Skin depth and cross section area

Question

I am really confused by a paragraph i am reading about skin depth. It gives the example of a conducting strip with height $a$, length $L$ and thickness $d$.

It states that for a direct current the area for the resistance is $A = ad$ giving $R=\frac{L}{\sigma a}$ where $\sigma$ is conductivty.

Thats easy enough to follow, but then it says for a much higher frequency the skin depth is much smaller than the cross sectional area so the area for the resistance becomes $A = 2a\delta$ where $\delta$ is skin depth. So we get $R = \frac{L}{2\sigma a \delta}$

I am struggling to understand how they got this second area for a higher frequency resistance, its not explained at all. Why $2\delta$, I can't figure it out.

Hope some one is able explain the derivation so I understand what is going on.

Thanks

Answer 1

Since you describe your shape as a “conducting strip,” I am guessing you have the constraint $d \ll a \ll L$, so that the whole strip is shaped like a bookmark or a stick of chewing gum.

In that case the skin effect, which limits the charge flow to within some distance $\delta$ from the conductor’s surface, would contribute cross-sectional area $2a\delta$ from the two long sides, but only a negligible $2d\delta$ from the two short sides.

In cross section, if your “strip” looks like

               width a
-----------------------------------
|                                 | thickness d
-----------------------------------
               area a·d

then when the skin depth becomes shorter than the thickness, the conducting area might become

               width a
=================================== upper: a·𝛿
||                               || thickness d
=================================== lower: a·𝛿
left: d·𝛿                 right: d·𝛿

and so the effective conducting area is $2(a+d)\delta ≈ 2a\delta$.

If your conductor were shaped more like a rod with $a\approx d$ than like a strip with $a\gg d$, you would get to include the skin effect volume on all sides. By using a conducting strip, you keep the total resistance roughly constant while the skin depth is larger than the strip’s thickness, $\delta \gtrsim d$.