The purpose of this post is to briefly describe the use of vectors in modern versions of Maxwell’s equations.
Vectors are a mathematical models whose conventions are (mostly) aligned with experimental evidence of electromagnetic fields.
Modern versions of Maxwell’s Equations can be written in several forms depending on the unit of measurement being used, or whether they are relativistic or not. The version of Maxwell’s Equations below are taken from Feynman (1965)
An example of a modern version of Maxwell’s Equations (Feynman 1965).
On the left hand side of Maxwells equations are symbols describing the “divergence” and “curl” of the electric and magnetic fields. On the right hand side of Maxwells equations are the “inputs” (for flux) into these fields.
The follow section explains some of the meaning of the notation, originally derived by Maxwell and Heaviside.
Fields and Vectors
A “field” is any physical quantity which takes of different values at different points in space…. the relationships between the values of the field at one point and the values at a nearby point are very simple. With only a few such relationships in the form of differential equations we can describe the fields completely – Feynman 1965. V2.5
Vector is a direction and quantity at a position in space, this is in contrast to a scalar which is a single value at a position in space.. Vectors E, B and j are shown in bold above.
A vector “field” is not an actual field. A vector field can only represent an electromagnetic field as infinitesimal points in space having both direction and magnitude. The space that is assumed can be Euclidian, three dimensional or a surface.
Operators
The ∇ symbol is a abstract three dimension operator, that represents the change in a vector or scalar. The ∇ is “abstract” because it does not define the actual operand (or thing) that has the location, direction and value in the vector.
Before describing left hand operations it’s worth listing right hand operators in the equations.
These are sometimes call the gradient, divergence (.) and curl (x) of the vector.
Gradient of a Vector, ∇T
The gradient operator ∇T means the vector T in two or three dimensions, known as the gradient of the field, where the gradient is a single value.
A example gradient looks as follows (IkamusumeFan 2014):-
Divergence of a Vector, ∇ . V
The divergence operator ∇ . V (or div V) means the three dimensional flow of the vector to or from a point.
For example (Guilmette 2012).
A positive value means a flow from point. A higher positive value means stronger flow outwards. A negative value means a flow toward a point. A higher positive value means stronger flow outwards. A zero value means a flow past a point.
Curl of a Vector, ∇ × V
The curl operator ∇ × V (or curl V) means the three dimensional rotation or twisting of field in a unit area.
For example (Guilmette 2012).
A positive value means an anti-clockwise. Higher positive values means stronger curl. A negative value means a clockwise. Lower negative values means stronger curl. A zero value means it is “conservative”, that is there is uniform flow (e.g. a straight line) and there is no twisting or circulation.
On the right hand side of Maxwells equations are a number of other operators. Addition and subtraction, is basically superposition. Conveniently forces matches vector addition. Multiplication and division is related how much of something is spread over a space and time. Importantly superposition and spread both take place across a three dimensional space which is the unit area for divergence and curl. These four operators define the underlying physics behind electric and magnetic fields.
Flux, Circulation and Force
The net outward flow – or flux – is the average outward normal component of the velocity times the area of the surface (Feynman 1965):
Flux = (average. normal component) . (surface area)
Where this flux flows in a circle [closed loop], in a tube bounded by a surface, then it is called the circulation. The circulation around any imagined closed curve is defined as the average tangential component of the vector multiplied by the circumference of the loop – Feynman 1965.
Circulation = (average tangential component) . (distance around)
The force of the vector field is the sum of the vector fields though superposition. Force is the measurable output effect of a field. This is true of electric and magnetic fields. Force is the output force.
In other words a fields can be seen as being made up of a flux (the inward flow), circulation (closed loops) and force (the outward flow).
Maxwell’s Equations
The modern version of Maxwell’s Equations, used by Feynman (1965), relies on defining only the electric and magnetic force and assumes the model of the vector fields is built into he maths.
In these equations E is the Electric field and B is the Magnetic field.
Equation 4.1 is an electrostatic equation because there is no change with respect to time (i.e. ∂B/∂t) on the right hand side. This equation shows the electric divergence, in otherwords how much electricity flows away (or toward) its source.
This equation is calculated as (p/ε0). Where p is electric charge density and ε0 is a constant. In other word the force of the electric field moving away from the source is directly related to the size of the density of the charge.
