Speed of Electricity Revisited

Speed of artificial lightning

Introduction

Some time ago I ran into this blog where they calculate the speed of “electricity” using this video, but there is a major issue with this, and that is the discharges they used for the calculation are completely harmless as it seen in the same video (8:17). Artificial lightning seen in fiction is anything but harmless, these discharges are usually powerful enough to stun/burn/kill relatively strong characters, so using inoffensive electrical discharges in order to determine the speed of artificial lightning is most likely wrong. Although I must admit that I never thought of using a slow-motion video and pixel scaling to calculate the speed of electrical discharges, so I must thank the author of the original post for the idea.

Here I’ve gathered 3 slow-motion videos which we can use in order to determine the speed of non-natural lightning/Artificial Lightning/Electrical discharges/Electricity (all these videos belong to The Slow Mo Guys). All of these discharges are powerful enough to at least cause pain to a regular human.

For the first part of the calculations and for the sake of simplicity we will be approximating the length of the electrical arcs as straight lines. Even though in most cases that approximation isn’t quite good, it is still good enough to provide a lower bound for the length of the arcs and therefore a lower bound for their speed. We will use a better approximation in the second part.

To determine the framerates I watched the discharges scenes frame by frame, you can do the same using the method shown in this video.

Calculations

First video

In the first video we can see a few discharges from a Van de Graaff generator. The video was recorded at 28,550 frames per second, which means that there are 35.03 microseconds (0.00003503 seconds) between each frame, this will be our timeframe. We can get the distance by using what is shown in Figure 1, I’ll call the distance depicted by the green line the “arm-elbow width”, as far as I know this measurement is not documented anywhere, so I used mine, 7.5 cm, for the calculation (you can use your own arm-elbow width to check the distance we are using here is correct). The discharge seen in Figure 1 is 343.8 pixels long and Dan’s arm-elbow width is 356.2 pixels, we can now pixel scale to get the length of the electrical discharge:

$d = \frac{7.5 cm}{353.7px} \cdot342.8px$

$d = 7.269 cm$

Now the timeframe. The discharge from Figure 2 appeared from one frame to the next, which means it reached Dan’s arm in less than 35.03 microseconds, now we can easily calculate the speed as follows:

$v = \frac{7.269cm}{35.03 \mu s}$

$v= \frac{0.07269 m}{0.00003503 s}$

$v = 2075 \left [ \frac{m}{s} \right ]$ (mach 6).

Notice that the fact the discharge took less than 1 frame to reach Dan’s arm means that this is just a lower bound for its speed since its actual movement couldn’t really be recorded.

Unfortunately, the video doesn’t show any other discharges we can use to calculate their speed (there are more electrical discharges on the video, but getting the distance is too complicated, maybe impossible), but we still have 2 more videos to go.

Figure 1: Arm-elbow width (green line), length of the electrical arc (red line).

Second video

Here they used a small Tesla coil and made some recordings at different frame rates. Important note: The video on YouTube has a frame rate of 50 fps, yet the slow-motion footage is displayed at 25fps, that’s why in Figure 2 you can see the “4,000x slower” note for the 100,000 fps take, since 100,000 is 25 x 4,000, this means that if you move 2 frames forward/backwards on the YouTube video you’re actually moving 1 frame in the slow-motion video.

The first take had a frame rate of 100,000 frames per second, which means there are 10 microseconds between each frame. In order to obtain the length of the discharges we will use the length of the stick that’s sticking out from the tesla coil, we can estimate this length by using the distance between Dan’s wrist and the knuckle of his little finger (blue line in Figure 2), I again used the measurements of my own body for this, and the distance between my wrist and the knuckle of my little finger is about 7 cm, using this value we can obtain the length of the stick in the tesla coil (green line) as it follows:

$\frac{l}{390.6px} =\frac{0.07 m}{358.4 px}$

$l = 390.6 px \cdot \frac{0.07m}{358.4px}$

$l = 0.0763m$

We can now use this value to obtain the distance covered by the electrical discharges and calculate their speed.

Figure 2: Wrist-knuckle distance (blue line), Tesla coil stick (green line).

