# Resistance = resistivity * L/A...

#### dtvonly

Joined Dec 14, 2012
43
Hi. the equation R=resistivity * (L/A) does make the resistance directly proportional to the length. It also make it inversely proportional to the cross sectional area. this does not make sense to me. Given the same wire and length, the bigger (cross sectional area) wire should have a larger resistance. But according to the equation, this works in reverse. Common sense doesn't "make" sense. I am a little confuse. please clarify. Thanks.

#### timwhite

Joined Apr 10, 2014
50
Common sense cannot really be applied to this case because electrons don't flow like we would think.

This quote from the Wiki page on resistivity explains this reasonably well:

In metals - A metal consists of a lattice of atoms, each with an outer shell of electrons which freely dissociate from their parent atoms and travel through the lattice. This is also known as a positive ionic lattice.4
This 'sea' of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (a voltage) is applied across the metal, the resulting electric field causes electrons to move from one end of the conductor to the other.
Near room temperatures, metals have resistance. The primary cause of this resistance is the thermal motion of ions. This acts to scatter electrons (due to destructive interference of free electron waves on non-correlating potentials of ions)[citation needed]. Also contributing to resistance in metals with impurities are the resulting imperfections in the lattice. In pure metals this source is negligible[citation needed].
The larger the cross-sectional area of the conductor, the more electrons per unit length are available to carry the current. As a result, the resistance is lower in larger cross-section conductors. The number of scattering events encountered by an electron passing through a material is proportional to the length of the conductor. The longer the conductor, therefore, the higher the resistance. Different materials also affect the resistance.

#### tshuck

Joined Oct 18, 2012
3,534
A larger cross-sectional area means more chances for electrons to move in the direction imposed by the applied stimulus/voltage.

Perhaps you could explain why the larger cross-section "should" have a larger resistance?

#### Sparky49

Joined Jul 16, 2011
833
Think of the wire as a water pipe, and the current as water flowing through the pipe. If you make the pipe's A smaller, it's going to be harder to get the same amount of water through in the same time. If you make it bigger, then the same amount of water can be pumped through, with less effort.

Technically, this analogy isn't 100% accurate, but the results are the same.

#### WBahn

Joined Mar 31, 2012
26,398
Hi. the equation R=resistivity * (L/A) does make the resistance directly proportional to the length. It also make it inversely proportional to the cross sectional area. this does not make sense to me. Given the same wire and length, the bigger (cross sectional area) wire should have a larger resistance. But according to the equation, this works in reverse. Common sense doesn't "make" sense. I am a little confuse. please clarify. Thanks.
Can you articulate why the larger diameter wire should have more resistance?

Imagine taking a small insulated wire that is, say, 1m long and, for discussion purposes, has a resistance of 100Ω. This is simply a resistor, right. Now, take two of these resistors and put them in parallel. Does it make sense that, like any two identical resistors in parallel that the net resistance is half, or 50Ω in this case? Now take 1000 of these resistors (wires) and put them in parallel. The net resistance would be 0.1Ω, right? Now imagine simply removing the insulation and putting all the wires up against each other. The result is really nothing more than a stranded wire made up of 1000 strands with the overall wire having 1000 times the area of a single strand and 1/1000 the resistance of a single strand. Now just imagine a solid wire that has the same 1000x the area as being 1000 individual wires, perhaps hexagonal in cross section, all butted up against each other.

#### WBahn

Joined Mar 31, 2012
26,398
Common sense cannot really be applied to this case because electrons don't flow like we would think.
Why can't common sense be applied to this case. It works just fine.

#### crutschow

Joined Mar 14, 2008
28,538
Hi. the equation R=resistivity * (L/A) does make the resistance directly proportional to the length. It also make it inversely proportional to the cross sectional area. this does not make sense to me. Given the same wire and length, the bigger (cross sectional area) wire should have a larger resistance. But according to the equation, this works in reverse. Common sense doesn't "make" sense. I am a little confuse. please clarify. Thanks.
If by "common sense" you mean a larger wire would have more resistance, then, by that logic, you could use an infinitesimally small wire to carry thousands of amps with little resistive power loss and power lines could be microscopic in size. Since that's obviously not the case, then it would seem your common sense meter needs some recalibration.