Sam Range

Before we begin, intuition check: if you’re swimming in the water and lightning strikes nearby, is it better to be in freshwater or saltwater? No cheating.

Some weeks ago, I found myself sitting in a Dairy Queen with a pen, a notebook, and some imspiration – it was physics time. Note: several months have actually passed since I wrote that sentence. However, I’m skipping the Harry Potter premiere and am determined to finally finish this.

I’ve wondered since my childhood if lightning actually posed a serious threat to swimmers. I spent enough time in the water to grow accustomed to blaring whistles and lifeguards ordering everyone out of the water every time a distant thundercrack sounded, but I had my doubts: if electricity could travel for miles through water to electrocute innocent beach-goers, why were there never news stories of swimmers tragically zapped by bolts of lightning from beyond their sight? Why were there not scores of dead fish washed up on the shores after every thunderstorm?

Lacking the means to test this experimentally, I let the uncertainty rest as an hazy malease in the back of my mind for many years.  I didn’t make any progress or even think of the problem again until a few weeks ago, sitting in that Dairy Queen. A few hours of scribbling, calculating, generalizing, and most of all researching actual values for constants later, I’m satisfied with the result. But, before I begin, you should understand this:

So, on to the physics. I decided to try out my new Wacom tablet for this.

First, assume a lightning bolt induces a constant current I into the water. The electrons flow in a uniform hemispherical distribution. As the electrons travel outwards, the current density J at radius R can be expressed using the current I and the surface area of the hemisphere.

During the middle of the strike, when current is assumed to be uniform, it is reasonable to assume that inductive effects are negligible, so current is proportional to voltage and current density is proportional to electric field. The proportionality constant is conductivity, in siemens per meter, sigma.

Substituting, we find the strength of the electric field at radius R in terms of I and conductivity.

Ultimately, we are interested in the worst-case voltage a swimmer could be subjected to at a given distance r from the strike. So, we integrate electric field over a distance l, where l is the length in meters of the swimmers armspan. We assume his arms are fully outstretched in line with the strike, as that represents the worst case scenario. We integrate with respect to R using the swimmer’s distance r as the midpoint of the integral to jive with the intuition of measuring to the swimmer’s body and to simplify the resulting antiderivative. Finally, solve for r.

My handwriting starts to shrink here. I’m fairly certain it’s involuntary.

Equation (3) shows what the minimum radius is to experience a voltage V given a certain conductivity, length l, and current I from the lightning. We wish to know what the minimum radius is to be lethal to the person, so we must express this in human terms.

An important concept here is how electricity kills. It does not matter what voltage one is subjected to (being shocked with thousands of volts of static electricity is harmless), but on the current through one’s body, and through which parts. To be lethal, one must experience a current of approximately 100 mA across the chest cavity. This lethal current I_L depends on the resistance R (recycled variable) of the person and the voltage V across his fingertips.

This equation shows the minimum lethal radius for this worst-case scenario. For convenience, it can be approximated by removing the l^2/4 term, as it will contribute little to any reasonably large radius and reasonably sized swimmer. Here, we see that increasing the conductivity actually require the lightning strike to be closer to induce death. It’s more dangerous to be in freshwater than saltwater.

Finally, we can plug in some variables. Of course, the length, lightning current, and especially conductivity vary considerably, but some reasonable assumptions allow some rough results. I assume a slightly taller than average swimmer with an armspan of two meters. I_L is the commonly accepted 100mA (.1 A). Conductivity depends on a whole host of factors, including depth, temperature, and salinity, but a fair reference value is 5.3 S/m. For large freshwater bodies, such as my neighbor Lake Michigan, median conductivity is about 29.5 mS/m.

It’s significantly more difficult to pin down a value for the resistance of a human from fingertip to opposite fingertip. The internal resistance of a human is exceedingly low, around 100 ohms, because of all the electrolytes in blood. However, skin greatly raises the resistance. For dry skin, the figure is about 1-2 mega ohms, but the figures I found for wet skin varied over several orders of magnitude, and I was unable to find any reference which distinguished between wet from fresh or salt water. So, I took a much-needed break from calculation and performed a small experiment.

Very simple, I got two vessels, filled them with water, held the probes from a multimeter in my hands, and set them in the baths. The only current path was through my arms. I waited for a little over 5 minutes to allow water to infiltrate my skin and provide the best conduction path possible to the bloodstream. I then repeated the experiment using salt baths (matching salinity by taste and memory, definite error source). I used “sea salt” from one of those grinders, so perhaps this helped the authenticity. When I was done, my fingers were all wrinkled and pruney, so it was a success. My observed resistances were 51.5 kilo ohms for freshwater and 16.6 kilo ohms for saltwater.

And, the grand table:

Figures are given for fresh water and for different sized lightning strikes. The largest listed is positive lightning, an uncommon type of bolt where electrons actually leap from the ground to the clouds which is significantly more powerful than a normal discharge.

These figures seem encouraging for surviving thunderstorms in the water, but be aware that some values for the resistance of a wet human are as low as 1100 ohms, which extends the maximum killing radius to 172 meters in freshwater. Regardless, it is a bad idea to swim anywhere during a thunderstorm.

If you’re still having a tough time accepting that conductive salt water is safer, remember that the same amount of electricity has to flow through the water no matter what the resistance. The more resistive the water, the greater the voltage difference from one point to another for that current (think V=I*R).