Why is resistance (inversely) proportional to area instead of circumference?

TL;DR From the Veritasium-vs-ElectroBOOM debate about the 1 m / c question, it seems to me that most properties about conductors depend only on surface charges. So shouldn't quantities like resistance $$R = \frac{\mathrm{resistivity} * \mathrm{length}}{\mathrm{cross\ sec\ area}}$$ depend on circumference instead of area?

Context

This comes from the following banter between Mehdi and Derek, if you're not aware:

1. "The Big Misconception About Electricity", Veritasium
2. "How Wrong Is VERITASIUM? A Lamp and Power Line Story", ElectroBOOM
3. "How Electricity Actually Works", Veritasium
4. "How Right IS Veritasium?! Don't Electrons Push Each Other??", ElectroBOOM. This is the one I care most about for this question, but the rest are context.
5. "Testing the LONGEST LOOP OF WIRE!!! to Turn a Lamp On", ElectroBOOM

Explanation

From what I understood, the argument concluded from the videos is that only the surface electrons (i.e., on the outer "cylinder" of the wire) contribute to the electric fields, and thus properties like current and energy flow.

If this is the case, it should follow that we don't care about what the electrons/charges in the middle of the wire are up to.

For example, it is understood that resistance is lower if there is more cross-sectional space (I chose an ambiguous word for a reason, please bear with me) in the wire. It is known that

Resistance $$R = \frac{\mathrm{resistivity} (\rho) * \mathrm{length} (l)}{\mathrm{cross\ sec\ area} (A)}$$ Or for our purposes, $$R \propto \frac{1}{A}$$ (if the other two are constant). But why - shouldn't it be $$R \propto \frac{1}{C}$$ (where C is the circumference of the wire), from the argument above?

I know I'm going wrong somewhere, simply because of how well established this $$R = \frac{ρL}{A}$$ relation is. Also, from my argument, multi-strand wires should have lower resistance than a single-core wire (both of the same gauge) since it has more surface area, but this isn't what we observe. What am I messing up?

• I believe the question is related to skin-effect. Current density distribution depends on the frequency of current. The formula neglects it or assumes DC. Commented Sep 16, 2023 at 9:24
• Surface charge gradients are what motivate current. These are set up by applied voltages. But the current itself is charge motion within the entire conductor (near DC anyway.) Commented Sep 16, 2023 at 10:15
• I liked this question. Particularly when we learn to derive expressions for capacitance of various shapes as well as skin effect when dealing with electromagnetic waves of sufficiently high frequency, we always keep mention absence of non-zero electric field inside a (perfect?) conductor. I suggest OP to consider reformatting this question here or ask it in the Physics SE. I would not be surprised if this has already been asked here or at Physics.
– AJN
Commented Sep 16, 2023 at 14:09
• 1, 2, 3, 4, 5, 6. You are definitely not alone in this line of thinking. there might be more questions asked along this line here as well as Physics SE.
– AJN
Commented Sep 16, 2023 at 14:16
• 1 If the physics SE questions already answer your question, then this question can be deleted IMHO. 2 I am not sure how to improve or reformulate this question (I didn't vote to close). To me, the question is okay since you provided context (via links); you provided a brief description of the take aways from the linked videos, so that people are not forced to watch the videos; You also asked a clear question (why formula A instead of B). 3 If you have chat or meta access, try asking for suggestion for improving this question at EE and Physics chat pages.
– AJN
Commented Sep 17, 2023 at 15:35

DC current flows equally through the conductor’s area.

AC current is “pushed” to the edge due to skin effect at higher frequencies. Look this up and you’ll find what you’re talking about.

Planar shapes and litz wire (wire with individually insulated strands) are used at higher frequencies to reduce R.