Thirteen different discharges are shown in the minute 2:04 of the video, but some of them share very similar trajectories and occur within pretty much the same amount of time, so they don’t really provide any relevant information. That’s why I picked 5 discharges with noticeable different trajectories in order to make the calculations.

First arc

The first arc is 849.5 pixels in length (Figure 3), since the stick of the Tesla coil is 585.4 pixels long and we already know that its actual length is around 7.63 cm we can obtain the length of the electrical discharge as follows:

$\frac{d}{849.5px} = \frac{0.0763m}{585.4 px}$

$d = 849.5px \cdot \frac{0.0763m}{585.4px}$

$d = 0.1107m$

This arc took 2 frames to achieve its maximum length, since there are 10 microseconds between each frame the discharge traveled 11.07 cm in 20 microseconds, therefore the speed is:

$v = \frac{0.1107m}{0.00002s}$

$v =5535\left [ \frac{m}{s} \right ]$ (mach 16.1).

Figure 3: Length of the tesla coil stick (green line), length of the electrical arc (red line).

Second arc

This arc is 878.3 pixels long and the stick is 592.3px (Figure 4), from now on I’ll skip the calculation of the discharge length since it’s complete analogous to the first case, hence:

$d = 0.1131 m$

This arc achieved max length in within 3 frames, which means our timeframe is 30 microseconds, therefore the speed is:

$v =\frac{0.1131m}{0.00003 s}$ $v = 3770 \left [ \frac{m}{s} \right ]$ (mach 11).

Figure 4.

Third arc

This discharge has a length of 980.7 pixels (Figure 5) and the stick of the Tesla coil is 588.3 pixels, then:

$d = 0.1272 m$

This arc also took 3 frames to achieve maximum length, therefore its speed was:

$v =\frac{0.1272m}{0.00003s}$

$v =4240 \left [ \frac{m}{s} \right ]$ (mach 12.36)

From now we’ll just skip the explanations because they’re quite repetitive.

Figure 5,

Fourth arc

See Figure 6.

$d = 0.09198 m$

The time frame is 30 microseconds again, then:

$v =\frac{0.09198m}{0.00003s}$

$v = 3066\left [ \frac{m}{s} \right ]$ (mach 8.939).

Figure 6.

Fifth arc

See Figure 7.

$d = 0.09329m$

Time frame: 30 microseconds.

$v =\frac{0.09329m}{0.00003s}$

$v = 3110\left [ \frac{m}{s} \right ]$ (mach 9.066).

Figure 7.

Sixth arc

I will now calculate the speed of a sixth discharge, what makes it different from the others is that this one landed on Dan’s hand.

See Figure 8. Here we can pixel scale by directly using Dan’s fist.

$\frac{d}{577.5px}=\frac{0.07m}{358.4px}$

$d = 577.5px \cdot \frac{0.07m}{358.4px}$

$d = 0.1128 m$

This arc achieved max length in 4 frames, therefore the timeframe is 40 microseconds, we now calculate the speed:

$v =\frac{0.1128m}{0.00004s}$

$v = 2820\left [ \frac{m}{s} \right ]$ (mach 8.22).

Figure 8: Electrical discharge landing on Dan’s fist.

Third video

On this video we can see fairly large electrical discharges from a big Tesla coil. According to Gav the video was recorded at 768 frames per second, which means there are 1.302 milliseconds between each frame. Unfortunately, the framerate of this video is quite low compared to the framerate of the previous video we analyzed, so the actual travel of electricity wasn’t recorded, yet this video is still good enough to provide a lower bound for the speed of artificial lightning.

Most discharges in this video occur within one single frame or less, so 1.302 milliseconds will be our timeframe. The electrical arcs we’ll use to calculate the speed are the ones from Figures 9 and 10, we can calculate their length by pixel scaling with the Tesla coil which, according to Dr. Megavolt’s website (the owner of the Tesla coil), is 8.5 feet or 2.591 meters tall.

Let’s calculate the speed of the discharge seen in Figure 9: We can see that its length is 404.6 pixels, while the Tesla Coil is 698 pixels tall, this way we can calculate the length of the electrical discharge as follows:

$\frac{d}{404.6px}=\frac{2.591m}{698px}$

$d = 404.6px \cdot \frac{2.591m}{698px}$

$d = 1.502m$

The speed is then:

$v =\frac{1.502m}{0.001302s}$

$v = 1153 \left [ \frac{m}{s} \right ]$ (mach 3.362).

