# Why do we prefer resistance over conductivity in describing Ohm's law?

In my mechanical engineering training , I've noticed that many constitutive relations (e.g. fourier law, poiseuille law, hooke's law) involve a flux which increases as the material parameter increases. In heat transfer and fluid flow, this is called a "conductivity". In solid mechanics, the stress increases as some elastic moduli increase.

Given this trend, I found it perplexing that in electrical engineering the preference is to use resistance rather than conductance, with current increasing as resistance decreases in ohm's law. Is there an advantage to using resistance instead of conductivity in electrical engineering in general? Or is it just traditional notation handed down the great minds who came (and published) before us?

• It may be because it is useful in analogy. In design thermal solutions for which neither the power to dissipate nor the temperature rise in the working fluid is known, the thermal resistance stack budget is used. It allows comparison and optimization of designs without required environment and system inputs. Jun 15 '15 at 21:17

I think your electrical/mechanical analogy is not quite right. If you think about DC electricity and mechanical statics, it seems reasonable that voltage ("electromotive force") is the analogy of mechanical force, and it is then tempting to say Ohm's law $RI = V$ is the analogy of $Ku = F$. But if you try to combine mechanical springs in series and parallel, and similarly electrical resistances, that "analogy" soon falls apart. Really, I think it is just a coincidence that two simple equations look similar to each other.

If you consider an AC electrical circuit and steady-state mechanical dynamics, and write the electrical equation in terms of charge not current, the real analogy becomes clear:

Mechanical: $(-M\omega^2 + Ci\omega + K)ue^{i \omega t} = Fe^{i \omega t}$.

Electrical: $(-L\omega^2 + Rj\omega + 1/C)qe^{j \omega t} = Ve^{j \omega t}$.

In other words, the analogy of electrical resistance is really mechanical damping, since they are the only things each equation that dissipate energy. Electrical capacitance is the analogy of mechanical compliance (the inverse of stiffness), and electrical inductance is the analogy of mechanical inertia.

The correct analogy of mechanical statics $Ku = F$ is electrostatics: $(1/C)q = V$.

The laws that you mention (e.g. Fourier law, Poiseuille law, Hooke's law), which involve a flux which increases as a material parameter increases, are interesting examples. Considering the original version of Ohm's Law was also phrased in terms of conductance, not resistance. This implies that in experimental physics, it's easier to watch something that's easy to change, and plot a linear relationship to something that changes, then it is to watch something that's difficult to change, and plot it against something that's easy to change.

It makes sense. You pull on a bar, and it grows. It's easy to change the "grow" by a set amount and stop, but difficult to get an exact pull force. You heat one end of the bar, and the heat spreads. It's easy to measure and you can stop when you reach a certain temperature. But you don't know how much heat you had to put in to make that happen.

It's also easy to apply different exact voltages to a wire, and watch the current change. Note the slope of that experiment would be the conductivity, not the resistance.

Why are you doing these experiments? So some day, you can make a bar that doesn't break when the load makes it grow. But, for now, you plot the thing that you changed v. the thing that also changed. And you call the slope "E", or "k", or "$\sigma$".

However, in engineering, it's different. While I use Hooke's law to figure out that bending my beam will lead to a displacement, inevitably I'm dividing by EI. Because I don't want my beam to deflect that much, I need a very large EI - but a small inverse would be better. I really wish I didn't have to divide - when I use a calculator it's easier to type $x_1 * x_2 * x_3$ v. $x_1 / (x_2 * x_3) * x_4$ - I'd rather we rate materials by 1/E, then by E, and the inverse of section modulus as well.

When calculating out fluid flow in a pipe, I'm always sizing my pumps not by dividing by the "flowability" of pipes, but adding up the friction factor resistance. I'm sizing my moving system to overcome this resistance, not to find out how my smooth pipes are helping the flow.

In heat transfer, I'm always dividing by the thermal conductivity. They even invented R-value on insulation so you could figure out the change in temperature between surfaces easier.

In electrical engineering, they didn't start with resistors - those are a manufactured device. So, they listed the value that mattered more to the engineers, and less to the physicist - the resistance. Unfortunately for Mechanical Engineers, we started with pipes and bars, so we keep with the same notation.

• Also, temperature always seemed off to me. It seems we really should use inverse of temperature instead. Especially since you can't even reach absolute 0 - just like you couldn't reach infinite thermal inverse.
– Mark
Jun 25 '15 at 23:50

I would suggest, without any analogy, that resistance is a more obvious and clear concept to explain and understand. How hard is it to push electronics through? While conductance seems less intuitive, at least to me. Additionally, it is not unusual to have 0 (or at least close enough to assume 0 for most purposes) of resistance. For example for typical analog circuits the wires, or other conductors, can be assumed to have 0 resistance. While working with conductance, you end up with a bunch of infinite values floating around. That can make the math a bit more sloppy.

• It is just as simple to explain the same phenomenon with a large conductance value where no resistance exists and zero conductance where infinite resistance exists. Are you suggesting that the case of zero resistance is more common than infinite resistance? Also, what makes the math so much more "easier/clearer" with resistance?
– Paul
Jun 25 '15 at 16:33
• @Paul In many (analog) systems, the component spacing as well as track/wire width means that the resistance between two parts can be safely assumed to be zero. It's also much more pragmatic in electrical engineering to view a circuit from the lens of "where do I have the greatest losses" rather than "where are the most conductive parts of my circuit". They are the same concept, but the standpoint and entire focus of optimization changes. Jun 25 '15 at 18:01

It seems most people here are tackling the subject from the standpoint of analogies, so as an electrical engineer who's trying to learn mechanical (the other way around) I'll take a different perspective.

It's more that resistance is much more readily usable in electrical engineering than conductance, from a practical standpoint. There is no doubt, the use of resistance gives way to many interesting derivations of Ohm's law (i.e. $P = I^2 R$) and this use of resistance rather than conductance allows for calculation of many practical, but mundane things, such as power dissipation of circuit elements. Here, there are two reasons to decreasing such dissipation - first, the increase in efficiency, and second, the decrease in heat generation, which is by far the driving factor, especially in the semiconductor era. There is no reason why you can't perform the equivalent operation $P = \frac{I^2}{G}$, but in practicality you'll almost never see that.

By virtue of that fact, the calculations are made somewhat simpler or more readily-understandable using resistance (to the extent that people don't like reciprocals). But, the formulas generally favor resistance and its AC/complex counterpart impedance, rather than conductance and admittance, although both varieties do exist.

It seems I finally hit the nail on the head in the comment above, but resistance has the foothold in EE because of not only historical perspective (people understood the concept of resistance first, conductance came rather second), but because the end goal of design is to release an optimal system. Because of this, one works to cut out or reduce losses rather than improve gains. It should also fit that if one is building an LED circuit, for example, it is easier to understand to fit the LED using a current-limiting 50 ${\Omega}$ resistor rather than the equivalent "20 mS conductor".

So, to finally answer your question, I suspect that a large part of it is tradition, granted that both concepts are simply right- and left-handed versions of the same principle. But, that tradition through Ohm's law and DC principle, at least has a grounding point.