The constant ε0 (ε0 = 8.854×10−12 ) (epsilon-naught), measures the permittivity of a vacuum to electric currents. ε0 is a constant that allows us to relate the length and time to the electric charge.
ε0 is defined (can be expanded) as follows.
Where u0 is The magnetic constant or vacuum permeability. u0 is defined as inductance 4π × 10−7 H/m, where H = Henry (m2·kg·s-2·A-2) and c2 is the constant speed of light 299792458 m⋅s−1 (Mohr 2008).
Equations 4.2 and 4.3 shows that the electric and magnetic fields only influence each other when the other is changing because the right hand part of the equation contains change with respect to time (i.e. ∂B/∂t) of E in the B equation (4.2) and B in the E equation (4.3). Also equation 4.2 and 4.3 both show that curl in the electric field results from movement in the magnetic field and that curl in the magnetic field results from the change in the electric field.
Equation 4.2, starting with ∇ x E, shows the divergence of the electric field. That is the force of the curl. This is the only equation starting with E that contains B, and that B is changing in respect to time, (hence the dt). Therefore a change in the magnetic field leads to a negative (clockwise) curl in the electric field. This is basically Flemings left-hand rule.
Equation 4.3, starting with ∇ x B, shows the divergence of the magnetic field. That is the force of the curl. This is the only equation starting with B that contains E, and that E is also changing in respect to time (hence the dt). The second part of equation 4.3 adds (j/ε0) to the changing value. Where j is the electric current density which is the rate at which the charge flows.
Equation 4.4 is a magnetostatic equation because there is no change with respect to time (i.e. ∂B/∂t). This equation shows that the magnetic field has no divergence. In other words it never flows away (endlessly) or towards (endlessly) a point. It always flows from a point and back to that same point. There is therefore no transfer of energy.
Fluxes and Forces
The modern version of Maxwell’s Equations differ from those developed by Heaviside because Heaviside used the term for D and B.
The Electric force (E) is conceived to be producing or produced by the density of a flux; called the electric displacement (D) [or electric induction]; and similarly magnetic force (H) as producing a second flux, magnetic induction (B).
However the value of the flux is dropped in the modern vector notation (see below), because the vector notation makes assumptions that make the values unnecessary, and only shows what is measurable and therefore only needs to correspond with force.
The flux is still needed to explain the phenomena Verbally (e.g. Feynmann 1965 – Chpt 28) but is hidden away using the convention of the operators.
Caveats
It is worth pointing out that Maxwell’s equations in their modern formats are not complete. For example, this incompleteness includes how energy transfers from electrostatics and magnetostatics to electromagnetic planar waves and how slow moving electric and magnetic waves behave. The reason this is important for a theory of consciousness is that slow moving ions are related, if not the basis of consciousness, so this aspect of the model needs to be examined in greater detail.
It is also worth pointing it that Maxwell’s equations are written as mathematical notation and the operators reference physical processes whilst the operands reference physical things. The operands (things) are ultimately defined (but not necessarily are) by measurable Standard Units of length, meter (m); mass, kilogram (kg); time, second (s); electric current, ampere (A); temperature, kelvin (K); amount of substance (mole- mol); luminous intensity (candela cd).
The operands are (somewhat) independent and related to one another by physical constants. For example the relationship between space and time involves the speed of light constant. Importantly the operators follow conventions, which are laws, which define how the values of the operands can change from one to another. These conventions, are not written into Maxwell’s Equations, and include laws such as the first law of thermodynamics that the total energy in the operands cannot change.
The combination of operands and operators in the Maxwell’s equations say as little as possible about the configuration of operands, so there is the possibility of complex configurations evolving to produce complex activity.
Consciousness
It is the contention of the Field Theory of Consciousness that consciousness is the field between the flux, and circulation and the force.
If consciousness is inside a field and the laws of physics will need to hold true, then the “lines of force” concept where there are direct lines between poles needs to be replaces with a lines of flux and circulation . Later posts will show how lines of force are accurate in several situations, including slow moving electric and magnetic fields, sub-wavelengths and the interaction between fields in multiple transmitters and receivers. The “lines of force” concept whilst parsimonious for the purposes of measurement, will be shown to need to admit the possibility of a lines of flux and circulation.
Fields of Consciousness 