2]      2

In this case, we might recognise that the result simply multiplies [1] by the surface area of the sphere, which leads to the result shown. However, we can also see that [2] no longer has any dependency on the radius [r] or for that matter the shape of the Gaussian surface, i.e. it can be an irregular shape. It can also be shown, although not necessarily here, that a series of charges [q=q1..qn] can be placed anywhere within an irregular Gaussian surface. As such, [2] essentially allows us to directly define Gauss’ Law and Maxwell’s 1st law in its integral form:

[3]      3

Where [q] represents the net charge integrated over the Gaussian surface and, as such, [3] is essentially the mathematical shorthand of the description of [2]. However, it is often quite useful to initially break these equations back down into the unit parameters:

[4]      4

So Maxwell’s 1st equation, as presented in the integral form, is describing the electric flux [φE=E.m2], i.e. the electric field [E] over a given surface area [m2], as the charge [q] divided by the electric constant [ε0]. Today, one can question the physical reality of the idea of an electric flux radiating through the conceptual surface enclosing the charge, but it is a logically consistent working model.

differentialBut what about the differential form?

The differential can often appear simpler in form, but tends to be more awkward to prove, but hopefully the following derivation is sufficient to provide the general idea. We have already generally proved that Gauss’ Law can be applied to any shape, such that we might define this shape to be a small elementary volume (dx dy dz) in the [xyz] planes. From the diagram, we can see the electric flux emerging from the surface [ABCD=dy*dz] at an angle [θ], such that we can define [E] through this surface as:

[5]      5

The substitution of [Ex] for [E.cosθ] is simply because this quantity is the electric field in the [x] plane. We might recognise that the flux flowing through the opposite surface [A’B’C’D’] must have a similar form, although carrying an opposite sign, i.e. [–Ex’ dy dz], such that we might we define the total flux though both surfaces as:

[6]      6

However, because we are dealing with a very small differential volume, we can make the following substitution:

[7]      7

As such, we can now substitute [7] back into [6] to give an expression rationalised to the elementary volume [dv]:

[8]      8

We can repeat this logic for the other planes [y,z], which we might reasonably expect to produce similar results, such that we end up with an expression for the total flux:

[9]      9

It is worth stopping at this point to understand that [9] is the logical equivalent of the integral form of Maxwell’s equation given in [3] and [4]. However, the differential form of Maxwell’s equation usually rationalises [9] by recognising that [q/dv] effectively represents the electric charge density [ρ] such that we end up with the following equation, which can be considered in terms of the electric flux per volume:

[10]    10

As such, [10] is the usual presentation of the differential form of Maxwell’s 1st equation. It can be seen that this equation is time-independent, i.e. it represents a static configuration. However, with reference back to [1], we can still see that the electric field [E] is directly proportional to the strength of the sum of the charges, contained within any Gaussian surface, and may initially be thought to ‘radiate’ outward to infinity, although with an ever-diminishing intensity based on an inverse square law.

Electric Displacement Field [D].

The topic of electromagnetism comes with a plethora of symbols that relate to all kinds of quantities. To some extent, this discussion has attempted to keep the number of quantities require to explain Maxwell’s equations to a minimum. However, there may be some value in introducing the concept of the electric displacement field [D]. In a dielectric material, the presence of an electric field [E] causes the bound charges in the atomic nuclei and their electrons to slightly separate, which induces a local electric dipole moment defined by:

[11]    11

 Again, the permittivity of free space [e0] has the value: 8.8544*10 −12 C2/Nm2