# Skin depth and cross section area

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

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, $$\delta \gtrsim d$$.

• Is there any chance you can add a visual diagram - I find it a lot easier to understand this topic with visuals.
– WDUK
Commented Apr 29, 2022 at 7:51
• Maybe? I’m not much for computer graphics, but I’ve given it a shot.
– rob
Commented Apr 29, 2022 at 9:16
• From your diagram which direction are we assuming the waves propagate ? I assume left to right or right to left. But if so why is there an upper and a lower cross section involved? Same question if its vertical propagation too. Assuming i've understood the diagram correctly.
– WDUK
Commented Apr 29, 2022 at 22:06
• Oh wait, i actually think i understand the diagrams now, its an outer layer around the whole strip that the current is flowing through at high frequency.
– WDUK
Commented Apr 29, 2022 at 22:10
• Yes, the “long” direction comes out of the page.
– rob
Commented Apr 30, 2022 at 3:45