Local to the surface, we define inward normal and tangential orthogonal coordinates, h and j, as in Figure 2.1. From Figure 2.1 it is clear that, when the curvature is positive, the scale factor corresponding to j in an orthogonal system hj is a decreasing function of h. In fact, when j represents the arc length and the contour of the conductor’s surface, the Lame coefficients can be written in the form

hj ¼ 1 h=d; hh ¼ 1; hx ¼ 1 (2:3)

where d¼ d(j) is the local radius of curvature. With these new coordinates, the curl, div, and curl curl operators applied

to an arbitrary vector function~f take the form

(r~f )x ¼ 1


@(hjfj) @h

@(hhfh) @j

¼ 1 1 h=d

@((1 h=d)fj) @h

@fh @j

¼ 1 1 h=d (1 h=d)

@fj @h fj

d @fh


¼ @fj

@h fj d h

d d h

@fh @j


(r~f )j ¼ 1


@(hxfx) @h

@(hh fh) @x

¼ @fx

@h (2:5)

(r~f )h ¼ 1


@(hj fj) @x

@(hxfx) @j

¼ d

d h @fx @j


r ~f ¼ 1 hxhjhh

@(hjhh fx) @x

þ @(hxhh fj) @j

þ @(hxhj fh) @h

¼ 1 1 h=d

@fj @j þ (1 h=d) @fh

@h fh


¼ d

d h @fj @j þ @fh

@h fh d h (2:7)




where d0 denotes the derivative of d with respect to j. The characteristic lengths associated with the coordinates j and h are the

characteristic size D of the conductor’s surface and the skin depth d, respectively. These lengths may be of different orders of magnitude due to the condition in Equation 2.1. Thus, it is natural to introduce nondimensional variables, ~j and ~h, that have variation ranges of the same order of magnitude and are related to j and h, respectively, as follows:

~j ¼ j=D; ~h ¼ h=d (2:11)


Here and below, the sign ‘‘’’ denotes nondimensional quantities. Switching to variables ~j and ~h in Equations 2.4 through 2.10, we obtain

(r~f )x ¼ d1 @fj @~h

fj d d~h

d d d~hD

1 @fh @~j


(r~f )j ¼ d1 @fx @~h


(r~f )h ¼ d

d d~hD 1 @fx

@~j (2:14)

r ~f ¼ d d d~hD

1 @fj @~j þ d1 @fh

@~h fh d d~h (2:15)

r (r~f ) h i

x ¼ d ~hdd

(d d~h)3D 1 @fx

@~j d

(d d~h)2D 2 @

2fx @~j2

þ 1 d d~hd

1 @fx @~h

d2 @ 2fx

@~h2 (2:16)

r (r~f ) h i

j ¼ d (d d~h)2D

1 @fh @~j þ d d d~hd

1D1 @2fh @~h@~j

d2 @ 2fj


þ fj (d d~h)2 þ

1 d d~h d

1 @fj @~h


r (r~f ) h i

h ¼ d ~hdd

(d d~h)3D 1 @fh

@~j d

(d d~h)2D 2 @

2fh @~j2

þ dd 0

(d d~h)3 fj

d (d d~h)2D

1 @fj @~j þ d d hD

1d1 @2fj @~j@~h


Using Equation 2.1, let us introduce a small parameter ~p proportional to the ratio of the penetration depth to the characteristic size of the conductor’s surface:

~p ¼ d=D ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t=(smD2)

p 1 (2:19)

The second relation in Equation 2.11 can be represented in another form using Equation 2.19:

~h ¼ h=(~pD) (2:20) The local radius of curvature, d, is directly related to the variation of the function,~f , in the direction tangential to the surface of the conductor. This leads to the following representation:

~d ¼ d=D (2:21)

Substituting Equations 2.20 and 2.21 into Equations 2.12 through 2.18, we get

(r~f )x ¼ d1 @fj @~h ~p fj

~d ~p~h ~p

~d ~d ~p~h

@fh @~j

" # (2:22)

(r~f )j ¼ d1 @fx @~h


(r~f )h ¼ d1~p ~d

~d ~p~h @fx @~j


r ~f ¼ d1 ~p ~d

~d ~p~h @fj @~j þ @fh

@~h ~p fh

~d ~p~h

! (2:25)

r (r~f ) h i

x ¼ d2 ~p3 ~h


(~d~p~h)3 @fx @~j ~p2


(~d~p~h)2 @2fx @~j2

þ ~p ~d~p~h

@fx @~h @