Question

Alternating current flows parallel to the surface of a semi-infinite slab of conducting material. At what depth is the current density reduced to (a) 10 per cent and (b) 1 per cent of the surface value? At what depth is the phase of the current density (c) 90°, and (d) 180 lagging on the surface value? (Answer (a) 2.34, (b) 4.64, (c) 1.574, and (d) 3.141.)

Answer 1

a) The amplitude of current inside a conducting material exponentiallu decreases becuse of the facot

$e^{-d/\Delta}$

where Delta is the skin depth

When this is 10 % = 10/100 = 0.1

$e^{-d/\Delta}=0.1$

$-d/\Delta=\log(0.1)=-2.3$

or

$d=2.3\Delta$

b) When the amplitude is 1 % = 1/100 = 0.01

$e^{-d/\Delta}=0.01$

or

$-d/\Delta =\log(0.01)=-4.6$

or

$d=4.6\Delta$

c) The phase is

$d/\Delta$

when this phase is 90 degrees = Pi/2 radians

$d/\Delta=\pi/2=1.57$

or

$d=1.57\Delta$

d) When this phase is 180 degrees = Pi radians

$d/\Delta=\pi=3.14$

or

$d=3.14\Delta$