Re: How does the length of a piece of wire affect its resistance, and why?

Mark,

The short answer is: the longer the wire, the larger the resistance.

More detail: Resistance is a measure of how difficult it is to push electrons (the particles that carry electrical current) through a wire. The larger the resistance, the more force you have to apply and the more energy you expend to produce a current. The process is analogous to the resistance you feel if you try to blow water through a straw or a hose. You have to blow harder to push water through a long hose, and you also have to work harder if the hose if very small in diameter. Many of the same equations we use to describe fluid flow ("fluid" meaning gas or liquid) through a system of hoses and pipes can be used to describe the flow of electrical current through a circuit.

The resistance of a wire is: R=r*L/A. L is the length of the wire, A is the cross-sectional area of the wire, and r is the"resistivity" of the material. Long, thin wires have the most resistance,just like long, thin pipes. All the subtle physics is neatly packaged in that little "r" symbol. (In most text books it is the Greek letter rho, but I don't know how to make HTML speak Greek.) Resistivity is basically a measure of how difficult it is for the electrons to travel through a given material.

The essential question is, how much do the electrons interact with the atoms that make up the material? This depends on things like the electronic structure of the material (conductor, semi-conductor, or insulator?), the physical structure (nice crystal, cracks and defects, or amorphous blob?), and the temperature (atoms "frozen" in place, or wiggling around madly getting in the way?). The more they interact with, or "bounce" off of, the atoms, the harder it is to keep them going in the desired direction.

Sidebar on the fluid analogy (sorry, but I find such things to be the most beautiful part of physics. Feel free to skip it if you are already overloaded):

For gas and water flow, traditionally we invert the equation above, and discuss "conductance" C, and "conductivity" c instead of "resistance" and "resistivity." Just a historical convention. I guess the first water engineers preferred to talk about how easy things were, while the electrical engineers dwelled on how hard life was. :-) Mathematically, C=1/R, c=1/r. Conductance can then be expressed as C=c*A*A/L. The area comes in twice for pipes because the resistance to water flow comes primarily from the surface of the pipe, rather than its interior like electricity, so having more open area for flow is a bigger help. Aside from that, "c" still holds all the subtle physics about the interaction of the fluid and the pipe: pressure, temperature, viscosity, and surface roughness, just like "r" held all the interaction between electrons and the wire.

Once you have the resistance or conductance calculated, the equations governing flow through pipes and water are exactly analogous. Take Ohm's law: V=IR. Voltage equals current multiplied by resistance. Replace "voltage" by "pumping speed," "multiplied by resistance" with "divided by conductance", and redefine current for fluid instead of electrons, and you have the equation governing a pipe and a pump instead of a wire and a battery.

It's true: all of life is about plumbing. :-)