Maxwell's Equations

This page now begins the specific discussion of Maxwell’s equations, although this first page of the discussion will only introduce the equations and some of the mathematical notation and terminology in use. Subsequent pages will then try to expand the discussion of each equation in-turn and the physical principles implied.  As indicated, there were originally 20 equations in Maxwell’s 1864 paper, which were later rationalised by Oliver Heaviside in 1884 to just 4 differential equations. Although the differential form is still in use today, they have in-turn been complemented, and some would say replaced, by an equivalent integral form. On first exposure to the equations, the question of format may initially seem to be a matter of personal preference as the focus is usually on the more pressing matter of what any of this mathematical shorthand is trying to tell you about electromagnetism. However, while the differential form can sometimes appear simpler, the following outline, of both forms, may change your mind.

Mathematical Notation

The intention of this ‘prelude’ to the main discussion of each of Maxwell’s equations is simply to outline some of the basic notation used in both the differential and integral forms. As such, the reader may wish to cross-reference the earlier discussion of Vector Calculus .

 Differential Form

  We will start with 2 examples from the differential form:

• The implied divergence of 2 vectors, e.g. ∇.E
• The implied curl of 2 vectors, e.g. ∇xE

We will initially describe the symbol [∇] as a ‘vector differential operator’, although this description is not usually that helpful in itself. So let us quickly expand this operator as follows:

The symbols i, j & k are the unit vectors associated with the scalar magnitude in each direction, i.e. x, y & z, and ∂/∂x signifies a rate of change of magnitude of some unspecified quantity written as a partial derivative in Cartesian coordinates. However, the Del [∇] symbol can also be interpreted as a separate vector linked to the partial derivative operators. Therefore, the mathematical operations, implied by [∇.E] and [∇xE] are really defined in terms of the dot (.) and cross (x) products of 2 vectors. With this very brief mathematical introduction out of the way, let us attempt to infer some initial meaning from the examples shown above.

• ∇.E: The Divergence of E
  Here we are really determining the dot product of 2 vectors, i.e. ∇ and E, which is a mathematical operation that always results in a scalar value. It might be suggested that the magnitude, and the sign, of the scalar value corresponds to the flow into or out of a point in space:
  
         
  
  In this case, the symbol [φ_E] represents the flow or the ‘flux’ associated with some vector, e.g. [E], flowing out of a unit volume. The actual physical meaning will be discussed further against the appropriate Maxwell equation.
  

• ∇xE:  The Curl of E:
  The curl of a vector is described as a vector operator that shows the rate of rotation of a vector field, i.e. both the direction of rotation and the magnitude of the rotation. However, let us first expand the form of ∇xE, as shown below, and then address the question mark (?) below against what this may be equated to:

The question mark (?) in this case has to equate to a quantity that would cause the vector [E] to circulate along a closed path, e.g. a circle. Without getting into the actual description of Maxwell’s equations, we might realise from earlier discussions that the electric field lines radiate directly out from a static charge and, as such, there can be no ‘curl’ in this static configuration. However, electric field lines are known to circulate around a changing magnetic field. As such, this may help to initially explain the use of ∇xE in Maxwell’s 3rd equation below.

Integral Form

Now let us turn our attention to the integral form. Again, we will simply introduce 2 basic expressions used by Maxwell’s equations and then try to provide some initial interpretation of the mathematical notation in use:

• The closed line integral:          
• The closed surface integral:     

Let us start by first looking at the vector dot product within the integral by making reference to an earlier discussion of vectors. The dot product of 2 vectors produces a scalar; the magnitude of which is defined by [E.cosθ], where [θ] is the angle between the 2 vectors, i.e. [E and dl] or [E and ds]. We can see, by making reference to the diagram on the left below that while the vector dot product of [E] and [dl] depends on the angle between them, the angle [θ] has to be integrated over the line or path A-B. As such, this process is described as the line integral.

In similar fashion, referencing the diagram on the right above suggests that the surface integral of a vector field [E] corresponding to the flux of [E] flowing through the surface [ds] subject to the dot product dependency on the angle [θ]. Specifically, we might define the electric flux, written [φ_E], as the amount of ‘flow’ of the electric field through an area [ds]. As such, the surface integral is the integral over all regions on the closed surface.

Note: The additional circle superimposed on the integral [∫]  symbol simply implies that the integral encompasses a closed path with respect to the line integral or a closed surface with respect to the surface integral.

Maxwell’s Equations of Electromagnetism

So having introduced some of the mathematical notation used by Maxwell’s equations, we are hopefully in a better position to reflect on the meaning of the four equations initially presented in both the differential and integral forms.

Equation [1] is based on Gauss’ law for electric [E] fields:

  [1]     

Equation [2] is based on Gauss law for magnetic [B] fields:

  [2]     

Equation [3] is based on Faraday’s law of induction:  

  [3]     

Equation [4] is based on Ampere’s law:

  [4]     

Unfortunately, any understanding of how electric and magnetic fields are actually unified under Maxwell’s equations still requires a fairly extensive understanding of an array of quantities that are deemed necessary to describe the various electromagnetic processes involved. Therefore, we also need to provide an introduction of the primary variables in use.

•  The electric field [E] also corresponds to force per charge. Basic units of measurement align to [volts/metre] or [m.kg/C.s²].
  
  
• The magnetic field [B] is sometimes referenced using several terms: magnetic induction, magnetic field density and also the magnetic flux density. Basic units of a magnetic field are the [tesla] or [kg/C.s].
  
  
• The magnetic flux [Φ_B] represents the number of magnetic field lines that pass through a given surface. It has units of [webers] or [m².kg/C.s].
  
  
• The charge density [ρ] , in this case, is the amount of electric charge in a given volume. It has units of [coulombs/metre³] or [C/m³].
  
  
• The current density [J] is a measure of the flow density of electric charge per unit area. It has units of [amperes/metre²] or [A/m²], where [A=C/s]
  
  
•  The permittivity of free space [ε₀] is also called the electric constant. It has units of [farads/metre] or [s²C²/kg.m³].
  
  
• The permeability of free space [μ₀] is also called the magnetic constant. It has units of [henry/metre] or [m.kg/c²]
  
  
• The differential element [ds] corresponding to a surface area [S]. Its units are simply [metre²] or [m²]
  
  
•  The differential element [dl] corresponding to the length [L]. Its units are [metres].
  
  
• The divergence operator [ ∇.] has units of 1/metre
  
  
• The curl operator [ ∇x] has units of 1/metre

This amount of terminology is not only quite daunting on first exposure; it can also be quite meaningless unless accompanied by some descriptive text that seeks to interpret some physical meaning from what is essentially a form of mathematical shorthand. This process will take place in the following sub-sections, but we might begin the process of simplification by reducing Maxwell’s equation to just the differential form for free space, where the charge density [ρ=0] and the current density [J=0]:

While, hopefully, these equations do not appear quite so daunting, each requires further explanation in order to provide some physical interpretation of what they are really trying to tell us about electromagnetism. The issue ‘EM Propagation’ is also the subject of another discussion, but we might also wish to just table a generic form of the EM wave equation for consideration in advance of any explanation of how it is derived from Maxwell’s equations:

[5]     

In [5a], the variable [A] can be substituted for the magnitude, i.e. amplitude, of either the electric or magnetic field at some given point in space and time; while [5b] alludes to a relationship between the propagation velocity of an EM wave in vacuum and the electric and magnetic constants of free space.