Page 52 - Maxwell House

32 Chapter 1

called magnetic field strength. This vector plays the same role in the magnetic field description
as the displacement vector D in electric fields. The equity (1.54) is the second constitutive
relation in Table 1.7. In a vacuum, the only difference between vectors H and B is the
dimension dictated by the chosen SI units. We explain the importance of vector H later
analyzing the interaction of EM fields with magnetic moments of materials. Loosely speaking,
this additional vector is not required, and everything can be expressed in term of B-field alone.
Nevertheless, it is worthwhile to point out that according to (1.48) B-fields depend on each
magnetic charge, i.e. on each of the myriad of binding in the matter magnetic moments as well
as the external free ones playing the role of EM field source. If so, it is convenient all these
additional fields induced by the bound magnetic moments in a matter to count and include into
H-vector the same manner as for E-field, i.e. putting = . The coefficients of
0
proportionality taking into account such secondary magnetic fields is called the relative

permeability of material. We will come back to this subject later in Chapter 2.
The factor in the front of parenthesis according to Table 1.5 has the unit dimension [V/s] that
can be interpreted as time variation of magnetic current Δ ⁄ Δ = Δ ( )⁄ 2 of the same
0
dimension. Finally, let us multiply both sides of (1.54) by Δ and keep in mind that Δ = Δ
where Δ is the path that the electric charge Δ passes for time Δ. Then we have

2

Δ x = −Δ � − x � [J] (1.55)

2
It turns out that all terms in (1.55) according to Table 1.5 has the unit dimension of energy in
Joules and thus are measurable. If so, the magnetic field strength | | [A/m] can be defined as
the limit

|| = lim ∆ ⁄ Δ (1.56)

Δ →0
2
3
Here Δ = Δ (Δ) ⁄ ((Δ) ) and all the values Δ , Δ and Δ can be measured

0

experimentally. Later we will give more natural definition of | | through the electric current
induced by magnetic field in a small loop. Note that for the first time and quite naturally we
could replace the electric charge with some equivalent in action magnetic current in order to
simplify the equation. As we have pointed out before, such use of “no existing” equivalent field
sources is widely practiced in electrodynamics.
2
2
Now, look at two terms in parenthesis (1.55). Since the ratio ⁄ is dimensionless both terms
have the same units of [A/m]. Therefore, they both are described the magnetic field and (1.55)
can be written as
2
Δ
x = − � + � (1.57)

Δ 2
Here is the vector of magnetic field is due to the electric field existence

= − x (1.58)

Eventually, we can define the magnetic potential in [Amps] in the same way as electric

potential
= ∮ ∘ = − ∮ ( x ) ∘ = ∮ ( x ) ∘ [A] (1.59)