# Dependence of resistance on cross section area

The electrons flow on the outer surface of the resistor, why then does the resistance of a resistor depend on it's cross sectional area and not on it's perimeter?

## Answers and Replies

DEvens
Gold Member
That is only true for certain kinds of resistors in certain regimes. For example, it may be true for high frequency electronics where the current is mostly at the skin of the conductor.

I though all metals have this property... Anyway if I take these types of materials, where the flow of electrons is at the skin, and double the width of the wire, will I get half the resistance or a quarter of the resistance? That is, is resistance linearly dependant on the cross section or on the perimeter?

I think it should be perimeter, but all textbooks show that R = ρ*L/S, where R is the resistance, ρ is specific resistance, L is the resistor length, and S is the cross section.

NascentOxygen
Staff Emeritus
I though all metals have this property... Anyway if I take these types of materials, where the flow of electrons is at the skin, and double the width of the wire, will I get half the resistance or a quarter of the resistance? That is, is resistance linearly dependant on the cross section or on the perimeter?
I presume roughly proportional to perimeter for frequencies/conditions where skin effect dominates. (I think I've read that skin effect can be seen on massive power transmission cables, such that current density near the core may be, say, half what it is nearer the surface. So a high tensile steel core need not reduce an aluminium cable's resistance appreciably.)

all textbooks show that R = ρ*L/S, where R is the resistance, ρ is specific resistance, L is the resistor length, and S is the cross section.
That's because they are dealing with ordinary bulk properties under DC and low frequency conditions.

Resistance is measured with DC current. When a conductor is subjected to DC, the current uses the entire cross section. It is only when you start using AC that the current stops using the inner part of the conductor (read up on skin depth)...in this cases, you need to find the relationship between your DC-resistance and AC-resistance to ease further calculations...this relationship will depend on the frequency you are working with and the cross-sectional geometry of your conductor.

I though all metals have this property... Anyway if I take these types of materials, where the flow of electrons is at the skin, and double the width of the wire, will I get half the resistance or a quarter of the resistance? That is, is resistance linearly dependant on the cross section or on the perimeter?

I think it should be perimeter, but all textbooks show that R = ρ*L/S, where R is the resistance, ρ is specific resistance, L is the resistor length, and S is the cross section.
If the skin depth is small compared with the diameter yes, the resistance is approximately inverse proportional to the diameter (and not diameter squared).
See here for example:
http://chemandy.com/calculators/round-wire-ac-resistance-calculator.htm

The electrons flow on the outer surface of the resistor, why then does the resistance of a resistor depend on it's cross sectional area and not on it's perimeter?
Consider the flow of electrons through a wire analogous to the flow of water through a pipe.
More the cross-sectional area, more the amount of water that can flow in a specific time. Lesser the CSA, lesser the amount of water, i.e., more the resistance.