Figure 9: Electrical discharge from a big Tesla coil.

And now we can calculate the speed of the second discharge (Figure 10), I will again skip the calculations since they’re analogous to the first ones.

$d = 1.691m$

Then:

$v = \frac{1.691m}{0.001302s}$

$v = 1299\left [ \frac{m}{s} \right ]$ (mach 3.787)

Figure 10: Second electrical arc from the third video. This arc is quite dim, if you are struggling to see it I recommend you to increase the brightness of your device.

A better approximation

Up until now we’ve calculated the speed of artificial lightning by approximating their length as straight lines, however the shape of these electrical arcs are far from being straight lines, therefore we now shall use a better method to estimate their length. What we will be doing now is approximating the length of these arcs by using several straight but short lines (a segmented line), I won’t go in to deep explanation on how this works since that’d take a lot of time and it isn’t the purpose of this blog, but if you are familiar with calculus you should already know how this approximation functions, and if you don’t, you can read about it here or watch this video.

In order not to prolong this further we will skip most calculations, since they’re trivial, so we can focus on the results.

First video

See Figure 11.

Length: 344.6px

Distance: 7.307 cm

Timeframe: 35.03 microseconds

$v = 2086\left [ \frac{m}{s} \right ]$ (mach 6.08).

Figure 11: Here we can see the straight and segmented line approximations yield pretty much the same result since this particular electrical arc is quite straight.

Second video

First arc

See Figure 12.

Length: 1055px

Distance: 13.75 cm

Timeframe: 20 microseconds.

$v = 6875\left [ \frac{m}{s} \right ]$ (mach 20).

Figure 12.

Second arc

See Figure 13.

Length: 967.6 px

Distance: 12.46 cm

Timeframe: 30 microseconds.

$v =4155\left [ \frac{m}{s} \right ]$ (mach 12.1).

Figure 13.

Third arc

See Figure 14.

Length: 1265 px

Distance: 16.41 cm

Timeframe: 30 microseconds.

$v =5468\left [ \frac{m}{s} \right ]$ (mach 15.9).

Figure 14.

Fourth arc

See Figure 15.

Length: 985 px

Distance: 12.82 cm

Timeframe: 30 microseconds.

$v=4275\left [ \frac{m}{s} \right ]$ (mach 12.5)

Figure 15.

Fifth arc

See Figure 16.

Length: 1061 px

Distance: 13.67 cm

Timeframe: 30 microseconds.

$v =4555\left [ \frac{m}{s} \right ]$ (mach 13.3).

Figure 16.

Sixth arc

See Figure 17.

Length: 641.8 px

Distance: 12.54 cm

Timeframe: 40 microseconds.

$v = 3134\left [ \frac{m}{s} \right ]$ (mach 9.136).

Figure 17.

Third video

First arc

See Figure 18.

Length: 515.7 px

Distance: 1.914 m

Timeframe: 1.302 milliseconds.

$v = 1470\left [ \frac{m}{s} \right ]$ (mach 4.29).

Figure 18.

Second arc

See Figure 19.

Length: 661.5 px

Distance: 2.48 m

Timeframe: 1.302 milliseconds

$v =1905\left [ \frac{m}{s} \right ]$ (mach 5.55).

Figure 19.

Which result should we use?

We can notice that only 1 arc yields a velocity that’s (barely) lower than mach 5, which is the lower bound for hypersonic speeds, yet this result comes from the video with the lowest framerate, meaning that the velocity is being underestimated quite a bit since the actual travel of electricity couldn’t be recorded, in addition, the second arc from the same video yields a velocity of mach 5.55, therefore it is completely safe to assume low hypersonic speeds as a lower bound for the velocity of artificial lightning. However, if we are looking for a general value, we should average over every result in each video, by doing so we obtain that the average speed of artificial lightning is 3769 m/s (mach 11). And as for the high end we can just take the highest result of the three videos, which is 6875 m/s (mach 20).

In summary:

Artifical lightning speed

	m/s	Mach
Lower bound	1715	5
Average	3769	11
Upper bound	6875	20

In general, we should use the average speed for artificial lightning.