PDE: Why do they have the wrong units?

Question

Take a look, for example, at the telegrapher's equations (let's look at the voltage one). They have the wrong units.

Equation

$u_{x} = Li_{t} + Ri$

*where $u$ is potential in volts $V$, $L$ is inductance in henries $H$, $i$ is current in amperes $A$, $R$ is resistance in ohms $\Omega$.

Unit Analysis

$[\frac{V}{m}] = [V]$

This appears to be consistent across many PDEs. What am I missing?

I don't understand your “unit analysis”. What are $V$ and $M$? — Harald Hanche-Olsen, May 10 '13 at 19:32
$L, R$ should be 'per unit length' quantities. $u$ should be in volts, and $i$ in amps. — copper.hat, May 10 '13 at 19:36
Oh, I get it. $V$ is supposed to mean Volts, and $M$ is meters? But meters are written with a lowercase m. — Harald Hanche-Olsen, May 10 '13 at 19:36
@copper.hat So are you implying that in all PDEs where the units appear off, that the multiplicative constants must be dimensionally normalized? — nick_name, May 10 '13 at 19:38
Are you talking about $L,C,R$? If so, they must represent the 'per unit length' quantities. — copper.hat, May 10 '13 at 19:40
@nick_name: That is a rather strange generalization, impossible to answer yes or no to. But if the units appear to be “off”, checking to see if they're really per length or per time seems like a good strategy. — Harald Hanche-Olsen, May 10 '13 at 19:41
@copper.hat Okay, maybe it's the case that the notation is often funky. Using $R,L,C$ really isn't correct (these are now linear densities). Thanks! — nick_name, May 10 '13 at 19:42
I'm not sure what you mean. Usually, $R,L,C$ are ohms, henry, farad, but in the context of transmission lines & guides, they are expressed in 'per unit length' quantities. This goes back a long way... — copper.hat, May 10 '13 at 19:43
There are only so many letters in the alphabet. Some reuse is unavoidable. — Harald Hanche-Olsen, May 10 '13 at 19:50

score 4 · Accepted Answer · answered May 10 '13 at 19:37

4

The units are fine. $L$ is inductance per length unit and $R$ is resistance per length unit.

answered May 10 '13 at 19:37

Harald Hanche-Olsen

31,960

Right, I guess the variable names are often not reflecting the units in these equations (implicit). – nick_name May 10 '13 at 19:43

zyx · Answer 2 · 2013-05-10T20:24:53.370

Changes of units, in any equation, modify the coefficients. If different coefficients are subject to different scaling exponents (which is what it means to be "dimensionally wrong"), it means that for the equation to not depend on choice of units, the coefficients must be dimensionful, and in a specific way. There is a unique set of units for the coefficients making the equation dimensionally consistent, up to multiplying the entire equation by some unit, like going from A=B to A m^2 = B m^2.

Equations that accurately describe physical processes usually are not dependent on choice of units, so it is likeliest in any particular example that the coefficients have the necessary units to cancel apparent discrepancies in the dimensionality of the different terms. As the first answer states, this is true for the telegraph equation, and the principle is more general.

I guess it depends on which two of harmed/harmful/harmless you consider it analogous to. Or formed/formful/formless, and the same suffices on shape. — zyx, May 11 '13 at 05:16

PDE: Why do they have the wrong units?

2 Answers2