238 Duals and analogues10.3 ANALOGUES
If an equation describing the operation of a physical system is identical in form
to one describing the operation of another physical system, then the corre-
sponding quantities in each equation are said to be analogous. The equations
themselves are also analogous. The form of the equations and the associated
mathematical manipulation are the important consideration, not the physical
similarity (or otherwise) between the systems.
There are two major advantages of analogous systems. One is the saving in
memory space resulting from the form of equations being identical in two or
more systems. The second is that, when dealing with mechanical/electrical
analogues, for example, it is possible to express the whole system in an
integrated form mathematically.
Electric, magnetic and conduction fields
The field vectors of the electric field are related by the equation
D = eE (10.5)
where D is the electric flux density, E is the electric field strength and e is the
permittivity of the medium of the field.
Similarly, for the magnetic field we have
B =/xH (10.6)
where B is the magnetic flux density, H is the magnetic field strength and ~ is
the permeability of the medium of the field.
For the conduction field
J : o-E (10.7)
where J is the current density, E is the electric field strength and ~r is the
conductivity of the medium of the field.
We notice that Equations (10.5), (10.6) and (10.7) are identical in form and
are said to be analogous. Similarly the corresponding quantities (D, B and J; E
and H; e, ~ and ~r) in each of the equations are analogues. Any one of these
equations could be obtained from one of the others by replacing each quantity
of the second system by the corresponding analogue from the first.
Example 10.3
The energy stored in an electric field is given, in terms of the field vectors, by
the equation W - DE/2 joules per cubic metre of the field. By consideration of
field analogies obtain an expression for the energy stored per cubic metre in a
magnetic field.