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Don’t be a resistor in parallel! (Stupid procrastination tricks.) July 13, 2010

Posted by mareserinitatis in electromagnetics, engineering.
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2 comments

We’re having a thunderstorm, so I thought it was a good idea to pull this post from the old blog.

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We’re having a thunderstorm right now. I’m supposed to be working on my thesis (I have decided that I will not go to bed until I am either thoroughly exhausted or have completed the changes on a couple chapters).

Being the highly distractable person I am when I am not happy doing something, I have been caught up in pondering the quantitative values that surround a “human resistor” in the presence of a lightning strike. I’m not concerned about the case where the person is the direct path to ground. I’m thinking about a comment someone made to me a couple weeks ago: You should never shower during a thunderstorm in case the lightning hits the plumbing. Here I always thought it was because you’d fall out of the tub when you lose power.

I have embraced a small part of me that thinks I am an engineer because I want to set up a simple model to determine what sort of currents would pass through a person.

So if you’re curious, here’s my solution…

This is going to be pretty basic model, since I really should get back to work. The source is the lightning, obviously, but we’re going to assume it’s a current source. It’s passing through two resistors in parallel to ground (literal ground, in this case…not too often that happens). Based on this info, let’s say the current is 10 kAmps.

According to (one particular) ESD human body model (HBM), the human body has a resistance of 1500 Ohms and a capacitance of 100 pF. I’m going to do something completely bogus because this would take a long time otherwise. The impedance due to the capacitance is going to be zero for this solution. I realize that lightning is a transient and is almost totally high frequency content…knowledge of which would take me a while to dig up. I’m going to ignore that knowledge, though, because I really don’t want to try to do anything close to Fourier analysis to figure this out. I’m lazy and I only want to look at the resistance. Aside from that, we’d have to have THz frequencies to provide any appreciable impedance.

The other resistance will be the plumbing from your shower head to the drain. We’ll assume a 3 cm diameter copper pipe that’s 2 meters tall. We’ll use this info to compute our resistance. The pipe resistance will be 17 nOhm m *length/area = 17e-9*6/(π*.0015^2)=14 mOhms. Now, I also realize that it’s not a solid copper pipe because (obviously) it’s filled with water. On the other hand, copper is lower resistance, so not as much energy will flow in the water as in the pipe. So I’m going to pretend that it is, in fact, a solid copper pipe, realizing that the resistance, in reality, would probably be higher (which is bad for the person in the shower…more resistance in the pipe means more current is shunted to the other resistor, the person).

(Note: in re-reading this, the counter to this argument is that a water filled pipe creates more surface area, i.e. the inside surface of the pipe, and so could potentially reduce the resistance of the pipe. However, the water inside the pipe will, as I said, have a higher impedance than the pipe. I’ll assume both effects are negligible.)

So we have two parallel resistors, one of 1500 Ohms and the other at 14 mOhms, giving us approximately six orders of magnitude difference. The equivalent resistance (R) will be 21 Ohms. Given a current (I) of 10 kAmps, we can whip out Ohm’s Law (V=IR) to find that the voltage drop across our human/shower parallel resistance is 210 kV.

But since we all know that current is what really kills, we need to figure out if enough current will be shunted through the pipes to make this a serious concern. Initially, it looks good since the person is such a high resistance compared to the pipe. We should note, however, that 100 milliamps is enough to shut down the heart.

What we need here is a current divider:

Ihuman=itotal*(Rshower/Rtotal)=10 kAmps * (.014/1500.014)=93 mA

Crap. At the very least that will feel terribly unpleasant, but with that sort of voltage, I imagine it could do a lot more. The deciding issues would be those little assumptions I mentioned above (the accuracy of the HBM model and the oversimplified impedance of the pipe).

Looking at this article, I’m inclined to think it might be a low estimate. This lightning strike was 7 miles away!

So yeah…no more showers during storms.

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