Drift speed of an electron in normal atmospheric conditions

[Merlight]

Homework Statement

How fast would a electric arc be when it ionizes through normal, unpressured air between say, two electrodes? Not the same as lightning as it is hypothetically from an artificial source.

Homework Equations

How would one calculate (please include your formula's and which units used in them and what they are with an example of solving a similar problem to mine please?) the velocity of the electric charge based on only knowing that the arc had 230 voltage and the arc length was 10 meters long?

https://www.physicsforums.com/showthread.php?t=500446

Bob S tried explaining a similar problem here but it was years ago and didn't the include relevant units somone would use in solving the problem? And he's not here anymore...using the variables in the equation, I can't really solve based on knowing only how to solve part of the problem. This is my attempt. :(

The Attempt at a Solution

I can give other specifications to my problem if this information is not sufficient enough to be able to solve this hypothetical problem.

Its for how fast an electric bolt would propagate at a human being if it was discharged in standard atmosphere pressure and temperature from 10 meters when mechanical source that had the electrical discharge from a cathode was about 230 volts?

Any questions about my question, I will be happy to respond, I am a college student and eager to learn, I have little to no background in physics, so please, be as simple and patient with me as you can be. I would like an answer preferably in m/s or however it would be translated to such units.

 

[az_lender]

I approach this through Ohm's Law (V = IR) and the rule that R = ρL/A. Here L is the 10m, ρ is the resistivity of air (maybe 2E16 Ωm), but what is A? Possibly the average cross-sectional area of a sphere where the ends of the arc are the poles. But my next step suggests it's not necessary to evaluate A, because the drift speed should be

v = I/(nAq) = (V/R)/(nAq) = V/(nqρL),

and the A has canceled out. Here n is the density of charge-carriers, and q is the charge on a charge-carrier; obviously V is the 230V. The charge carriers are ionized oxygens and nitrogens, presumably singly-ionized so that the q is presumably 1.6E(-19) C, but I don't know if it's "fair" to calculate n as the number density of air molecules (which is pretty easy to get).

 

[Merlight]

Ok, so this is as far as I'm following your answer?
I got how you canceled out variables and got this equation:

v = 230/(n*(1.6E-19)*(2E16)*(10))

Where:

230 = in volts

1.6E-19 = in C, a unit of relative current for batteries

2E16 = in Ωm

10 = length in meters

Where n should be:

http://en.wikipedia.org/wiki/Number_density

Using dry air,

n = 0.02504

Plugging that back into the equation:

230/(0.02504*(1.6E-19)*(2E16)*(10)) = ?

The "answer" is: ~287040

But why is this answer not fully correct to you with using n?

In what units is the speed measured in right now with the answer I got?
"v" should be in m/s or meters/second, correct, like how it is normally calculated?
And did I calculate this correctly / mathematical errors?

 

[Merlight]

Hmm.. the equation above gives extraneous results with the one variable the dude above was concerned about if it was ok to use in place?

I'll try from a different angle.

Okay, does anyone have a clue what variables should I use if I'm using this formula instead how it is rewritten for solving for the drift velocity of the described conditions as above?

Main link:http://en.wikipedia.org/wiki/Drift_velocity#Mathematical_formula

In terms of the basic properties of the right-cylindrical current-carrying metallic conductor, where the charge-carriers are electrons, this expression can be rewritten as[citation needed]:

http://upload.wikimedia.org/math/6/6/d/66d5facff74f898f925a13a4d1541a9c.png

where,
v is again the drift velocity of the electrons, in m·s−1;
M is the molar mass of the metal, in kg·mol−1;
V is the voltage applied across the conductor, in V;
NA is Avogadro’s number, in mol−1;
d is the density (mass per unit volume) of the conductor, in kg·m−3;
e is the fundamental electric charge, in C;
ρ0 is the resistivity of the conductor at 0°C, in Ω·m;
α0 is the temperature coefficent of resistivity of the conductor at 0°C, in K−1;
T is the temperature of the conductor, in °C,
ℓ is the length of the conductor, in m; and
f is the number of free electrons released by each atom.

I can't see how to solve for a few of these variables, as I'm trying to find f, e, NA, and the molar mass of "air", as well as mainly incorporating the rest of them in.

 

[Merlight]

Actually, disregard my questions about the above equation, I realized it should be correct as long as the formula is not giving speeds exceeding the speed of light. Thanks guy. :)