By R. Belmans
To obtain solutions for reallife field problems, a subsequent treatment of the Maxwell equations is necessary. The potentials are introduced to reduce the mathematical dimensions of the field problem. This approach results, for example for a line integral, in building a simple difference. To improve understanding of the field equations, a scheme is introduced representing the Maxwell equations. The various potentials are implemented in this scheme (Hafner47).
The four field equations in differential form can immediately be put into a schematic:
To keep magnetic and electric field quantities in the same column, the div and curl operators point to the right or left respectively. Consider the four following possible chain or arrow structures, with s a scalar and v a vector potential field.
The arrows represent the differential operators div, curl or grad respectively. A chain consists of at maximum two arrows. The second arrow always points to zero. Two arrows in one direction indicate a double derivative in the same direction in space.
Comparing this structured scheme with the structure of the Maxwell equations indicates the places where the scheme has to be completed by a potential. It is obvious that the fourth arrow chain can be completed by the vector potential A in the form B = V x A because the flux density B satisfies the zero condition for a divergence free field eq.(iii), (3.28).
Applying this relation to the first Maxwell equation yields
Completing with the electric scalar potential of a gradient field V and in arrow notation it is:
The complete scheme with the arbitrary potentials can now be written by:
To study the field equations further and to see the interdependences between the source terms and the potentials, an improved graphical scheme can be used, the diagrams of Tonti (Bossavit16).
Using the same schematic as in the last section for the differential operators, and introducing an additional arrow for the time dependency of the quantities, the following arrow system is obtained (Table 4.3):
Table 4.3. Tonti’s arrow system notation.
interdependency 
operator 
arrow 
geometry 
vertical 

material 
horizontal 

derivative with respect to time 
perpendicular to material arid geometrical arrow 

Fig. 4.6. Maxwell equations in Tonti’s arrow notation for a) the magnetic field and t>) the electric field.
The strong equivalence of the electric and magnetic fields is obvious using this notion (Fig. 4.6). The magnetic flux density B and the electric flux density D connect both field types. This results in the diagram of Tonti for the electromagnetic Field (Fig. 4.7) (Bossavit16).
The strong equivalence of the electric and magnetic fields is obvious using this notion (Fig. 4.6). The magnetic flux
Fig. 4.7. a) Tonti’s diagram for the electro magnetic field, b) by considering Ohm’s law.
If conducting material is assumed, the dependency of electrical field strength and current density can be considered in the diagram of Tonti directly by Ohm’s law (Fig. 4.7)
This equation, and respectively its arrow, represents the time independent current from eq.(ii), (3.27).
For ideal conductors we have the zero divergence condition, and therefore it can be written:
To complete the diagrams, the defined potentials can be added to it as indicated (Fig. 4.8).
Fig. 4.8. Diagram of Tonti with potential definitions.