Teach Your Physicist to Suck Eggs

Recently I was reminded of a neat science demo where burning a match inside a bottle can suck in a boiled egg that otherwise wouldn't fit:

I saw similar demos growing up, and I believe I was given the incorrect explanation of the match using up the oxygen to lower the pressure. As the video above points out, it's actually the heat from the fire that makes the air expand and escape from the bottle. After the air cools, the pressure is lower and the atmosphere pushes the egg into the bottle. I was curious whether I could determine the pressures involved in this demo.

Burning the match releases heat, which raises the temperature of the air in the bottle. According to the Energy Information Administration, burning a match releases 1 BTU of energy. Using air's heat capacity, density, and a volume of 1 L, we can calculate that this will raise the temperature by an impressive 1220 K. We can use the ideal gas law to see what that does to the air:

Increasing the temperature, T, will cause the pressure, P, to to go up. This pushes up on the egg allowing some air to escape. If we suppose that the pressure required is very small, we can instead assume all the energy from heating the air goes into expanding the volume, V. We could use this to find the final volume of the air, and thus how much escapes from the bottle, but we don't really care about the air that escapes. Instead, we can just suppose our 1 L bottle of air at a room temperature of 300 K is now at its new temperature of 1520 K, with the pressure remaining constant.

Now we can flip the relation, and suppose the volume and number of particles, N, stays constant. We can write

where the subscripts specify the initial and final states. Using the temperature we calculated and 1 atm of pressure, returning to room temperature would bring the pressure to approximately 0.25 atm. That implies the atmosphere is pushing down with 0.75 atm of net pressure. We can get the area of the egg's cross-section and multiply by the pressure for a force of 110 N, or about 25 lbs. Not enormous, but it's easy to see how applying that force unevenly by pushing with your fingers would just mush the egg.

Turning to the latter part of the video, how about getting the egg out again? The maximum pressure a person can blow is about 0.1 atm, 40% of the pressure that pushed it in. If you watch the video above though, you can see the egg goes into the bottle with a great deal of speed, while getting it out isn't nearly as violent, so these results appear to be consistent. Cool experiment to try if you've never seen it before!

Window's Vista

This week, Futility Closet had a post about the Vista Paradox. The paradox is connected to an effect I discussed before, where the angular nature of our vision causes objects to move in unexpected ways. In this case, we're viewing a distant tower through a window. As we approach the window, the window takes up proportionately more of our vision that the tower, causing it to appear to shrink. I tried making an animation of this to test my understanding:

As expected, the tower (red) changes very little, while the closer window (blue) expands rapidly. Initially, I was concerned with how the window's proportions appear to change, but I checked with my mother Sally, and she gets the same expression I did. There still may be an error in my implementation though.

After reading the Futility Closet post, I was reminded of an effect often used in filmmaking called a Dolly Zoom: The camera moves forward (backward) while zooming out (in), causing the edges of the frame to stay where they are, but the center to contract (expand). The scene that always comes to my mind for this is one from Fellowship of the Ring:

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Using the same framework as above, I plotted a series of rings at different distances, but the same size. Then I moved the camera toward them, while fixing the edges of the plot to the nearest ring:

This effect makes the center appear to shrink, the opposite of the motion in the film, but exactly the same technique. It's neat to see this artifact of human perception appear in both architecture and filmmaking!

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!