Pilfered Power

Some time ago I saw an article about the concept of harvesting power from high-voltage transmission lines. The article reported disturbingly large voltages obtained from the equipment they set up, which led them to stop the experiment for the sake of safety (and legal protection). I was curious if I could figure out a way to predict the voltage we could expect from such a situation.

The electric system transmits power with alternating current (AC) at high voltages, which gets stepped-down by transformers when it is delivered to our houses. To see why the transmission is done at high voltage, we can imagine sending out a power P through lines with resistance R:

The power lost is inversely proportional to the square of the voltage, meaning raising the voltage can quickly decrease the power we lose to resistance.

When current travels through a wire, it produces a magnetic field in circles around the wire. The equation for this field is

where μ is the permeability of free space, I is the current in the wire, and ρ is the radial distance from the wire. For alternating current we can write

where I0 is the amplitude of the current and f is the oscillation frequency, 60 Hz in the US. If we set up our fence directly below the transmission line with the z-axis pointing along the fence and the y-axis pointing up toward the wire, then the B-field at this point will be

where h is the height of the transmission wire, and we've replaced I0 with the power & voltage carried by the line. Now to get the voltage induced in the fence, we can use Ampere's Law:

Plugging in our B and integrating to get E gives

Electric field is measured in volts/meter, so this will tell us the relation between the length of our fence and the voltage we can expect between the two ends. The example case discussed in this Wikipedia article uses P = 1000 MW and V = 765 kV. This page gives the minimum height for wires carrying that voltage as 45 ft. Plugging in those values, along with the 60 Hz frequency I mentioned, gives 330 megavolts/meter, which is pretty insane! Now I did assume best (or maybe worst) case numbers in this calculation, but if you watch the video in the article that started this, the presenter seems a bit concerned with the size of the spark he gets at the end. Good (grounded) fences make good neighbors!

Duck and Cover

[I considered "Don't Look Up," but I have a soft spot for Red Scare propaganda.]

This week there was news of an asteroid with a 1-in-82 chance of hitting the Earth in 2032. Most reports seemed to spin this as a reassuringly low chance, but as someone well-versed in low probabilities, I found it to be uncomfortably high, so I wanted to dig into it a bit. NASA's Small Body Database has an entry for this asteroid, which gives its orbital parameters and a nice interactive plot of its path through space. In order to do some calculations with this info though, I found a tool developed by MIT called poliastro, which can connect to the database and read the orbit definitions.

As a first pass, I tried duplicating the type of plot NASA included on its page, showing the orbit of the asteroid compared with a few of the planets around the time of the possible collision:

This shows the orbits of Earth in blue, Venus in cyan, Mars in red, and the asteroid in magenta. The planets' orbits are all fairly circular, while the asteroid's is sharply elliptical, and crosses the paths of the planets. The predicted flyby is around December 2032, and if you watch the animation you can see the asteroid skirts behind the Earth with a fair gap.

All predictions include errors though, so what happens if this orbit isn't quite the right one? Poliastro offers an object called a maneuver that can perturb a given orbit. Starting with the orbit above, I applied a random change in velocity with magnitude up to 10% of the initial value. Then I measured the distance between the Earth and the asteroid in time for each variation in the orbit:

The x-axis is the Modified Julian Date, a measure of time often used in astronomy. The black line is the predicted orbit, which only gets down to around 200 million km separation. For reference, that's about 1,000 times the distance from Earth to the Moon. However, some of those variations get a lot closer. To see how many of them are getting close, we can look at a histogram comparing the closest approach for each of the variants to the original:

Values below zero indicate the variant gets closer than the base prediction, and the red line marks the distance that would result in a collision with the Earth. Several of those variants are getting awfully close, so much like my pressure cooker post, I'm not sure I feel better at the end of this!

Water Wedge

The streets around our house have many potholes, which fill up with ice:

Water expands when it freezes, which means when a hole fills with water that freezes, the ice can potentially make the hole bigger. To do this, the ice puts pressure on the asphalt, and vice versa. Adding pressure can make it more difficult for ice to freeze, so I looked up some info on asphalt strength. This paper gave a plot comparing the bond strength for different aging times:

Figure 8

These are all around 2-4 MPa. We can look at the phase diagram for water to see how this would affect freezing:

Wikipedia

This is well within the range where the freezing point is still around 0°C, so we don't need to worry about the pressure we're applying. Water expands by about 9% when it freezes, so we can imagine water filling a crack, then pushing out in all directions:

That's the general idea I had in mind, but getting into the details, the steps I used are

1. Begin with a flat road. Put a dent in the center with depth y0 (*parameter).
2. Find all points below the road, and use them to define a polygon.
3. Find the centroid (center of mass) of the polygon and push all points out by 9%.
4. Push the polygon down according to its weight (*parameter).
5. Add points anywhere the separation has passed a threshold.
6. Adjust points along line for equal spacing.
7. Repeat from 2.

I added steps 5 and 6 since otherwise the crack just scaled up from the initial dent. We can look at the type of crack these steps give as we cycle through freezes/melts:

This seems a little extreme, but we can take a look at how this changes with the two parameters I marked above. The two qualities I was mainly interested in were the width and depth of the crack. We can see how those change in time with different y0 and weight values:

It makes sense these curves look exponential, since the increase depends on the current size. What really surprised me though was how well-separated these two qualities are according to the input weight and y0. These plots suggest that the width is almost wholly determined by the weight, and the depth by the initial depth. For the depth, this can easily be tied to the exponential behavior – Starting deeper increases the rate the depth increases. A little less obvious is the weight, but if we look at the initial diagram of the setup, the wedge shape of the ice means pushing down also pushes out. A quality of exponential systems is that they can very quickly get away from their starting point if left unchecked – A reminder to keep those roads smooth!