Page 165 - Maxwell House

POYNTING's THEOREM 145

1
′
are ∝ and ∝ and therefore , ∝ −1 (). Here () = ()

,

and () = (). If −∞ < < 1 , → ∞ as → 0 and if −∞ < < 0 → ∞ as →

0. It means that the solution of Maxwell’s equations is not unique and we can get the infinite
choice of fields with different degree of singularity. Something is not quite right. To remove
this uncertainty let us recall one of the fundamental principles of electrodynamics stating that
the electromagnetic field energy within any finite domain is all the time finite. So look back at
equations (3.12) and (3.13) defining the energy stored in electric and magnetic fields and rewrite
them for the tiny cylinder (blue area in Figure 3.3.4a). The energy stored per length ∆ of this
cylinder is
2− 2 2 2+2
()/∆ ∝ �∫ 0 ()� ∫ ∝ (3.73)

0
Here the term in the braces is the constant independent of cylinder radii while the infinitesimal
volume element of the cylinder is = ∆. Making the similar assessment for magnetic
energy density we obtain

()/∆ ∝ ∫ 2−2 ∝ 2 (3.74)

0
Eventually, any physically reasonable solution of Maxwell’s equations must lead to the finite
energy volume meaning that according to (3.73) and (3.74) ≥ 0. The exact value of this
parameter depends on the internal angle of the wedge and boundary conditions on its surface.
In case of PEC wedge, -component is tangential to the top ( = 0) and bottom ( = )

wedge surface and thus it must be zero (see Table 2.2). The function

() ∝ sin () (3.75)

satisfies both conditions if

= (3.76)
2−
Then we can write the electric and magnetic field around the wedge vertex ( → 0) as

−
−
∝ 2−, , ∝ 2− (3.77)

A similar derivation can be carried out for the case of the magnetic field is parallel to the wedge
vertex

−
−
∝ 2−, , ∝ 2− (3.78)

According to these equations the tangential to sharp wedge vertex components of electric and
magnetic fields → 0 as → 0 whereas all normal components → ∞ or singular at the same
spot while < . Example of some typical wedges and the radial field variation around their
vertex are shown in Table 3.1.